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Emergent spacetime in matrix models Yeh, Ken Huai-Che 2018

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Emergent Spacetime in Matrix ModelsbyKen Huai-Che YehB.Sc., National Tsing Hua University, 2007M.Sc., The University of British Columbia, 2011a dissertation submitted in partial fulfillmentof the requirements for the degree ofDOCTOR OF PHILOSOPHYinthe faculty of graduate and postdoctoral studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2018c© Ken Huai-Che Yeh, 2018The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the thesis entitled:Emergent Spacetime in Matrix Modelssubmitted by Ken Huai-Che Yeh in partial fulfillment of the requirements for the degree ofDOCTOR OF PHILOSOPHY in Physics.Examining Committee:Joanna L. Karczmarek, PhysicsSupervisorGordon W. Semenoff, PhysicsSupervisory Committee MemberDouglas Scott, PhysicsUniversity ExaminerKai Behrend, MathematicsUniversity ExaminerAdditional Supervisory Committee Members:Mark Van Raamsdonk, PhysicsSupervisory Committee MemberKris Sigurdson, PhysicsSupervisory Committee MemberiiAbstractWe study the non-commutative geometry associated with matrices of N quantum particles in matrixmodels. The earlier work established a surface embedded in flat R3 from three Hermitian matrices.We construct coherent states corresponding to points in the emergent geometry and find thatthe original matrices determine not only the shape of the emergent surface, but also a uniquePoisson structure. Through our construction, we can realize arbitrary non-commutative membranesembedded in R3.We further conjecture an embedding operator that assigns, to any 2n + 1 N -dimensional Her-mitian matrices, a 2n-dimensional hypersurface in flat (2n+ 1)-dimensional Euclidean space. Thiscorresponds to defining a fuzzy D(2n)-brane corresponding to N D0-branes. Points on the hyper-surface correspond to zero eigenstates of the embedding operator, which have an interpretation ascoherent states underlying the emergent non-commutative geometry. Using this correspondence, allphysical properties of the emergent D(2n)-brane can be computed.Many studies have been carried out exploring the geometry emerging from the matrix configu-ration, but they have not always produced consistent results. We apply two types of point-probemethods, as well as the supergravity charge density formula to the generalized fuzzy sphere S2so(4).Its tangled structure challenges the applicability of these probing methods. We propose to disen-tangle blocks of S2so(4) regarding the geometrical symmetry and retrieve S2so(4) as a thick two spherewith coherent layers consistently in three methods.The Yang-Mills (YM) matrix model with mass term representing a cutoff radius generates re-markable spherical solutions of the emergent universe, but it is unsolvable, unlike for matrix modelsdominated by a Gaussian potential. By coarse-graining the dimension of matrices, quantum gravityis reproduced by the Gaussian model at the fixed point of the dimensional-renormalization groupflow. We approach the unsolvable YM model using the same dimensional-renormalization and dis-cover a non-trivial fixed point after imposing spherical topology. This fixed point might lead toa new duality between quantum gravity and the massive YM model, and its existence also sets adensity condition on the generalized fuzzy sphere.iiiLay SummaryGeneral relativity explains the motion of planets around Sun through distortions of space. Quantummechanics explains an electron’s motion by a probability distribution. However, the two theoriescannot explain each other consistently. The solution lies in string theory with a new understandingof space, which resembles a pixelized picture made up of many dots. These dots exist in a probabilitydistribution encoded in a set of matrices, and a probe dot recovers the space from these matrices,with an interaction reflecting the distance to other dots. This thesis aims to reconstruct a smoothspace from a macroscopic view of this pixel space. We establish the smooth limit of pixel spacesand refine the point-probe method. The refined method reveals the geometry of a set of matricesrepresenting a simplified universe.ivPrefaceA version of Chapter 3 has been published by Mathias Hudoba de Badyn, Joanna L. Karczmarek,Philippe Sabella-Garnier, and Ken Huai-Che Yeh [1]. The paper was based on algebraic and numer-ical works conducted by Joanna L. Karczmarek, Philippe Sabella-Garnier, and the thesis author.Mathias Hudoba de Badyn contributed to the numerical breakthrough for raising the precision toa convincing level. The thesis author also carried out the computations for area and Laplacian ofnon-commutative surfaces.A version of Chapter 4 has been published by Joanna L. Karczmarek and Ken Huai-Che Yeh [2].The paper was based on collaborative works of Joanna L. Karczmarek and the thesis author. Thethesis author proposed the generalized embedding operator. Joanna L. Karczmarek solved theeigen-equations for higher dimensional planes and spheres and constructed the non-commutativeunit cell with coherent states.The thesis author wrote chapters 5 and 6, based on shared works of him and Joanna L. Kar-czmarek. The thesis author established the geometry of the generalized fuzzy spheres consistentlywith the embedding operator and the modified Laplace operator in which the supergravity chargedensity formula confirmed the modification, and he is responsible for the algebraic and numericalresults for the generalized fuzzy sphere and the fuzzy three sphere. Joanna L. Karczmarek computedthe SO(3) orbit of the coherent states of the fuzzy three sphere and pointed out that the radiusconflict is due to the shape of the non-commutative unit cell, and she corrected several mistakesmade by the author and led him in the right direction.Chapter 7 is the sole work of the thesis author, although it has been improved significantly fromdiscussions with Joanna L. Karczmarek.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Emergent spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Non-commutative geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Matrix models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.1 D-brane dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 BFSS and IKKT matrix models . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.3 Emergent Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Point probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Fuzzy sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.1 The coadjoint orbit of SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.2 Spherical D0-D2 bound state . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.3 One-loop stabilization in IKKT . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.4 Point probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.5 Fields on the sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Continuous limit of the emergent membranes . . . . . . . . . . . . . . . . . . . . 273.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Basic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 Coherent state and its properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32vi3.3.1 Example: non-commutative plane . . . . . . . . . . . . . . . . . . . . . . . . . 323.3.2 Example: non-commutative sphere . . . . . . . . . . . . . . . . . . . . . . . . 333.3.3 Looking ahead: polynomial maps from the sphere . . . . . . . . . . . . . . . . 333.3.4 Example: non-commutative ellipsoid . . . . . . . . . . . . . . . . . . . . . . . 343.3.5 Polynomial maps from the sphere . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.6 Local non-commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3.7 Coherent states overlaps, U(1) connection and Fµν on a D2-brane . . . . . . . 463.3.8 Non-polynomial surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.4 Large N limit and the Poisson bracket . . . . . . . . . . . . . . . . . . . . . . . . . . 503.5 Area and minimal area surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.6 The torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.7 Open questions and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 Matrix embeddings on flat R2n+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.1 Introduction and conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Conventions and a recursive property of Ed . . . . . . . . . . . . . . . . . . . . . . . 604.3 Non-commutative R2n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.4 Even dimensional spheres S2n and non-commutative space with SO(2n) invariance . 654.5 More examples in d = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.6 Even dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 Noncommutative layered spherical shells S2dso(2d+2) . . . . . . . . . . . . . . . . . . 735.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2 Embedding operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.3 Laplace operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.4 Supergravity charge density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.4.1 so(3) sphere S2J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.4.2 so(4) sphere S2Jl,Jr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.4.3 D2 charge density with fixed angular momentum . . . . . . . . . . . . . . . . 955.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.5.1 Fourier transform of the Green’s function of the Helmholtz equation . . . . . 976 Fuzzy three sphere in R4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.2 Embedding operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.3 Laplace probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047 Critical phenomena of the massive Yang-Mills matrix model . . . . . . . . . . . 1107.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107.2 Renormalize group flow of N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113vii7.3 The thickness of the generalized fuzzy sphere . . . . . . . . . . . . . . . . . . . . . . 1168 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122viiiList of FiguresFigure 1.1 Fig. 1.1(a) illustrates a particle moving on a plane by a string with endpointsrestrained on a D2-brane while Fig. 1.1(b) illustrates the same setting effectivelyas a string with endpoints restrained on any two of N D0-branes, where theirlocations are uncertain within an area of ~ if the plane is described by the Moyalplane [X1, X2] = i~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Figure 3.1 Difference between x3 at finiteN (obtained numerically) and c (its largeN asymp-totics), as a function of N . The line represents equation (3.3.43), which has nofree parameters and appears to be an excellent match to the numerical data. Inthis figure, (a, b, c) = (1.5, 0.5, 3), c1 = 2, c2 = 5 and c4 = 4. For these values,equation (3.3.43) implies that c− x3 = 2.71875/J . . . . . . . . . . . . . . . . . . 40Figure 3.2 Magnitude, ∆, of the difference between the approximate eigenvector and theexact eigenvector as obtained numerically, for the ellipsoid in figure 3.1. Thestraight line, shown to guide the eye, is a best fit to the last few points andcorresponds to ∆ = 1.12√J. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 3.3 The difference between the actual eigenvalue x3 and the classical (large N)position c for a generic surface given by x1 = 1 + w1 + 0.5w3, x2 = 2w2,x3 = w3 + 0.2w1w2, at a point given by (w1, w2, w3) = (1/2, 1/4,√11/4). Theline shows equation (3.3.57). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 3.4 θ23/θ12 for the example in figure 3.3. This ratio appears to decrease like J−1. . . 46Figure 3.5 Magnitude of the overlap between the eigenstate corresponding to the point p atthe north pole and the eigenstate corresponding to a point p’ a distance |d| away.The green H corresponds to points p’ with x2 = 0, while the blue N correspondsto p’ with x1 = 0. The dashed line corresponds to equation (3.3.70). Plotted foran ellipsoid with c1=1, c2=0.75, c=12, with N=16,384. . . . . . . . . . . . . . . 47ixFigure 3.6 Angle φ between the normal vector ~n computed using equation (3.2.7) and thenon-commutativity vector ijkθjk, for the surface in equation (3.3.76) at a pointgiven by x = 0.5, y = 0. The blue N corresponds to N=3000 and the red H toN=12 000; the agreement between plots at different N shows that the plottedquantities scale with N in the expected way. On the horizontal axis we havea derivative of the non-commutativity along the surface scaled by√θ, whichincreases as µ is increased in equation (3.3.76). . . . . . . . . . . . . . . . . . . . 49Figure 3.7 Relative error in the non-commutative area as given in equation (3.5.1) comparedto the classical area, for an ellipsoid with major axes 6, 3 and 1. The error fallsoff with J like J−1; a best fit line, 1.02/J , is shown to guide the eye. . . . . . . . 54Figure 5.1 The upper figure plots the reciprocal radius versus the shell number k in irrep(Jl, Jr) = (200, 40). We expect the emergent radius distributed from R0 = 160to roughly R80 ≈ 240 with 81 layers. 20 samples of Rk solved from the eigen-equation are presented in plus signs and Rapp in cross signs. The dot line is thelinear fit with respect to Rk, and it is fairly close to the approximation 1 − k240 .In the lower plot, we see the radial error (Rapp −Rk)/Rk increases toward outershells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 5.2 The upper figure plots the reciprocal radius versus the shell number k in irrep(Jl, Jr) = (51, 50). We expect the emergent radius distributed from R0 = 1 toroughly R100 ≈ 101 with 101 layers. 20 samples of Rk solved from the eigen-equation are presented in plus signs and Rapp in cross signs. The dot line is thelinear fit with respect to Rk, and it is fairly close to the approximation 1− k101 . Inthe lower plot, we see the radial error (Rapp−Rk)/Rk increases much dramaticallytoward outer shells comparing to fig 5.1. . . . . . . . . . . . . . . . . . . . . . . . 81Figure 5.3 Radii versus shell number k in irrep (Jl, Jr) = (200, 40). The cross sign indicatesthe emergent radius, and the error bar refers to the variances 〈Λk|(Σ34−zk)2|Λk〉.We find that the variance as the thickness of most shells covers incoherent mar-gins, and coherent shells intercept with each other. . . . . . . . . . . . . . . . . . 82Figure 5.4 This data took so(4) generators in irrep (Jl, Jr) = (100, 40) for the lowest twoeigenvalues of the Laplace operator at different given radii. . . . . . . . . . . . . 85Figure 5.5 In so(4) irrep (Jl, Jr) = (200, 40), we present 6 samples at k = {0, 13, 27, 40, 54, 67}shells. For the upper plot, we draw black vertical lines for the Dirac radii, plussigns for the Laplace radii, and dash lines for λk,1(z). For the lower plot, wepresent ζ = RLk−RDkRDk+1−RDk−1showing the difference of Laplace and Dirac radius overthe distance between two adjacent shells. . . . . . . . . . . . . . . . . . . . . . . 88xFigure 5.6 Coefficients of coherent states at k = 16 shell in irrep (Jl, Jr) = (200, 40); co-efficients at larger indices are zeros and not presented; non-zero coefficients arefound only at states respecting the total azimuthal momentum= Jl + Jr − k.~a denotes Dirac’s upper half zerostate, ~b denotes Dirac’s lower half zerostate, ~cdenotes Laplace’s coherent state, and ~d denotes the approximate zerostate withperturbation method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 5.7 Parallelogram area between ~a and ~c in irrep (Jl, Jr) = (200, 40); ~a and ~c agreeeach other better at inner shells. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Figure 6.1 Given (Nf = 5, 15;R4 = 1), we plotted the minimum eigenvalue of LS3(x4) atvarious probe location x4. Enum is the lowest eigenvalue computed numerically,and Eλ is computed analytically with eq.(6.3.3). We found Eλ = Enum for anygiven {Nf , x4} and confirmed Eλ is the lowest eigenvalue. . . . . . . . . . . . . . 105Figure 6.2 GivenNf = 11, the black dots are the complete eigenvalues of LS3(x4 =R42Nf+1√Nf(Nf+4)),and the red line segment indicates the lowest eigenvalue Eλ(x4). There are12 = Nf + 1 ground states supporting our argument of (Nf + 1)-fold degener-acy in the ground state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Figure 6.3 The dashed circle depicts the three-sphere SL with radiusRL found by the Laplaceprobe. The dashdotted circle depicts the three-sphere SQ with radius RQ dueto the squared root sum of coordinates. The red dotted line represents the non-commutative cell at the x4 North pole, which is a ball with diameter 2√Θt inx1,2,3. The fuzzy three-sphere is the region bounded by SQ and SL as a result ofthe dispersion Θt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109xiChapter 1Introduction1.1 Emergent spacetimeFor over a century, physicists have been seeking the elementary understanding of spacetime fromboth the dynamical aspect in general relativity and the probabilistic aspect in quantum mechanics.General relativity considers spacetime geometry as the dynamical theory of gravity while quantummechanics considers the gravity as a quantum field. However, physically meaningful computationsof gravity can not be done in the unifying framework of quantum mechanics. Currently, in themodern view of string theory, both aspects of spacetime merge in the matrix models, where thephysical spacetime arises effectively from a set of matrices rather than being given as an a priorimanifold. This effective description of spacetime is called the emergent spacetime.For example, Moyal plane [3, 4] is an emergent plane made up by the position X and the mo-mentum P matrices, which parameterize the quantized phase space. From Heisenberg’s uncertaintyprinciple[X,P ] = XP − PX = i~, (1.1.1)one can explicitly represent X and P by infinite dimensional matrices asX =√~20√1 0 0 . . .√1 0√2 0 . . .0√2 0√3 . . .0 0√3 0 . . .............. . .and P =√~20 −i√1 0 0 . . .i√1 0 −i√2 0 . . .0 i√2 0 −i√3 . . .0 0 i√3 0 . . .............. . ..(1.1.2)The coordinate (x, p) on the Moyal plane comes from the expectation values,x = 〈α|X|α〉 and p = 〈α|P |α〉, (1.1.3)with respect to the coherent state |α〉, which is the eigenstate of the annihilation operator aˆ = X+iP1and satisfies aˆ|α〉 = |α〉 for α = x+ip. Therefore, we establish the x-p plane emerging from matricesX and P.As well as the coordinate, by taking the expectation with respect to the coherent state, a symbolmap correlating any analytic function f(x, p) on the plane to 〈α(x, p)|f(X,P )|α(x, p)〉 on the emer-gent plane is established. As indicated in the uncertainty principle, the algebraic computation onthe emergent plane is not commutable, and an observable O is uncertain within a certain amountof variance σO =√〈O2〉 − 〈O〉2. The uncertainty,~2≤ σxσp, (1.1.4)enlarges the coordinate (x, p) from a point to a cell with roughly the area of ~, which reflects thenon-commutativity of how fuzzy the emergent plane is. The Moyal plane converges to the classicalplane in the continuous limit whenlim~→0〈α(x, p)|f(X,P )|α(x, p)〉 = f(x, p), ∀(x, p) ∈ R2. (1.1.5)At the scale where the non-commutativity is relatively irrelevant, i.e. ∆x∆p >> ~, we expect thephysical observables on the emergent spacetime converge to experimental observations while thematrix interpretation of spacetime resolves the problems in quantum gravity.To determine physically meaningful matrices, we focus on solutions of equation of motion inmatrix models, e.g.,[Xµ, [Xµ, Xν ]] = cXν . (1.1.6)In general, matrix models have the form of quantum mechanics of N×N Hermitian matrices. Thereare many types of self-consistent string theories, and they are all contained in a mysterious, notfully understood M-theory as different limiting cases. Matrix models are interesting because theyare conjectured to be the complete dynamical description of M-theory in the large N limit. Moredetails are given in Section 2.2. In addition to the view of M-theory, matrix models can also berelated to the low energy description of worldvolume of a discretized membrane, which is assembledwith all possible loci of particles collectively. The illustration of discretized spacetime in matrixmodels arises from string theory.String theory replaces the fundamental constituent of point-like particles with one-dimensionalstrings. Various particles, like electrons, gravitons and any other elementary particles, are repre-sented by vibrational modes of the string. Instead of the worldline action of the particle trajectoryin spacetime, the dynamics of the string is described by the worldsheet action of a string sweepingthrough spacetime. A string can be an open line or a closed loop. The hypersurface of the collectionof open string endpoints is found to be a dynamical object named the D-brane [5,6]. In other words,string endpoints are restricted in the directions normal to the D-brane and move freely along thedirections tangential to the D-brane.2In the brane-world scenario, the Universe appears as a D-brane with attached strings representingmatter living therein, and the empirical physics of the Newtonian force, general relativistic gravity,and the standard model of particle physics are reproduced in the large-scale limit [7–10]. This pictureof the Universe has a discretized effective description because of the democratic characteristic ofD-branes [11–16]. The D-brane democracy means that infinitely many lower dimensional D-branesare collectively equivalent to a higher dimensional D-brane in such a way that the dynamics of lowerdimensional D-branes and the attached open strings converge to the commutative scenarios on thecorresponding higher dimensional D-brane, see illustration in Fig. 1.1.(a) D2-brane with a string (b) N D0-branes with a stringFigure 1.1: Fig. 1.1(a) illustrates a particle moving on a plane by a string with endpoints restrainedon a D2-brane while Fig. 1.1(b) illustrates the same setting effectively as a string with endpointsrestrained on any two of N D0-branes, where their locations are uncertain within an area of ~ if theplane is described by the Moyal plane [X1, X2] = i~.With N D-branes closely distributed at a distance far shorter than the string length, two end-points of an open string each has distinctive N choices [17,18], and the spacetime coordinates of ND-branes in their transverse directions are promoted to be N ×N matrices [19]. So the open stringdynamics at the substring-length scale is in the discretized worldvolume of N D-branes [14,20]. Thispoint will be clarified when we identity the open string effective field theory as Yang-Mills gaugetheory in Section 2.2.The challenge in this thesis lies in depicting a geometric object derived from a set of matricesand determining whether the continuous limit of matrix algebra exists. Early work on matrixconfigurations was mostly restricted to highly symmetric objects like planes, spheres, or tori, wherethe matrices are the representation of the symmetry group corresponding to the geometry, e.g.,the matrix representation of so(3) algebra generates the emergent sphere. Recently, new methodsfor generating emergent surfaces from general matrices have been proposed. In this thesis, weexamine these methods and make modifications for more complicated objects in higher dimensions,and we realize the necessary and sufficient conditions for an emergent surface to have a continuouslimit locally. For the analysis of D-brane dynamics or the interpretation of numerical results incosmology [21], it is important to study general matrix configurations with arbitrary shapes.3To probe the fine structure of a higher dimensional D-brane or a group of D-branes, refer-ence [20] computed the non-relativistic scattering of a lower dimensional D-brane colliding beam.Subsequently, reference [22] constructed a codimension-one two-dimensional surface from three col-lective coordinates with an effective Hamiltonian Heff(~x) measuring the distance from the targetsurface to a point-like probe at ~x. In particular, one can construct the coherent state as the groundstate of Heff(~x) on the emergent surface from the general matrix configuration [1,23]. In addition tothe surface equation, the coherent states reveal the internal structures of the matrix configurationfrom the non-commutativity of local fields, see references [1, 2, 24]. Thus, the emergent surface isestablished from given collective coordinates.The desire to probe the Universe emerged from general matrices motivates the work collectedin this thesis. To clarify the criteria in the correspondence about how to recover the empiricalspacetime in matrix models, we study emergent surfaces with inhomogeneous D-branes distributionsin Chapter 3. More explicitly, we examine an arbitrarily distorted sphere to determine the numberof D-branes on a curved patch required to reach a continuum limit. In Chapter 4, previous works onthe point probe are generalized for hypersurfaces embedded in higher dimensions. The hyperspheredemonstrates that coherent states are often highly degenerate, and they form the representationspace for the local symmetry. In Chapter 5, we carefully examine a spherical matrix configurationS2kso(2k+2) out of so(2k + 2) generators with several probing methods including the point probe,where S2kso(2k+2) or its Wick rotation Hm+nso(n,m+2) with the signature (n,m+ 2) are classical solutionsof the matrix theory. In particular, H4so(2,4) represents an expanding 3+1-dimensional Universe[25–27]. Conceptually, one can picture the worldvolume of an emergent de Sitter Universe as anon-commutative hyperboloid. Given that the D-brane constituents are much thinner towardsboth ends, the empirical world is captured by the almost commutative middle segment. Outsidethe middle segment, e.g., the Universe before the big bang, extremely non-commutative spacetimedeviates far from our empirical realization and requires a new non-perturbative description dueto the divergent string length scale, which is proportional to the non-commutativity. In Chapter6, we examine the fuzzy three sphere and resolve the issue of inconsistent radius with the non-commutative dispersion via coherent states constructed in the previous chapters. In Chapter 7, westudy the critical phenomena of the massive Yang-Mills matrix model, which generates the solutionsof generalized fuzzy spheres examined in Chapter 5, and we determine the critical mass when themodel could potentially have a dual picture in the gravity theory.In this introduction, we illustrate an emergent space constructed from matrices by the exampleof Moyal plane and motivate physically meaningful matrices as a solution of equation of motionin matrix models, and we consider the Universe effectively as a discretized D-brane constructedfrom a set of matrices. The values of physical observables are taken via the coherent state-symbolmap from matrix computations. In the next chapter, we briefly introduce the point-probe methodsmentioned above and give more technical details about D-brane dynamics and matrix models.4Chapter 2Background2.1 Non-commutative geometryNon-commutative geometry [28, 29] has become a vast field in mathematics ever since A. Conne’sspectral triple(AHD) [4, 13], which consists of a matrix algebra A of bounded operators on theHilbert space H and a Hermitian operator D encrypting the geometric information in its spectrum.Non-commutative geometry is of great interest to string theorists because it is the language ofM-theory [13,14].From a physics perspective, advancing from geometry to non-commutative geometry is compa-rable to the transition from classical mechanics to quantum mechanics. Classical observables arefunctions on a phase space, which is a symplectic manifold with a natural Poisson structure. In otherwords, phase space is a smooth manifold with closed non-degenerate 2-form ω = ωijdxi ∧ dxj (sym-plectic form), where a Lie bracket {·, ·} = ωij ∂∂xi∂∂xjis defined as Poisson bracket. The quantumobservables are Hermitian operators acting on the Hilbert space and obeying the non-commutativealgebra, which replaces Poisson brackets by commutators. The quantized phase space can then berealized as the non-commutative description of the symplectic manifold in classical mechanics [30].The non-commutative geometrical description of the emergent spacetime shall bring us many bizarreyet fascinating discoveries, like one’s first step into the quantum world from classical physics.A series of non-commutative geometries arise from coadjoint orbits of Lie groups in the orbitmethod [31, 32]. The coadjoint orbit of a compact connected Lie group G means the locus in thevector space of the dual algebra g∗, and the locus is resulted from the adjoint action of G, i.e.,Ad∗Gi(g∗c ) = G−1i · g∗c ·Gi for g∗c ∈ g∗ and Gi ∈ G. (2.1.1)One finds coadjoint orbit {Ad∗G(g∗c )} is a G-invariant symplectic manifold [33]. The coadjoint orbitis isomorphic to the homogeneous coset space G/Stab(g∗c ), where Stab is the stabilizer subgroupof G, and the topology of a coadjoint orbit is more transparent in the view of its isomorphismG/Stab(g∗c ). The orbit method states that G/Stab(g∗c ), which we think of as the phase space of a5classical mechanical system, can be quantized by the unitary irreducible representation (unirrep) ofG, and the Poisson bracket defined with respect to the symplectic form of G/Stab(g∗c ) correspondsto the commutator in the quantization. Unfortunately, this method is not precisely a theorembecause there are counter-examples of groups with their coadjoint orbits failing to correspond toany of their unirreps [34].The correspondence principle maps a quantum operator to a classical observable, such that thequantum system converges to the classical one in the infinitesimal Planck-length limitation, whichis achieved by taking the representation dimensions to be infinite in the studies of non-commutativegeometry. The Weyl-Wigner correspondence [35,36] is a conventional treatment bridging a classicalfunction and an operator introduced in the Weyl quantization as the following.Let us consider an example of a Euclidean space Rn and its Weyl quantization, following thepopular lecture of reference [37]. The quantized Rn is described by a set of generators {xˆi} ofthe algebra A, and we have the correlation from the generator to the coordinate, xˆi → xi. TheWeyl symbol was introduced to map the function f(x) in a Fourier series with coefficients f˜(k) =1(2pi)n/2∫Rn dnxf(x)e−ikixi to the operator Wˆ [f ] in A asWˆ [f ] =1(2pi)n/2∫Rndnk f˜(k)eikixˆi. (2.1.2)The differentiation of functions has its corresponding operation in the non-commutative regime.Differentiation with respect to xˆi is taken to be the commutator with a skew-Hermitian matrix ∂ˆi,i.e., ∂ˆ†i = −∂ˆi, defined by the algebra[∂ˆi, xˆj]= δij ,[∂ˆi, ∂ˆj]= 0, (2.1.3)which leads to[∂ˆi, eikj xˆj]= ikieikj xˆj , and therefore the differentiation on the operators Wˆ [f ] isderived as the Weyl symbol of the differentiated function,[∂ˆi, Wˆ [f ]]= W [∂if ] . (2.1.4)The volume integral is captured by the traceTr(Wˆ [f ])=∫dnxf(x), (2.1.5)which follows from Tr(∫Rndnk(2pi)n e−ikixieikixˆi)= 1 for the normalization of xˆi.In the canonical example with the algebra [xˆi, xˆj ] = iθij , where θij is an invertiable constantmatrix, one finds the inverse mapping of the Weyl symbolf(x) = Tr(Wˆ [f ]∫Rndnk(2pi)ne−ikixieikixˆi). (2.1.6)6Therefore the Weyl-Wigner correspondence becomes a one-to-one mapping. In this example ofconstant θij , the algebra of symbols Wˆ [f ? g] = Wˆ [f ]Wˆ [g], known as the Moyal star product,follows straightforwardly from the inverse mapping asf ? g(x)= Tr(Wˆ [f ]Wˆ [g]∫Rndnk(2pi)n e−ikixieikixˆi)= 1(2pi)n∫Rn∫Rn dnpdnqf˜(p)g˜(q)e−i2piqjθijei(pi+qi)xi= ei2∂∂yiθij ∂∂zj f(y)g(z)|y,z→x,(2.1.7)where we have applied Tr(ei(pi+qi)xˆi ∫Rndnk(2pi)n e−ikixieikixˆi)= ei(pi+qi)xi .In another less trivial example when the non-commutative structure follows a Lie algebra,[xˆi, xˆj ] = Ckij xˆk, the star product can be obtained similarly. From the Baker-Campbell-Hausdorffformula, we haveeikixˆieipj xˆj= exp[i (ki + pi + gi(k, p)) xˆi], (2.1.8)where gi depends on the structure constant Ckij of the Lie algebra, and we find the star product asWˆ (f ? g)= 1(2pi)n∫Rn∫Rn dnkdnpf˜(k)g˜(p)ei(ki+pi+gi(k,p))xˆi= Wˆ(1(2pi)n∫Rn∫Rn dnkdnpf˜(k)g˜(p)ei(ki+pi+gi(k,p))xi)= Wˆ(eixjgj(i ∂∂y,i ∂∂z)f(y)g(z)|y,z→x).(2.1.9)For our interest of general matrix configurations in this thesis, e.g., the algebra is not closed forthe generalized fuzzy sphere studied in Chapter 5, we relaxed the one-to-one symbol map and tooka different approach to assign the local value of an operator on the emergent surface like a functionon the surface. In the Berezin quantization [38], every point on an emergent surface is associatedwith a coherent state |αp〉, and the symbol map correlates any operator Aˆ to a function on theemergent surface via the expectations(Aˆ) = 〈αp|Aˆ|αp〉. (2.1.10)The collection of coherent states forms a complete representation of the Hilbert space, and the starproduct can be defined accordingly.Historically, coherent states were introduced by Schrödinger as a quantum oscillator’s classicalstates satisfying the correspondence principle. Later, the concept was generalized in reference [39]for coherent states existing in homogeneous spaces generated by the orbits of arbitrary Lie groups,which include most of the non-commutative geometries studied in this thesis. We consider thecoherent states either as the zero modes (eigenstates with zero eigenvalue) of the Dirac operatordefining the non-commutative geometry in Chapters 3 and 4, or as the optimally localized states7in Chapter 5. Coherent states have been shown to be a powerful tool on analyzing fuzzy spaces inmany recent studies [1, 23, 24, 40–42], and most of the work in this thesis is devoted to searchingcoherent states for various non-commutative geometries.2.2 Matrix models2.2.1 D-brane dynamicsBy restraining the loci of open string endpoints on a hypersurface in the spacetime, the open stringboundary is in the Dirichlet/Neumann condition along the direction transverse/tangential to thesurface, respectively. This hypersurface with p spatial dimensions is denoted by a Dp-brane. Inresponse to the influences of string fields, Dp-branes have tension Tp, which is reciprocal to thestring coupling gs.In the weak string-coupling limit, D-branes appear as rigid massive solitonic objects restrainingopen strings. In the strong string-coupling limit, D-branes vibrate and have masses as light asstrings. The massless field in the open string spectrum in ten-dimensions has components paralleland perpendicular to the D-brane, and they can be described by a low-energy field theory on theworldvolume of the D-brane. non-perturbatively, one interprets the 9− p transverse components Φiassociated with the fluctuation of theDp-brane as scalar fields and the p+1 longitudinal componentsAα as a U(1) gauge field on the Dp-brane.The dynamics of Dp-branes is described by its p+1-dimensional worldvolume in the Dirac-Born-Infeld action (DBI action) [43–45]SBI = −Tp∫dp+1σ(e−φ√−det (P [G+B]αβ + 2pil2sFαβ)). (2.2.1)This action captures the dynamics of a single low energyDp-brane and slowly varying weakly coupledfields on the brane. The notation P [. . . ] refers to the pullback of a tensor from the bulk spacetimeto the Dp-brane worldvolume, e.g., the induced metric of the p+ 1-dimensional worldvolume fromthe bulk metric Gµν :P [G]αβ = Gµν∂xµ∂σα∂xν∂σβ= Gαβ + 4pil2sGi(α∂β)Φi + 4pi2l4sGij∂αΦi∂βΦj , (2.2.2)where ls is string length scale. The gauge field Fαβ = ∂αAβ − ∂βAα and scalar fields Φi = 12pil2s xiare the massless excitations of the open strings attached on the Dp-brane. While the backgroundis filled with the closed strings, SBI is affected by the massless modes of closed strings includingthe dilaton φ, the graviton G, and the Kalb-Ramond field B. By considering vanishing fieldsB = F = 0 for simplicity, SBI becomes the p+ 1-dimensional worldvolume swept by the Dp-brane,and the worldvolume times the tension Tp gives the familiar Nambu-Goto type p-brane action, whichcomputes the worldvolume swept by a p-dimensional surface.8A Dp-brane also carries Ramond-Ramond(RR) charges interacting with the (p + 1)-form RRpotential Cp+1 from the closed string spectrum [6]. The coupling with RR fields stabilizes theconfiguration of a D-brane if a certain RR field is given in the background, e.g., spherical D2 (fuzzysphere) [16]. This coupling is captured in the Wess-Zumino action [12,45]SWZ = µp∫P [ΣnC(n)eB]e2pil2sF . (2.2.3)The summation of n-form RR potentials implies possible structures of branes within branes [12].The non-commutative characteristic arises from the multiplicity of (nearly) coincident D-branes.Degenerate open string states among N2 combinations of attachments on N coincident D-branespromotes Aα to a U(N) gauge field and Φi to the adjoint scalar fields [18]. Instead of presenting thenon-abelian DBI action of N D-branes [16,46], which is beyond of the scope of this introduction, weintroduce its low energy approximation as the dimensionally reduced Yang-Mills (YM) gauge theory,where the non-commutative characteristic is rooted in non-abelian fields. The ten-dimensionalsupersymmetric YM theory of a U(N) gauge field Aµ is given bySYM = − 14g2YM∫d10x Tr(14FµνFµν +12Ψ¯ΓµDµΨ), (2.2.4)where Fµν = ∂µAν−∂νAµ+i[Aµ, Aν ] and Dµ = ∂µ+i[Aµ, ·]. By assuming fields depend on xα=0,...,ponly and reassigning Ai=p+1,...,10 = Φi, SYM has the following dimensional reductionSredYM =−14g2YM∫dp+1x Tr(14FαβFαβ +12DαΦiDαΦi − 14∑i 6=j [Φi,Φj ]2+ i2Ψ¯Γi[Φi,Ψ] +12Ψ¯ΓαDαΨ).(2.2.5)On the other hand, we can expand SBI in the low energy limit at length scales L  ls, so theexpansion can be truncated by slowly varying fields (O(F 4) = O((∂x)4) ≈ 0). In flat space withB = 0, the leading order of SBI with non-abelian Φi and Aa recovers the bosonic part (14F2) in SredYMby tuning the coupling constant asg2YM =gsTp(2pil2s )2, (2.2.6)where the string coupling gs = eφ is governed by the dilaton φ.The emergent spacetime composed of N D-branes is a non-commutative geometry because thetransverse coordinates of N D-branes are grouped in terms of matrices [13,19,47], which are recog-nized as the adjoint scalar field Φi in the U(N) gauge theory. The matrix coordinate is motivatedby the T-duality [5, 18], which is a duality between two string theories each with a dimensioncompactified by the radius R or R′ = α′/R.In the compactified direction, T-duality relates the momentum of an open string to the endpointsof the fixed open string in another theory, and we see that a wrapped Dp-brane is exchanged into aDp+1/Dp−1 brane in the theory T-dualized in the direction perpendicualr or parallel to theDp-brane,9respectively. In addition, if a magnetic field appears in the background, the vector potential as a partof the conjugate momentum is T-dualized to be the endpoints of the open string. Subsequently,the U(N) gauge field on N coincident Dp-branes results in the matrix coordinate in the adjointrepresentation of SU(N) for embedding N Dp−1-branes spread across the compactified dimensionin the dual theory.Furthermore, T-duality relates a theory to another with a reciprocal radius, and N separatedD-branes in the theory compactified by sub-string radius R < lp are dual to N nearly overlappedD-branes of the non-commutative theory in the uncommpactified limit R′  1. Incidentally, theinterchange of background fields and the non-commutativity is explicit in the low energy descriptionof N D-branes in terms of Yang-Mills gauge theory, see references [14,48,49]. Known as the Seiberg-Witten map [14], T-duality connects ordinary YM theory with a background magnetic field to thenon-commutative YM theory.2.2.2 BFSS and IKKT matrix modelsBanks-Fishler-Shenker-Susskind model (BFSS model) proposed in reference [50] is a ma-trix quantum mechanics theory obtained by dimensionally reducing the supersymmetric Yang-Millstheory from 9 + 1-dimensions to 0 + 1 dimension. The actionSBFSS =∫dt Tr(12DtXµDtXµ + Ψ†DtΨ− 14[Xµ, Xν ]2 − 12Ψ¯Γµ [Xµ,Ψ])(2.2.7)describes a system of non-relativistic N D-particles(D0-branes) in type IIA string theory with nineN ×N bosonic matrices Xµ and sixteen Grassmann N ×N matrices Ψ.It has long been suspected that different types of ten-dimensional superstring theories dual toeach other are different limits of a unified eleven-dimensional M theory, where the fundamental con-stituents are generalized from strings to higher dimensional membranes. A membrane compactifiedon a light-like circle has its Hamiltonian in the large N gauged quantum mechanics [51], which is thesame as the Hamiltonian for a group of D-particles at distance smaller than the string length [52].Then, reference [50] conjectured that the light-cone M theory could be interpreted as a system ofD0-branes with the action SBFSS in the IIA string theory when these D0-branes are boosted alongthe x11 direction in the frame with infinite momentum.Precisely, through a boost, the light-like compactification can be viewed as a limitation of aspace-like compactification with infinitesimal radius Rc → 0 and a divergent compact momentumPc = N/Rc →∞. The infinitesimal compactification reduces the x11-dependence of fields and wrapsup the two-dimensional membrane in M theory into the fundamental string of type IIA theory.The type IIA string coupling g and string length scale ls are related to the compactificationradius Rc and Planck length l11 in eleven-dimensions byg2 =R3cl311and l2s =l311Rc, (2.2.8)10and the limit of Rc is taken while the coupling and the string scale are small. On the other hand,at infinite compactification radius, M theory can be regarded as the strong coupling limit of typeIIA string theory, and furthermore, references [53–55] argued that the BFSS matrix theory can stillbe interpreted as the compactified M theory at finite N .Ishibashi-Kawai-Kitazawa-Tsuchiya (IKKT model) is proposed in reference [56] as a non-perturbative and constructive definition of type IIB string theory. This IIB matrix model has theactionSIKKT = Tr(−14[Xµ, Xν ]2 − 12Ψ¯Γµ [Xµ,Ψ]), (2.2.9)which can be regarded as the completely dimensionally reduced super Yang-Mills theory.Unlike the BFSS model, where time is a number in the time-dependent matrices in the BFSSmodel, IKKT model sets time as a Hermitian matrix as well as other spatial collective coordinates.The manifest Lorentz invariance and N = 2 supersymmetry support the framework of dynamicalemergent spacetime as collective coordinates of D-instantons (D−1-branes) from the solutions in themodel [7, 8].The IKKT model can also be viewed as the matrix-regularized worldsheet of superstrings [56].In the type IIB superstring theory, the Green-Schwartz action with gauge-fixed κ symmetry isSGS = −T∫d2σ(√−det∂aXµ∂bXµ + 2iab∂aXµΨ¯Γµ∂bΨ) . (2.2.10)SGS is the Nambu-Goto type integration of the superstring worldsheet. As shown in references[57–59], SGS is equivalent to the Schild actionSSchild =∫d2σ[α√g(14{Xµ, Xν}2 − i2Ψ¯Γµ{Xµ,Ψ})+ β√g], (2.2.11)where the Poisson bracket is defined as {X,Y } ≡ 1√g ab∂aX∂bY , and g is an auxiliary field that canbe identified as g = detgab from the worldsheet metric gab. The variation over√g yields√g =12√αβ√(ab∂aXµ∂bXν)2 =12√αβ√−2det∂aXµ∂bXµ, (2.2.12)so we see that SSchild is equivalent to SGS up to a normalization of Ψ.To recognize the IKKT model as the matrix-regularized Schild action, one treats Xµ and Ψ asN ×N Hermitian matrices and replaces the Poisson bracket and the integration respectively by{ , } → 1i[ , ] and∫d2σ√g → Tr, (2.2.13)and one finds the IKKT model with a cosmological term inSSchild → αTr(−14[Xµ, Xν ]2 − 12Ψ¯Γµ [Xµ,Ψ])+ Trβ, (2.2.14)11where the cosmological term Trβ = βN produces the relative weights for different matrix size N ,which is also the number of branes. The role of the auxiliary field √g is here indirectly played byN .Similar interpretation of SIKKT as a quantized worldsheet applies as well to SBFSS such thatSBFSS can be viewed as the worldvolume of quantized membranes [51, 60]. The IKKT model isdeeply related to the BFSS model, just like the fact that type IIA and IIB string theories aretwo perturbative expansions under the united M-theory. Type IIA and IIB superstring theoriesare interchangable using toroidal compactification, and similarly the Wick-rotated IKKT modelwith a compact Euclidean time can be regarded as the BFSS model at finite temperature [13]. Inreference [61], the duality between two matrix models is further examined in the decompactifiedlimit using Monte Carlo simulations.2.2.3 Emergent UniverseIn the brane-world scenario [9, 10], the empirical 3+1-dimensional Universe is illustrated by a D3-brane embedded in a bulk spacetime with higher dimensions required by string theories. As the extradimensions are still beyond our abilities to measure, this scenario offers a macroscopic alternativecomparing to the idea of compactifying all extra dimensions by vanishingly small radii, and thisscenario explains the hierarchy problem about gravity being significantly weaker than other forcesbecause the closed strings carrying the gravity escape into the bulk while open strings carryingother forces are restrained on the D-brane.The quantized brane world arises naturally in the non-commutative picture of a D-brane decom-posed into N D-particles collectively described by matrices in matrix models. The Universe thatemerges in matrix models is a set of matrices as the solution to the equations of motion, while thematter in the Universe arises from perturbations of the matrices. The concept of emergence meansthat the matrix theory formulated without references to the background spacetime appears to bethe theory in a spacetime manifold with certain topology given by matrices. Furthermore, by con-sidering both space and time dynamically emerged in the covariant IKKT matrix model [7, 8], thenon-commutative worldvolume of the Universe is encoded in a set of matrices since the beginningof time, and one can study the very origin of the Universe.The matrices of the emergent Universe in the IKKT model shall depict the evolution of expandingthree spatial dimensions and exhibits the spatial symmetry spontaneously broken from SO(9) toSO(3) because our Universe appears to have only three rather than nine spatial dimensions. Matriceswith these features have been numerically found from the partition functionZIKKT =∫dXdΨ eiSIKKT (2.2.15)by Monte Carlo simulations in references [21,62,63]. The expansion radius R(t) starts exponentiallyas inflation and slows down to R(t) ∼ √t, in agreement with the Friedmann-Robertson-Walker12Universe in the radiation dominated era.However, ZIKKT is not well defined because the Lorentzian metric allows for divergence in boththe temporal and spatial directions, so cutoffs must be added by hand. If one makes the Wickrotation X0 → iX0, ZIKKT with a Euclidean metric is proven to be convergent [64]. Although thesponteneously broken symmetry of SO(10) is also observed [65] in which the SO(3) symmetric vacuahave the minimum free energy among other subsymmetries SO(d<10), the extent of spacetime islimited in the Euclidean version, and the real time dynamics can not be extracted.The numerical study [21] of the Lorenzian ZIKKT with infinite extent requires temporal andspatial cutoffs,1Ntr (X0)2 ≤ κL2 and 1Ntr (Xi)2 ≤ L2, (2.2.16)where the cutoff parameters κ and L can be sent to∞ in the large N limit by rescaling the matricessuch that the computation with enlarged κ and L remains consistent. Applying the cutoffs andintegrating out the fermionic matrices Ψ, the partition function is regularized as∫dXL2∫0dr δ(1NTr(X2i )− r)θ(κL2 − 1NTr(X0)2))eiTrF2PfM(X), (2.2.17)where F 2 = −14 [Xµ, Xν ]2, PfM(X) is the Pfaffian1, and θ is the Heaviside step function.Rescaling the matrices, X → ρX transformsF 2 → ρ4F 2,dX → ρ10(N2−1)dX,PfM(X) → ρ8(N2−1)PfM(X),(2.2.18)so the integration of the scale,∫dr, essentially converts δ(1NTr(X2i )− r)eiTrF2 into the constraintδ(TrF 2)δ(r − L2) in the large N limit. In reference [21], Monte Carlo simulations are applied onthe approximate partition function∫dX δ(1NTr(X2i )− L2)θ(κL2 − 1NTr(X0)2))δ(1NTrF 2)PfM(X). (2.2.19)The spacetime constructed in reference [21] is based on the following observations. In the basisof the diagonalized temporal matrix arranged in a strictly increasing order, the spatial matricesare nearly in a band form such that elements decrease rapidly in the off-diagonal direction. Thenreference [21] divides the temporal matrix into blocks and identified the corresponding spatial blocksas the collective coordinates of the emergent Universe at the time approximated by the average of thetemporal eigenvalues in the block. With the truncated matrices X¯i, the expansion of the Universe1Pfaffian is a shorthand notation for the terms appearing from Gaussian-integrating out fermions in the partitionfunction. By simply referring to the scaling of Pfaffian in equation (2.2.18), we explain the large N approximation ofthe regularized ZIKKT in equation (2.2.19).13is approximated by the radius R(t) =√∑9i=11nTrX¯2i , where n < N is the rank of the block. Witha proper SO(9) rotation, one finds that 1nTrX¯2i is significantly larger in three out of nine directionsand identifies the spontaneous symmetry breaking of SO(9) down to SO(3).The constrains in the approximated partition function (2.2.19) suggest that the emergent Uni-verse can be approached by the stationary points of the bosonic part 1NTrF2 with fixed 1NTrX20 =κL2 and∑i1NTrX2i = L2, and one can instead work on the actionS˜b = tr(−14[Xµ, Xν ][Xµ, Xν ] +λ˜2(X20 − κL2)−λ2(XiXi − L2)), (2.2.20)with Lagrangian multipliers λ and λ˜. the equations of motion[Xµ, [Xµ, Xν ]] = ρ(ν)Xν , (2.2.21)where ρ(0) = λ˜ and ρ(i) = λ, admits many interesting analytic solutions, in contrast to the masslessmodel ρ = 0, which has no finite-dimensional non-commutative solutions. Extending the early stageemergent Universe [21] to a much longer time scale, references [62,66] studied an almost commutativesolution with a similar exponential expansion behavior.Another intriguing solution representing a 3+1-dimensional expanding Universe with five collec-tive coordinates of so(2,4) generators in a matrix representation is given in references [26, 67]. Theso(2,4) Universe is a two-sheet hyperboloid H4so(2,4) featuring big bang and inflation, where the bigbang arises from the signature of H4so(2,4)’s metric changing from Euclidean to Lorentzian. Surpris-ingly, the signature-changing manifold is a valid solution of the Einstein’s equation [68]. In theHartle-Hawking proposal [69], the Universe is created from the quantum tunneling as a Euclideanfour-sphere with no boundary, and only when the metric becomes Lorentzian after the big bang isthe time meaningful.From m + n + 1 generators of the Lorentz algebra2 so(n,m+2), equation (2.2.21) admits solu-tions of m + n-dimensional fuzzy spaces, such as spheres, hyperboloids of one-sheet or two-sheet,depending on the signature (n,m + 2) and the Lagrangian multiplier ρ(n,m). For either a sphereor a hyperboloid embedded in the Minkowski background, it is natural to find the induced metric(pullback) with signature change as time grows. However, references [26, 67] pointed out that thenon-commutative structure on the brane is crucial for the early rapid expansion. Instead of takingthe brane metric as the induced metric on the emergent surface, one shall take the brane metricas the effective metric, which exhibits a behavior of inflation [27, 70], where the effective metricis read off by comparing the kinetic terms of the scalar field in the non-commutative/semi-classicdescriptions. S2so(4) as the simplest of its kind, is studied thoroughly with various probing methodsin Chapter 5 for its intriguing non-commutative structure and abundant physical applications.2Throughout this dissertation, Lie groups are denoted by capital letters, and Lie algebras are denoted by lowercaseletters.142.3 Point probesTo map the occupation of D-branes from their matrix coordinates, one considers a point probemoving in the background space and measures the length of a string stretching from the probe tothe target. The emergent surface is the collection of the probe locus where the connecting stringfinds a massless mode, which reflects the zero length, from the effective Hamiltonian consisting ofthe matrix coordinates and the probe locus. In this section, we introduce two effective Hamiltoniansproposed for such point probe methods.The Dirac-type point probe is proposed in reference [22]. The authors considered the probebrane interacting with a stack of N static D0-branes at an orbifold in the BFSS model, and the di-mension of the space transverse to theD0-branes is reduced to three, i.e., the embedding backgroundis R3.A fermionic string stretching from the stack to the probe is described by the fermion mass matrixgiven by the following effective Hamiltonian:Heff(xi) =∑i=1,2,3σi ⊗ (Xi − xi) , (2.3.1)where σi are the Pauli matrices, Xi are Hermitian N×N matrices corresponding to the positions ofthe stack of D0-branes, and xi are the positions of the probe brane. The stretched string has a zeromass fermionic mode when Heff has a zero eigenvalue. Thus, from three Hermitian matrices Xi, theco-dimension one surface in flat R3 is given by the locus of points where Heff has zero eigenvalues.Heff is derived from the Dirac operator term312Ψ¯ΓI [φI,Ψ] , (2.3.2)which gives the Dirac mass of the fermionic string in the matrix models. φI represents the collectivecoordinates of the target D-branes and the probe in the direct sumφI = XI ⊕ xI =(XIN×N 0N×101×N xI)(N+1)×(N+1). (2.3.3)Considering only the fermionic strings connecting the target to the probe, the fermion matrix isdefined asΨ =(0N×N ψN×101×N 0)(N+1)×(N+1). (2.3.4)Referring to references [71,72], one can justify the restriction on fermion states by fractional branesin orbifold models.3The matrix Dirac operator ΓI⊗ [φI, ·] comes from the standard Dirac operator ΓIDI in the dimensionally reducedYM models, see equation (2.2.5), where the notation of direct product ⊗ is omitted in the standard textbooks.15To replace nine gamma matrices Γ in the BFSS model by three Pauli matrices σ, one simplychoose a representation of Γ manifesting the subgroup structure SO(3)⊗ SO(6) with respect to thebranching rule of the spatial symmetry SO(9). By demanding that φJ = 0 except for the threeφi=1,2,3 holding the target D-branes, it is straightforward to derive equation (2.3.1) from equation(2.3.2).For D-branes in higher dimensions, it is natural to construct a similar effective Hamiltonianfrom the Dirac operator. In Chapter 4, we conjecture an embedding operator assigning any 2n+ 1hermitian matrices to a 2n-dimensional hypersurface in R2n+1 by generalizing equation (2.3.1).With nine spatial dimensions, we let the indices {i, j, . . . }, {α, β, . . . } denote the 2n+ 1 embeddingdimensions and the rest 8 − 2n dimensions. In the restricted representation manifesting SO(2n +1)⊗ SO(8− 2n), Γ is written asΓα =(0 gα† ⊗ 1gα ⊗ 1 0), and Γi =(1⊗ γi 00 −1⊗ γi), (2.3.5)where γ and g satisfy {{γi, γj} = 2δij ,gαg†β + gβg†α = 2δαβ.(2.3.6)These Γ matrices give rise to so(p+1), so(8-p) algebras asΣij =14i[Γi,Γj] ∈ so(2n+ 1) and Σαβ = 14i[Γα,Γβ]∈ so(8− 2n), (2.3.7)and the desired symmetry is manifest. To be explicit, the reduced gamma matrices with the so(d)symmetry can be written asγi = σ3 ⊗ γ˜i, i = 1, . . . , d− 2,γd−1 = σ1 ⊗ 1,γd = σ2 ⊗ 1,(2.3.8)where γ˜i are constructed inductionally from {σ1, σ2, σ3}. Henceforth, 12Ψ¯Γµ [φµ,Ψ] reduces toψ¯γi [φi, ψ] in the 2n+ 1-dimensional subspace, and we define the embedding operator asE2n+1(x1, . . . , x2n+1) =2n+1∑i=1γi ⊗ (Xi − xi) (2.3.9)for the codimension-one fuzzy object embedded in R2n+1.Besides locating where Heff has zero eigenvalue, we study the corresponding zero eigenvector|Λ(~x)〉 in E2n+1(~x)|Λ(~x)〉 = 0 in Chapter 3. |Λ(~x)〉 plays the role of the general coherent states [39],and the expectation 〈Λ(~x)|Oˆ(X)|Λ(~x)〉 assigns the value of any local operator Oˆ(X) at ~x on theemergent surface in the same manner as Berezin’s symbol map [38]. For example, the expectation16of matrix coordinates with respect to |Λ(~x)〉 gives us the coordinate of the surface point〈Λ~x|1⊗Xa|Λ~x〉 = xa. (2.3.10)This equality comes from the vanishing expectation0 = 〈Λ~x|γa ⊗ 1 · Ed(~x)|Λ~x〉= 〈Λ~x|1⊗ (Xa − xa) |Λ~x〉︸ ︷︷ ︸real+∑i 6=a〈Λ~x|γaγi ⊗ (Xi − xi) |Λ~x〉︸ ︷︷ ︸imaginary, (γaγi)†=−γaγi. (2.3.11)In the large N limit, one expects that the general coherent state |Λ〉 establishes the symbol mapof an analytic function f like〈Λ~x|f(X)|Λ~x〉 = f(~x) +O(N−1) (2.3.12)for ~x ∈ {~x|det(Ed({X})) = 0}. The issue of mismatch of dimensions between |Λ〉 and matrixcoordinates X is addressed in Chapter 3, where we split |Λ〉 into copies of parallel vectors, eachwith the same dimension as X, and the symbol map becomes trivial.In order to quantify the non-commutative structure on the fuzzy object, the non-commutativityis measured by the variance of the expectation,Θ ≡ 〈Λ|∑i1⊗ (Xi − xi)2 |Λ〉. (2.3.13)We may define the non-commutativity operator Θˆ in terms of commutators asΘˆ =∑i 6=jSij ⊗ 1i[Xi, Xj ], (2.3.14)whereSij =14i[γi, γj ] ∈ so(d). (2.3.15)From the vanishing quadratic expectation0 =〈Λ|E2d |Λ〉=∑i〈Λ|1⊗ (Xi − xi)2 + 14∑i 6=j[γi, γj ]⊗ [Xi, Xj ]|Λ〉, (2.3.16)we see that the non-commutativity has the same value as the expectation of the non-commutativityoperator Θ = 〈Λ|Θˆ|Λ〉.The Laplace type point probe is proposed in reference [24]. Another effective Hamiltonianis defined asL =∑i(Xi − xi)2 (2.3.17)17for the collective coordinates {X} and the probe locus ~x. Instead of finding zero modes on thestring spectrum, one recognizes the emergent surface point at the probe locus where the groundenergy EL(~x) of L is minimized, and the corresponding ground state |λ〉 generates the symbol map.Similar to the embedding operator, one expects 〈λ|Xi|λ〉 = xi +O(1/N) from the symbol map,and 〈λ|L|λ〉 tells us the dispersion of the emergent point, which converges to the standard deviation∑i〈λ|X2i |λ〉 − 〈λ|Xi|λ〉2 in the large N limit. The minimization of 〈λ|L|λ〉 indicates the groundstate |λ〉 as the coherent state in a general sense such that |λ〉 is optimally localized.Not every arbitrary set of matrices admits a convergent configuration. For a matrix configurationM to be convergent, one shall find a distinguishable hierarchy of eigenvalues of the Hessian matrixH(~x) = ∇∇EL(~x), and the eigenvectors corresponding to the smaller eigenvalues are tangent toM while the rest of the eigenvectors with larger eigenvalues are transverse to M according toreference [24]. In other words, one expects that the ground energy EL(~x) varies slowly along M,but varies dramatically away fromM.However, minimizing the ground energy is less trivial then finding the zero mode of the em-bedding operator. If the emergent surface is wrinkled, close layers introduce a difficult situationof tracking the eigenvalue of L belonging to a specific layer. Say that two eigenvalues EL(~x) andE′L(~x) correspond to the layer A and B respectively, it is possible to find EL( ~xA) > E′L( ~xA) on thelayer A, so EL( ~xA) does not appear as the ground energy and contradicts the definition.In Chapter 5, we study an example with close layers and resolve the emergent geometry explicitlywith geometric symmetries. For one matrix configuration, one expects the emergent surface andthe coherent states based on the Dirac type and the Laplace type point-probes should converge toeach other in the large N limit. The detailed comparison is presented in Chapter 5.2.4 Fuzzy sphere2.4.1 The coadjoint orbit of SU(2)Following its introduction in reference [73], our first realization of the fuzzy sphere starts with thequantized coadjoint orbit of SU(2) as mentioned in Section 2.1. According to the orbit method[31, 32], the coadjoint orbit of SU(2) is quantized by the irreducible representations of su(2) or itsuniversal enveloping algebras. The inner product of the Lie algebra g = su(2) and its dual g∗ isdefined as〈g∗, g〉 = Tr (gˆ∗ · gˆ) (2.4.1)for gˆ and gˆ∗ being the matrix representations of g and g∗, respectively, and we simply find thedual algebra of g is just itself g = g∗, which is true for simple compact Lie groups. Therefore,the coadjoint representation is isomorphic to the adjoint representation in SU(2), and the coadjointorbit of SU(2) through gc ∈ su(2) isΩgc = {g˜c = GigcG−1i |∀Gi ∈ SU(2)}. (2.4.2)18The stability group stab(gc) is simply U(1), so Ωgc is isomorphic to the coset space SU(2)/U(1) ≈CP1, which is exactly a sphere in the stereographic projection. Thus, a sphere is quantized properlyby the unitary irreducible representations of su(2). Explicitly, we haveXˆi = αJˆi, Jˆi=1,2,3 ∈ su(2) spin J irrep (2.4.3)for the fuzzy sphere with radiusR = α√J(J + 1), (2.4.4)which is based on the Casimir invariant∑Xˆi = R2.2.4.2 Spherical D0-D2 bound stateIn the context of the BFSS matrix model reviewed in Section 2.2, the fuzzy sphere can be either astatic solution of the equations of motion with a specific background field or an oscillating solutionwithout any background field. These two views are elaborated in the following.In the presence of a background Ramond-Ramond field, the non-abelian Wess-Zumino (WZ)action of equation (2.2.3) introduces a stable distribution of N D-branes [16]. Comparing to thedielectric effect, which describes that the external field induces charges redistribution in the polarizedneutral material, the fuzzy sphere appears as N D0-branes distributed spherically under a constantbackground RR field F (4). These N D0-branes resemble a D2-brane associated with F (4) just likethe polarized material is turned into an electric dipole or other multiples in the dielectric effect.To construct the fuzzy sphere as N D0-branes, the constant background RR field is given asF(4)tijk = −2fijk, (2.4.5)where i, j, k = 1, 2, 3, and other forms of RR fields F (n) are zero. By considering the leading orderterms expanded from the non-abelian Born-Infeld and Wess-Zumino action, we have the scalarpotentialV (Φ) = −Λ2T04Tr([Φi,Φj ]2)− i3Λ2µ0Tr(ΦiΦjΦk)F(4)tijk. (2.4.6)The variation equation δV (Φ)/δΦi = 0 gives[[Φi,Φj ],Φj]+ ifijk[Φj ,Φk] = 0, (2.4.7)and a solution of equation (2.4.7) is the fuzzy sphere,Φi = fJˆ i, where Jˆ i ∈ su(2) spin J irrep. (2.4.8)Comparing to the trivial solution of commutative Φi, which represents separated D0-branes, thefuzzy sphere is favored because of the lower energy in the potential of equation (2.4.6).19In the largeN limit, the fuzzy sphere converges toN D0-branes bounded on a sphericalD2-brane,and these N D0-branes dissolve into a U(1) gauge field,Fθφ =N2sin θ, (2.4.9)living on the D2-brane, where (θ, φ) denotes the spherical coordinate. The continuous limit isdescribed by the abelian Born-Infeld action of the spherical D2-brane with the U(1) field Fθφ andthe WZ term with the background RR field F (4)trθφ, and this agrees with the non-abelian BI actionof Φi plus the WZ term with the same RR field F(4)trθφ up to a 1/N2 correction.On the other hand, without the presence of background fields, the variation of the bosonic partin SBFSS in the light-front gauge leads to the equations of motionX¨i +[Xj , [Xj , Xi]]= 0 (2.4.10)and the Gauss constraint [X˙i, Xi]= 0. (2.4.11)It was pointed out in reference [74] that the fuzzy sphere is a time-dependent solutionXi = R(t)Jˆi (2.4.12)with oscillating R(t) being the solution of the differential equationR¨(t) + 2R(t)3 = 0. (2.4.13)The general solution, which can be found in reference [75], is written inversely ast = ±∫dR√C1 − R42+ C2. (2.4.14)The physical picture of R(t) can be pictured as pendulum motion. Consider the trajectory(x(t), y(t)) on the ellipsex2 +y22= 1 (2.4.15)traced by the angular arc length t. One special solution of equation (2.4.13) is written asR(t) = |cn(t)| = |x|√x2 + y2, (2.4.16)known as the Jacobi elliptic function.202.4.3 One-loop stabilization in IKKTWith proper coefficients, the fuzzy sphere and its higher dimensional generalizations can be consid-ered as solutions to the classical equation of motion in the infrared regulated IKKT model [21,76],e.g., S4so(6) in R1,4 is studied as an expanding Universe with Minkowski metric in reference [26].For the D-dimensional background with Minkowski metric ηµν , the variation of the action(2.2.20)δS˜b/δ(Xω)ij gives the equations of motionηµν [Xµ, [Xν , Xω]]− ρ(ω)Xω = 0 (2.4.17)in which the Lagrange multipliers ρ(0) = λ˜ and ρ(i = 1, . . . , D − 1) = λ respect the SO(1,D-1)symmetry. For matrices Yµ respecting the algebra[Yµ, [Yµ, Yν ]] = Yν , (2.4.18)equation (2.4.17) can be solved byX0 =(D − 2D − 1 λ˜− λ)1/2Y0 and Xi =(λ˜D − 1)1/2Yi. (2.4.19)The fuzzy sphere Yi = Jˆi satisfies the algebra of equation (2.4.18) at D = 3. In higher dimensionsD > 3, the algebra (2.4.18) is satisfied by the generalized sphere SD−1so(D+1) with collective coordinatesYa=1,...,D = Σa D+1 (2.4.20)from the generators Σ in the so(D+1) algebrai[Σab,Σcd] = δadΣbc + δbcΣad − δacΣbd − δbdΣac. (2.4.21)More details about the generalized sphere SD−1so(D+1) will be given in Chapter 5.We can also view S˜b at (2.2.9) as the dimensional reduced SU(N) Yang-Mills model with anegative mass term −ρ2XµXµ. The negative mass implies an instability, and negative eigenvaluesappear consequently in the second order variation:δ2S˜bδXjδXj= − (X0 ⊗ 1N − 1N ⊗X0)2 +∑i 6=j (Xi ⊗ 1N − 1N ⊗Xi)2 − λ1N2 ;δ2S˜bδX0δX0= −∑i (Xi ⊗ 1N − 1N ⊗Xi)2 + λ˜1N2 . (2.4.22)The quantity δ2S˜bδXjδXjhas negative eigenvalues when λ > 0. Hence, the stable Hermitian solution inthe Minkowski background strictly requires the Lagrange multipliers λ < 0 and λ˜ > 0.If one considers a purely spatial fuzzy sphere by embedding it in the Euclidean background, i.e.,ηµν → δµν in equation (2.4.17), one has X0 =(λ− D−2D−1 λ˜)1/2Y0 instead. Because Hermitian X021requires λ > 0, the fuzzy sphere is unstable in the Euclidean background. Although the fuzzy sphereis not a classical solution with positive mass, i.e., λ < 0, it can still be stabilized via quantum effects.Reference [77] computed the IKKT model (2.2.9) with a positive mass term for infrared regulationand found that by taking the vacuum expectation value as the fuzzy sphere with a critical radius,the one-loop effective action has a local minimum because of the soft (low energy) supersymmetricbreaking effect. The SUSY breaking suppresses the bosonic fluctuations and leaves the fermionicfluctuations unaffected, and the suppressed bosonic strings, which bound the D0-branes, lead to theexpansion of the sphere until the SUSY breaking diminishes at larger radius. Eventually the fuzzysphere with quantum fluctuations is stabilized at a physical radius, which depends on the givennegative value of λ.2.4.4 Point probesHere, we demonstrate two types of point probes on the fuzzy sphere. As proposed in Chapter 3, thecoherent state and the probe reveals the local geometry on the emergent objects. For the simplesu(2) structure of S2so(3), the coherent states over the sphere are derived easily, and we have thepicture of the fuzzy sphere partitioned into N non-commutative cells by the symbol map. Thecoherent natures of narrow overlapping and over-completeness assure the symbol map converges tothe classic picture in the large N limit.The fuzzy sphere (2.4.3) applied in the Dirac-type emergent operatorE3(~x) =∑i=1,2,3σi ⊗(αJˆi − xi),where Jˆi ∈ su(2) spin J irrep, (2.4.23)has the zero state at the North pole as the highest weight state:|ΛN 〉 = |1/2, 1/2〉 ⊗ |J, J〉 ≡ |lh〉 ⊗ |λh〉 . (2.4.24)Here, E3(x3 = R)|ΛN 〉 = 0 determines the radius of the sphere asR = αJ. (2.4.25)Comparing to the estimation from the Casimir invariant√∑Xˆi = α√J(J + 1), there is a 1/Jdiscrepancy.Having a coherent state identified at one point, the coherent state at any other point can be ob-tained by spherical symmetry. In the following, we discuss the general condition of SO(d) symmetryin the embedding space Rd as seen by the point probes. For any point probe operator, either anembedding operator or Laplace operator, given a known coherent state |Λ0〉 at x0 as an eigenstateof the point probe operator Ed(~x0) |Λ0〉 = EG |Λ0〉, a (d−1)-dimensional sphere obeys the following22condition for ~x ∈ Sd−1. There must exist a similarity transformation UˆSO(d) such thatEd(~x = Rˆ · ~x0) = Uˆ−1SO(d)Ed(~x0)UˆSO(d), (2.4.26)whereRji = exp(Φji)= δji + Φji +12ΦjkΦki +13!ΦjkΦklΦli + . . . (2.4.27)is the d-dimensional rotation for xi = Rjix0j , and the corresponding coherent state is|Λx〉 = Uˆ−1SO(d) |Λ0〉 . (2.4.28)In the case of the embedding operator or the Laplace operator, the SO(d) symmetric conditionequation (2.4.26) is simplified to∃Vˆ such that RjiXˆj = Vˆ −1XˆiVˆ . (2.4.29)This condition is convenient when we examine general spherical objects that are less trivial thanSdso(d+2), e.g., the three sphere discussed in Chapter 6. For the embedding operator∑γi⊗(Xˆi − xi),UˆSO(d) can be rewritten asUˆSO(d) = Lˆ⊗ Vˆ , (2.4.30)whereLˆ = exp(i2ΦijSij)and Sij =14i[γi, γj] ∈ so(d). (2.4.31)For the Laplace operator L = ∑i (Xˆi − xi)2, UˆSO(d) can be rewritten asUˆSO(d) = Vˆ−1. (2.4.32)It is clear that the fuzzy sphere satisfies the spherical condition of equation (2.4.29). With thealgebra [Ji, Jj ] = iijkJk and the BCH formula, we finde−iθ ~J ·nˆJieiθ~J ·nˆ =∑j(cos θδij + sin θijknk + (1− cos θ)ninj) Jˆj ≡ RjiJˆj . (2.4.33)Without lost of generality, rotating from the North pole zero state at ~xN , we can read the arbitraryzero state at xi = RjixNj as|Λ(θ, ~n)〉 = exp(iθ~n · ~σ2)⊗ exp(iθ~n · ~J)|lh〉 ⊗ |λh〉 . (2.4.34)|Λ(θ, ~n)〉 has the same form as the Bloch coherent states [78] studied in quantum optics. Given nˆ =(sinφ,− cosφ, 0) for ~x = (cosφ sin θ, sinφ sin θ, cos θ), we may expand equation (2.4.34) explicitly23as|Λ(θ, ~n)〉 =(cos θ2 |λ(θ, φ)〉eiφ sin θ2 |λ(θ, φ)〉), (2.4.35)where|λ(θ, φ)〉 =J∑m=−J(2JJ +m)1/2(cosθ2)J+m(sinθ2)J−mei(J−m)φ |Jm〉 . (2.4.36)This form of |λ(θ, φ)〉 introduces two convenient properties of the Bloch coherent state, the narrowoverlap,| 〈λ1|λ2〉 | =(cosξ2)2Jfor ξ = cos−1 ~n1 · ~n2, (2.4.37)and the completeness,12J+1 =2J + 14pi∫dΩ|λ〉〈λ|. (2.4.38)The non-commutativity (2.3.13) at the North pole isΘ = 〈ΛN |∑i1⊗ (Xi − xi)2 |ΛN 〉 = α2J, (2.4.39)and it is the same over the fuzzy sphere by symmetry. For the fuzzy sphere with a fixed radius R,it converges to the classical commutative sphere, while Θ = R2/J converges to zero in the largeN limit. When the coherent state splits as the direct product |Λ〉 = |l〉 ⊗ |λ〉, the expectation ofSij ∈ so(d) in equation (2.3.15) gives us the vector normal to the sphere:〈Λ|Sij ⊗ 1N |Λ〉 = 〈l|Sij |l〉. (2.4.40)On the fuzzy sphere, Sij = ijkσk/2 leads to a vector normal to the sphere〈l|Sij |l〉 = 12Rijkxk. (2.4.41)Thus, the non-commutativity Θ is in fact the projection ofθij =〈λ|1i[Xi, Xj ] |λ〉(2.4.42)on the tangent plane, and θij reflects a non-commutative cell of the quantized sphere by analogywith a quanta ~ in quantum mechanics. Taking the flat limit of the fuzzy sphere at the North pole,we find the non-commutativity θ =〈λ|1i [X,Y ] |λ〉is the same as the one computed on the Moyalplane tangent to the sphere.Th Laplace operator at the North pole of the fuzzy sphere is diagonalized in the spin J repre-24sentation:L(~x = (0, 0, R)) =∑i(Xˆi − xi)2= α2Jˆ2 − 2αRJˆ3 +R2. (2.4.43)Every eigenstate |J,m〉 of L(~x = (0, 0, R)) has the eigenvalue, i.e. displacement energy,Em = α2J(J + 1) +R2 − 2αRm= (R− αm)2 − α2m2 + α2J(J + 1), (2.4.44)which is minimized at R = αm and m = J . So we obtain the same radius αJ and the same coherentstate |J, J〉, just like the embedding operator. By rotational symmetry, we see that the arbitrarycoherent state on the sphere is the Bloch coherent state |λ(θ, φ)〉 in equation (2.4.36), and it projectsidentical results comparing to the embedding operator.2.4.5 Fields on the sphereIn the brane-world of matrix configurations, the matter of attached string modes is the set ofperturbations of collective coordinates. Parallel to the Weyl quantization on the non-commutativeplane in Section 2.1, here we introduce the fuzzy spherical harmonics Yˆlm for expanding the fields onthe fuzzy sphere. The fuzzy spherical harmonic expansion is also applied on the arbitrary distortedsphere in Chapter 3.The regular spherical harmonics are polynomials of coordinates in R3, and the fuzzy sphericalharmonics are polynomials of non-commutative coordinates Xi with the same coefficients asYˆlm = R−l∑cf (lm)c1,...,clXc1 . . . Xcl . (2.4.45)With the same coefficient F (lm) in the regular spherical harmonics expansion of the F (θ, φ) on thesphere, the operator Fˆ on the N truncated fuzzy sphere is written asFˆ =N∑l=0l∑m=−lF (lm)Yˆlm, where F (lm) =∫dΩF (θ, φ)Y ∗lm(θ, φ). (2.4.46)The local value of Fˆ on the fuzzy sphere is taken to be the expectation value 〈λ(θ, φ)|Fˆ |λ(θ, φ)〉in the symbol map [38]. The expectation value 〈λ(θ, φ)|Yˆlm|λ(θ, φ)〉, converging to Ylm(θ, φ) in thelarge N limit, guarantees the convergence limN→∞〈λ(θ, φ)|Fˆ |λ(θ, φ)〉 = F (θ, φ) if the order of F (θ, φ)does not exceed N .The fuzzy spherical harmonics have the orthogonality condition1NTr(Yˆ †l′m′ Yˆlm)= δll′δmm′ . (2.4.47)in analogy to∫dΩY ∗l′m′Ylm = δll′δmm′ . With this orthogonality, another way of retrieving the field25F (θ, φ) on the fuzzy sphere isFN (θ, φ) =1NN∑l=0l∑m=−lTr(Yˆ †lmFˆ)Ylm(θ, φ), (2.4.48)which agrees with the symbol map up to 1/N corrections:FN (θ, φ) = 〈λ(θ, φ)|Fˆ |λ(θ, φ)〉+O( 1N). (2.4.49)Consequently, there are two ways to construct the star product,F1 ? F2(θ, φ) =1NN∑l=0l∑m=−lTr(Yˆ †lmFˆ1Fˆ2)Ylm(θ, φ) (2.4.50)andF1 ? F2(θ, φ) = 〈λ(θ, φ)|Fˆ1Fˆ2|λ(θ, φ)〉. (2.4.51)For the su(2) algebra of the fuzzy sphere, the non-commutative derivative of Fˆ can be definedparallel to the adjoint action Ad(·) of the rotation generators on the regular sphere,Ad(Li)Fˆ ≡ [Li, Fˆ ]→ 1iijkxj∂kF. (2.4.52)Just like how rotation generators ~L = −i~x×∇ act on the spherical harmonics, the fuzzy sphericalharmonics under the adjoint action satisfy the same relationships:Ad(L)2Yˆlm = l(l + 1)Yˆlm; (2.4.53)Ad(L±)Yˆlm =√(l ∓m)(l ±m+ 1)Yˆlm±1; (2.4.54)Ad(L3)Yˆlm = mYˆlm. (2.4.55)In this chapter, we introduce the language of non-commutative geometry and the physics back-ground of D-branes in order to interpret spacetime as non-commutative world-volumes of D-branes.In Section 2.1, we introduce the basic non-commutative geometry deriving from quantization. InSection 2.2, to realize the spacetime in string theory in terms of matrices, we introduce the matrixmodels in string theory, where their solutions of the equations of motion describe the groups ofD-branes with matrices. In Section 2.3, we describe the method of point probes. The spectrum ofthe string stretched from the probe to the emergent surface reveals the length of the string. Theprobing method is our main tool for investigating emergent geometry from a set of matrices. InSection 2.4, we flesh out the ideas described above by examining a robust example of the fuzzysphere.26Chapter 3Continuous limit of the emergentmembranes3.1 IntroductionString theory contains many hints that spacetime might be a more complicated object—possiblyeven an emergent one—than a manifold. Most of our understanding about non-perturbative stringtheory comes from the study of D-branes, extended objects that strings are allowed to end on. WhenN identical D-branes are considered, their coordinate positions are described by N ×N hermitianmatrices. If these matrix coordinates are simultaneously diagonalizable, their eigenvalues are easilyinterpreted as the positions of the D-branes. When they are not, as is the situation generically,the D-brane positions are not well defined, even in the classical ~ → 0 limit. Thus, D-branesdo not ‘view’ spacetime in the same way that ordinary point particles do. The standard stringtheoretic interpretation of such ‘fuzzy’ configurations is through the so-called dielectric effect [16],where lower dimensional D-branes ‘blow up’ to form higher dimensional D-brane. Lack of localityis related to the lower dimensional D-branes being ‘smeared’ over the worldvolume of a higherdimensional emergent object.In most previous work, explicit geometric interpretation of the matrix coordinates as a higherdimensional object has been limited to simple and highly symmetric geometries, such as planes,tori and spheres.1 In their paper, reference [22], take this one step further: using the BFSS modelthey found a geometric interpretation of three matrix coordinates as a codimension-one surfaceembedded in three dimensions. Their argument, explained in Chapter 1 and generalized in Chapter4, led to the following effective Hamiltonian:Heff(xi) =∑i=1,2,3σi ⊗ (Xi − xi) , (3.1.1)1One example of an attempt in a more general setup is reference [79], where a matrix configuration correspondingto a given surface was constructed using string boundary states, if zero energy states of a certain Hamiltonian arisingfrom the boundary action can be found.27where Xi for i = 1, 2, 3 are Hermitian, N×N , matrices corresponding to the positions of the stack ofD0-branes in a three dimensional flat transverse space, and xi are the positions of the probe brane.The fermionic mode is massless when Heff has a zero eigenvalue. Thus, the surface correspondingto the three matrices Xi is given by the polynomial equation det(Heff(xi)) = 0. This defines aco-dimension one surface in flat R3 space parametrized by (x1, x2, x3).We use equation (3.1.1) as the starting point for a concrete and explicit study of geometry ofthe emergent surface, identifying zero eigenvectors of Heff with coherent states underlying non-commutative geometry of the emergent surface [38]. We focus on configurations where a smoothand well-defined surface arises from matrices with a large size N . Rather than assume it a priori, weprove a correspondence principle between matrix commutators and a unique Poisson bracket on theemergent surface arising from the matrix configuration (X1, X2, X3). This explicit correspondencemakes the usual procedure of going from matrix models to surfaces much less ad hoc, and might beof use when quantizing membrane actions by replacing them with a matrix model. We demostratehow easy it is to construct surfaces with desired properties using our approach on several non-trivialexamples, including the torus.Our approach is most similar to that espoused in reference [80] (see also reference [70] andreferences therein), but with an explicit construction for the coherent states associated with pointson the surface. The results can also be thought of as a concrete realization of the abstract idea inthe classic work by Kontsevich, reference [30]. Other work includes references [81, 82], though ourconstruction appears more general as it allows us to vary the local non-commutativity independentof the shape of the surface.For most of this chapter, we focus on the following questions: under what conditions would asequence of non-commutative geometries, each arising from a matrix configuration (X1, X2, X3) andlabeled by an increasing matrix size N , converge to a smooth limit? Which quantities characterizethe surface in this limit?Since the polynomial equation det(Heff(xi)) = 0 has degree 2N , generically, the locus of itssolutions does not need to be smooth in the large N limit. When some generic matrices Xi arescaled so that the range of their eigenvalue distributions remains finite at large N , the resultingsurface is generically quite complicated and does not have a large N limit. As a simple (but notgeneric) example, let Xi = diag(σi + a1i , . . . , σi + aNi ), where σi are the Pauli matrices and aki arereal numbers. The resulting surface is a union of N spheres of radius 1 each centered at (ak1, ak2, ak3)for k from 1 to N . There is no sense in which the surface achieves a well-defined large N limit. Inthe degenerate case where all aki are zero, the surface is a single sphere of radius one centered at theorigin. However, it still does not correspond to a smooth geometry, rather, it is a very fuzzy spherewith SU(N) symmetry. To obtain a smooth geometry, we can instead consider Xi = Li/J , with Liforming the irreducible representation of SU(2) with spin J (this is the standard construction of thenon-commutative sphere, see section 3.3.2 for details). This sphere has radius 1 independent of J .As N = 2J + 1→∞, the non-commutative sphere reproduces the ordinary one.28When the the large N limit exists and is smooth, the emergent surface will be characterized byits geometry (the embedding into flat R3 space) and by a Poisson structure defining (together withN) a non-commutative geometry in the large N limit. In section 3.2, we will make some definitionsand introduce our approach. In section 3.3, we will analyze, analytically and numerically, a series ofexamples from which a general picture will emerge. In section 3.4 we will prove the correspondenceprinciple and discuss smoothness conditions which determine how large N has to be for a givennon-commutative surface to be well described by the corresponding matrices. In section 3.5, we willdiscuss the issue of area and derive the matrix equation for minimal area surfaces. In section 3.6,we construct a smooth torus embedded in R3. Finally, in section 3.7 we discuss topics for futurework.3.2 Basic setupSince our emergent surface is given by the locus of points where the effective Hamiltonian Heffin equation (3.1.1) has a zero eigenvalue, for each point p on the surface Heff has a (properlynormalized) zero eigenvector |Λp〉:Heff|Λp〉 = 0 . (3.2.1)The above equation defines (in non-degenerate cases) a two dimensional surface embedded in threedimensional space. We will take the three dimensional space to be flat; the metric on the emergenttwo dimensional surface will then just be the pullback from the flat three dimensional metric.It is instructive to rewrite the above equation in a slightly different way:∑i=1,2,3(σi ⊗Xi) |Λp〉 = ∑i=1,2,3(σi ⊗ xi) |Λp〉 . (3.2.2)This equation can be thought of as an analogue of an eigenvalue equation: while the three matricesXi cannot be simultaneously diagonalized, the above equation says that if we double the dimen-sionality of the space under consideration, there are special vectors |Λp〉 on which the action of Xiis described by only three parameters. In analogy with the Berezin approach to non-commutativegeometry [38], we would like to think of these states as coherent states corresponding to points onthe non-commutative surface as mentioned in Chapter 1.2 In the Berezin approach, every pointp is associated with a coherent state |αp〉. One then defines a map from any Aˆ to a function onthe non-commutative surface via s(Aˆ) = 〈αp|Aˆ|αp〉. This function is usually called the "symbolmap". From it one can find the corresponding star-product and the rest of the usual machinery ofnon-commutative geometry.The first difficulty we see with Λp being the coherent state is that our operators Xi (and theirfunctions) cannot be seen as acting on |Λp〉, due to dimension mismatch. We can simply ‘double’2A somewhat similar approach but with a different effective Hamiltonian, and applicable only in the infinite Nlimit, was recently made in reference [23].29these operators by using 12⊗Xi instead (1k will denote the k× k identity matrix). However, whileit is true that〈Λy| 12 ⊗Xi |Λp〉 = xi(σ) , (3.2.3)this approach is somewhat artificial. We will see that there is a more natural solution: for largeN , when the emergent non-commutative surface is smooth in the sense discussed in this chapter’sIntroduction, the eigenvector |Λp〉 is approximately a product, |Λp〉 = |a〉 ⊗ |αp〉, where |αp〉 isN -dimensional and |a〉 is 2-dimensional. In the next section, we will examine examples in which thezero eigenvectors of Heff do factorize in this manner when N is large. A way to measure the extentof the factorization is to write any (2N)-dimensional vector as|Λp〉 =[|α1〉|α2〉], (3.2.4)with ||α1||2 + ||α2||2 = 1, and to defineAp =√||α1||2||α2||2 − |〈α1|α2〉|2 , (3.2.5)which can be thought of as the area of the parallelogram defined by the two vectors |α1〉 and |α2〉.We will be arguing that, in the large N limit, Ap is of order N−1/2, implying that |α1〉 and |α2〉 areindeed approximately parallel and we can write|Λp〉 =[a|αp〉b|αp〉]+O(1/√N) . (3.2.6)(By O(1/N−1/2) we mean that the norm of the correction vector decreases with increasing N like1/N−1/2.) It will then be the N -dimensional vector |αp〉 that will play the role of a coherent statecorresponding to point p.The complex coefficients (a, b) of the 2-vector |a〉 determine the direction of the normal vector n atpoint p given by (x1, x2, x3). To see this, consider moving p slightly to (x1 +dx1, x2 +dx2, x2 +dx3),where (dx1, dx2, dx3) is an infinitesimal tangent to the surface. First-order perturbation theoryimplies that to maintain the condition that Heff has a zero eigenvalue, we must have 〈Λp|dHeff|Λp〉 =〈Λp|σi ⊗ (−dxi)|Λp〉 = 0. Thus dxi 〈Λp|σi ⊗ 1N |Λp〉 = 0, implying thatni := 〈Λp| σi ⊗ 1N |Λp〉 (3.2.7)is a vector normal to the surface at a point p. This is an exact statement and does not rely on ourfactorization assumption. Incidentally, we have the formula |n|2 = 1− 4A2p, so the normal vector isclose to being a unit normal when the factorization condition holds. When we use equation (3.2.6),we obtain that the normal vector is (a¯b+ ab¯, i(ab¯− a¯b), a¯a− b¯b). Thus, the coefficients (a, b) fix the30direction of the normal vector. Conversely, the normal vector fixes the coefficients (a, b) up to anoverall irrelevant phase.Next, we will try to define local non-commutativity given in Chapter 1 on the surface. The localnon-commutativity can be thought of in two different ways: the size of ‘fuzziness’ (or uncertainty) ofthe operatorsXi in the state |Λp〉, or the size of the commutators of theXis when acting on |Λp〉. In acoherent state, these two notions should be equal, and they turn out to be equal here, strengtheningour case that |Λp〉 can be thought of as a coherent state. Using σiσj = iijkσk = −σjσi for i 6= jand σ2i = 1, we have a nice little identity(Heff)2 = 12 ⊗∑i(Xi − xi)2 + 12iijkσi ⊗ [Xj , Xk] . (3.2.8)Then, since 〈Λp|(Heff)2|Λp〉 = 0, we have〈Λp|12 ⊗∑i(Xi − xi)2|Λp〉 = −12iijk 〈Λp|σi ⊗ [Xj , Xk]|Λp〉 . (3.2.9)When the vector |Λ〉 is indeed a product, we can use equation (3.2.6) to make the following definition:the local non-commutativity on the non-commutative surface isθ = 〈αp|∑i(Xi − xi)2|αp〉 = 12ijk θij nk , (3.2.10)whereθij := 〈αp| − i[Xi, Xj ]|αp〉 . (3.2.11)The LHS of expression (3.2.10) is a sum of squares of uncertainties in the operators Xi, whilethe RHS depends on the commutators. The particular combination of commutators is of interest:with our factorization assumption, the commutator term picks up only the contributions that aretransverse to the normal, for example, if the normal vector n is pointing in the x3 direction, only[X1, X2] contribute to θ. In fact, it will turn out that, in the large N limit, ijkθij is nearly parallelto nk. Thus, we can also write θ asθ = 〈αp|√∑i 6=j−[Xi, Xj ]2 |αp〉 . (3.2.12)As for the first expression in equation (3.2.10), it will turn out that if we take the normal vectorto point along the x3 direction, we have 〈α|(X1 − x1)2|α〉 ≈ 〈α|(X2 − x2)2|α〉  〈α|(X3 − x3)2|α〉,so the coherent state is ‘flattened’ to lie predominantly in the 1-2-plane and balanced (‘round’).To flesh out these ideas, we will examine a series of increasingly complex examples. In theprocess, we will construct the approximate eigenvector |αp〉 and study corrections to the large Nlimit described above.313.3 Coherent state and its propertiesWe will make the following choice for the Pauli matrices σiσ1 =[0 11 0], σ2 =[0 −ii 0], σ3 =[1 00 −1]. (3.3.1)In this convention, we can write Heff in a natural way in terms of N ×N blocksHeff =[X3 − x3 (X1 − iX2)− (x1 − ix2)(X1 + iX2)− (x1 + ix2) −(X3 − x3)], (3.3.2)We will now examine a series of examples of increasing complexity, always focusing on a pointwhere the normal vector to the surface is pointing straight up (in the x3 direction).Our final conclusion will be that at such a point, the zero-eigenvector of Heff has the form givenin equation (3.2.6):|Λ〉 =[|α〉0]+ O(N−1/2). (3.3.3)|α〉 with 〈α|α〉 = 1 will be the coherent state associated with this particular point on the surface,−i〈α|[X1, X2]|α〉 will correspond to the local value of non-commutativity at this point. This resultis easily generalizable to any orientation of the surface using an SU(2) rotation of the Pauli matrices.3.3.1 Example: non-commutative planeConsider the example of a non-commutative plane: let X3 = 0, and let [X1, X2] = iθ. Out ofnecessity, X1 and X2 are infinite dimensional operators. This will not be the case when we areconsidering compact non-commutative surfaces. We haveHeff =[−x3 A† − α¯A− α x3], (3.3.4)where A = X1 + iX2, A and A† are the lowering and raising operators of a harmonic oscillator with[A,A†] = 2θ, and α = x1 + ix2. The lowering operator A has eigenstates |α〉, called the coherentstates, corresponding to every complex number α: A|α〉 = α|α〉. We thus have a zero eigenvectorfor Heff with x3 = 0:|Λ(α)〉 =[|α〉0]. (3.3.5)The non-commutative plane is flat and has constant non-commutativity. The normal vector is〈Λ|σi ⊗ 1|Λ〉 = (0, 0, 1) and we have −i〈α[X1, X2]α〉 = θ.The importance of this example is that, locally and in the large N limit, any non-commutativesurface should look like the non-commutative plane. This is the observation that will allow us to32write our definition of a large N (smooth) limit.3.3.2 Example: non-commutative sphereHere we have Xi = Li/J where Li form the N -dimensional irrep of SU(2): [Li, Lj ] = iijkLk andwhere J = (N − 1)/2 is the spin. It is useful to consider the usual raising and lowering operators,L± = L1 ± iL2. Without loss of generality, consider that point on the non-commutative surfacewhich lies on the x3 axis. With x1 = x2 = 0, Heff isHeff =[L3/J − x3 L−/JL+/J −(L3/J − x3)]. (3.3.6)We will use as a basis the eigenvectors of the L3 angular momentum, |m〉:L3|m〉 = m|m〉 , m = −J . . . J , 〈m|m〉 = 1 , J = N − 12. (3.3.7)It is easy to see that|Λ〉 =[|J〉0](3.3.8)is a zero eigenvector of Heff if x3 = 1. Thus, the non-commutative sphere has radius 1.3 The otherzero eigenvectors on the sphere can be written in Bloch states as introduced in Chapter Looking ahead: polynomial maps from the sphereA large class of surfaces that can be studied using our tools are surfaces that are generated frompolynomials of the normalized SU(2) generators considered above:Xi = polynomial(L1/J, L2/J, L3/J) , (3.3.9)where the polynomials in three variables have degrees and coefficients that are independent of N .In this case, we expect that at large N the non-commutative surface will approach an algebraicvariety given by the image of the unit sphere under the polynomial maps used to construct Xi.Concretely, consider a surface S in R3 constructed as follows: let p1, p2 and p3 be three poly-nomials discussed, in three variables w1, w2 and w3. Then, consider the image in R3 under thesethree polynomial maps of the surface∑i(wi)2 = 1, ieS ={(x1, x2, x3) | xi = pi(w1, w2, w3) and∑i(wi)2 = 1}. (3.3.10)3This is a different definition of the radius of the non-commutative sphere than the usual one, which is based onthe quadratic Casimir of the SU(2) irrep, and which gives the radius to be√J(J + 1)/J =√N+1N−1 .33We will restrict our considerations to surfaces which are non-self-intersecting, meaning that thepolynomial map is one-to-one. The corresponding non-commutative surface is specified by threeN ×N matrices Xi which can be written as corresponding polynomial expressions in Li:Xi = sym (pi(L1/J, L2/J, L3/J)) , (3.3.11)where, to avoid ambiguity, the ‘sym’ map completely symmetrizes any products of the three non-commuting matrices Li. This symmetrization will turn out to play little role in what follows:re-ordering the terms of order k leads to small—suppressed by a power of J—corrections in thecoefficients of the polynomials of order less than k.Now, consider an arbitrary point p = (y1, y2, y3) on the surface S. Acting with SO(3) on thespace (x1, x2, x3), arrange for the normal vector to S at the point p to point along the positivex3-direction, and acting with SO(3) on the space (w1, w2, w3), arrange for the pre-image of thepoint p to be the north pole. It is then necessary that the polynomial maps take a formx1 = y1 + c1w1 + c2w2 + a(w3 − 1) + p(2)1 (w1, w2, w3 − 1) ,x2 = y2 + c3w1 + c4w2 + b(w3 − 1) + p(2)2 (w1, w2, w3 − 1) , (3.3.12)x3 = y3 + c(w3 − 1) + p(2)3 (w1, w2, w3 − 1) ,where ci, a, b and c are real numbers and where p(2)i (·) are polynomials of degree at least 2. Toavoid a coordinate singularity, we should have c1c4 − c2c3 6= 0. Then, using a rotation of w1 andw2 (in other words, rotating the unit sphere around the north pole), we can set c3 zero and c4 > 0.Finally we can take c1 > 0 by adjusting the sign of w1 if necessary.The four coefficients c1, . . . , c4 determine the metric on the surface in terms of the metric on thesphere. If the metric on the sphere is gS2 , then the induced metric on the surface isgab :=(CT gS2C)ab, where C =[c1 c2c3 c4]. (3.3.13)This implies that√det g/√det gS2 = detC, which is a useful fact to keep in mind.Without loss of generality, we are interested in the eigenvector of Heff at a point such that thenormal to the surface is pointing along the 3-direction. We now want to show that the correspondingzero-eigenvector of Heff has the form shown in equation (3.3.3).Before we plunge into analyzing this rather general setup, we will narrow the example down toa simpler one which nonetheless contains most of the salient features of our general approach.3.3.4 Example: non-commutative ellipsoidHere, we will consider a stretched non-commutative sphere. The most generic closed quadraticsurface in three dimensions is an ellipsoid, with three orthogonal major axes positioned at some34arbitrary position in the three dimensional space under consideration. In other words, we will allowXi to be arbitrary linear combinations of L1/J , L2/J and L3/J . Under the general frameworkdescribed above, this amounts to setting the higher degree polynomials p(2)i to zero:Xi = AijLj/J, where A =c1 c2 a0 c4 b0 0 c . (3.3.14)The classical, or infinite N , surface is given by xi = Aijwj with∑i(wi)2 = 1. It is easy to checkthat at a point x = (a, b, c), this surface has a normal vector which is pointing along the positivex3-direction. We will therefore consider finding the exact location of the surface at a point withx = (a, b, x3) where we expect x3 to be close to c. We haveHeff(x3) =[cL3J − x3 A† + (a− ib)(L3/J − 1)A + (a+ ib)(L3/J − 1) −(cL3J − x3) ] , (3.3.15)whereA =(c1 + c4)− ic22JL+ +(c1 − c4) + ic22JL− . (3.3.16)What we need to do is find a good approximation to the zero eigenvector of Heff(x3), togetherwith an estimate for the (hopefully small) difference x3 − c. We conjecture that such a vector isin some way similar to that in equation (3.3.8): the ‘top part’ is large compared with the ‘bottompart’ and is dominated by components with the largest eigenvalues of J3. To achieve this, writeHeff as a sum of two parts:Heff(x3) =[0 A†A 0]+[cL3J − x3 (a− ib)(L3J − 1)(a+ ib)(L3J − 1) − (cL3J − x3)]. (3.3.17)If we focus on vectors whose N -dimensional sub-vectors are dominated by components with largeL3 eigenvalues, then the first part can be thought of as being of order N−1/2 while the second partis of order N−1. Our attempt to find an approximate eigenvector of Heff(x3) will treat the secondpart as a small perturbation on the first part, suppressed by N−1/2.Consider now a vector—which we will show to be either a zero eigenvector of A or very close tosuch, and which will thus be an approximate zero-eigenvector of Heff(x3)—given by[|α〉0], (3.3.18)35where4|α〉 = 1√KbJc∑m=0ξm√√√√ m∏k=1(2k − 1)(2J − 2k + 2)(2k)(2J − 2k + 1) |J − 2m〉 , (3.3.19)with ξ is given byξ = −c1 − c4 + ic2c1 + c4 − ic2 . (3.3.20)The normalization constant, for which 〈α|α〉 = 1, can be computed in the large J limit asK =bJc∑m=0(|ξ|2mm∏k=1(2k − 1)(2J − 2k + 2)(2k)(2J − 2k + 1))(3.3.21)≈ 1 +∞∑m=1(|ξ|2m (2m− 1)!!(2m)!!)= 1 + 2∞∑m=1|ξ/2|2m (2m− 1)!m!(m− 1)! (3.3.22)= 1 +|ξ|21− |ξ|2 +√1− |ξ|2 = 1√1− |ξ|2 , (3.3.23)where it is important that |ξ| < 1, which can be seen from the explicit form in equation (3.3.20).For completeness, let us state that1− |ξ|2 = 4 detC‖C‖2 + 2 detC , (3.3.24)or1− |ξ|21 + |ξ|2 =2 detC‖C‖2 . (3.3.25)Writing |ξ| in terms of rotational invariants of the matrix C gives a clear geometric interpretationthis is quantity: it is a measure of how much the map in equation (3.3.14) distorts the aspect ratioat the point we are interested in.With a short calculation5 we see that A|α〉 = 0 for integer spin J , and that for half-integer spinJ , we haveA|α〉 = −c1 − c4 + ic22J√KξJ+1/2√√√√J−1/2∏k=1(2k − 1)(2J − 2k + 2)(2k)(2J − 2k + 1)√2J | − J〉 (3.3.27)= K−1/2(c1 − c4 + ic2) ξJ+1/2 (2J − 2)!!(2J − 1)!! | − J〉 . (3.3.28)4Some standard notation we will use: the ‘floor’ function, bxc = the largest integer not exceeding x; the doublefactorial, (2n)!! = (2n)(2n− 2) . . . (4)(2) and (2n− 1)!! = (2n− 1)(2n− 3) . . . (3)(1) for n a natural number.5Recall thatL−|k〉 =√(J − k + 1)(J + k) |k − 1〉 , L+|k〉 =√(J − k)(J + k + 1) |k + 1〉 . (3.3.26)36This is very small: the norm-squared of A|α〉 is bounded above byb(J) :=((c1 − c4)2 + (c2)2) |ξ|2J+1 . (3.3.29)Since |ξ| < 1, the above quantity goes to zero like exp(−(2 ln |ξ|)J) for large J . Further,(L3J− 1)|α〉 = − 1√KbJc∑m=02mJξm√√√√ m∏k=1(2k − 1)(2J − 2k + 2)(2k)(2J − 2k + 1) |J − 2m〉 (3.3.30)and the norm-squared of this vector is equal to1KbJc∑m=0(2mJ)2 (|ξ|2mm∏k=1(2k − 1)(2J − 2k + 2)(2k)(2J − 2k + 1)), (3.3.31)which is bounded above by6bJc∑m=0(2mJ)2 (|ξ|2m)< J−2∞∑m=0(2m)2(|ξ|2m):= u(J) . (3.3.35)Thus, the bound has the form u(J) = (function of ξ) · J−2.When Heff(x3 = c) acts on the normalized vector[|α〉0], the resulting vector’s norm is, in thelarge J limit, bounded by√(a2 + b2 + c2)u(J) + b(J), which is itself bounded by a constant timesJ−1. To summarize, ∥∥∥∥∥Heff(c)[|α〉0]∥∥∥∥∥ < C(ci)J , (3.3.36)where C(ci) does not depend on J and therefore on N .6We need to provide a bound onm∏k=1(2k − 1)(2J − 2k + 2)(2k)(2J − 2k + 1) (3.3.32)Consider, for m a positive integer less or equal than bJc,F (m) :=m∏k=1(2k − 1)(2J − 2k + 2)(2k)(2J − 2k + 1) =(2m− 1)!!(2J − 2m− 1)!!(2J)!!(2m)!!(2J − 2m)!!(2J − 1)!! . (3.3.33)F (1) = J2J−1 < 1 and F (bJc) can also be easily shown to be less than 1 (we need to consider two cases, with J integeror half-integer). Finally, we notice that F (m+ 1) < F (m) for m smaller than roughly J/2 and F (m+ 1) > F (m) form larger than than. This implies that F (m) has a minimum near J/2 and that for 1 < m < bJc it is less than thelarger of F (1) and F (bJc) which are both less than 1. Therefore,m∏k=1(2k − 1)(2J − 2k + 2)(2k)(2J − 2k + 1) < 1 . (3.3.34)37It follows that[|α〉0]is an approximate eigenvector of Heff(c) and we can place a bound on thecorresponding eigenvalue: there exists a vector Λ˜ such thatHeff(c) Λ˜ = Λ˜ , with || < C(ci)J. (3.3.37)One can ask the following question: is Λ˜ close to[|α〉0]? To answer this question, we examine theargument that guarantees the existence of Λ˜ as above: consider the length squared of Heff[|α〉0]as expanded in eigenvectors of Heff:HeffΛi = λiΛi ,[|α〉0]=2N∑i=1ciΛi ,∥∥∥∥∥Heff(c)[|α〉0]∥∥∥∥∥2=2N∑i=1|ci|2|λi|2 . (3.3.38)With the bound in equation (3.3.36), it is clear that at least one of the eigenvalues λi must beless than C(ci)/J . Further, if none of the other eigenvalues are small enough, then the eigenvectorcorresponding to the unique small eigenvalue (which we denoted with Λ˜) is very close to[|α〉0]itself. For example, if the next smallest eigenvalue λj of Heff is of order N−1/2 (as numerical studiessuggest), then the corresponding coefficient cj must be of order N−1/2 as well. Therefore, thedifference between Λ˜ and[|α〉0]has length of order N−1/2.Further, we would like to conclude that there exists a third vector Λ, such thatHeff(c− ζ) Λ = 0 , with |ζ| of order 1/J , (3.3.39)with Λ close to Λ˜ and therefore[|α〉0]. It is possible to argue for this in first-order perturbationtheory: as we deform x3 from c to c − ζ, the eigenvalue of interest changes from  (in equation(3.3.37)) to 0, while the eigenvector changes from Λ˜ to Λ. Since  is of order N−1, ζ should alsobe of order N−1. Making this analysis rigorous is difficult because, effectively, we are trying to doperturbation theory in 1/N while taking a large N limit. Since any sums we take would be over Ncomponents, these sums can easily overwhelm any 1/N suppression factors. For example, to showthat Λ is close to Λ˜, it is again necessary to bound the remaining spectrum of Heff(c) away fromzero. This is the same bound as was necessary above: the remaining eigenvalues must be boundedaway from zero by at least const/√N , which seems to be the case when examined numerically.Instead of attempting a rigorous proof, we will obtain some analytic estimates based on the as-sumption that the 1/N expansion is valid and then confirm these estimates with numerical analysis.Our idea will be to obtain an analytic result for the leading order contribution to x3 − c (which38will turn out to be of order 1/N as predicted above) and confirm its correctness by comparingwith numerical results. We will also confirm that our approximate eigenvector[|α〉0]is a goodapproximation to the exact zero eigenvector of Heff(x3). Crucial to this approach are two facts:that the eigenvector |α〉 has components which fall off exponentially with m, so that only thosecomponents with spin close to the maximum spin J are appreciable, and that the second term inequation (3.3.17) is small (of order 1/N) when acting on these components. Further analysis willthen reveal that when the first-order correction to the approximate eigenstate is included:[|α〉|β〉],the vector |β〉 also has components which fall off exponentially with m. We will interpret this as a‘quasi-locality’ feature of the non-commutative surface.Now, return to our way of writing Heff as a sum of two parts in equation (3.3.17). Our specialvector[|α〉0]is an approximate zero eigenvector of the first of these two operators (and an exactzero eigenvector for odd N). Thinking of the second term in equation (3.3.17) as a small perturbationin first-order perturbation theory, we obtain, to first-order, that the change in the eigenvalue is equalto[〈α| 0] [ cL3J − x3 (a− ib) (L3J − 1)(a+ ib)(L3J − 1) − (cL3J − x3)] [|α〉0](3.3.40)= 〈α|L3Jc− x3|α〉= 〈α|L3J− 1|α〉c + 〈α|α〉(c− x3)= − cKbJc∑m=02mJ(|ξ|2mm∏k=1(2k − 1)(2J − 2k + 2)(2k)(2J − 2k + 1))+ (c− x3)= −F (ξ, J)KJ+ (c− x3) .On the last line, we can make an approximation by adding an exponentially small ‘tail’ to the sum,so that the function F (ξ, J) will no longer depend on J , making c− x3 be of order J−1. Explicitly,we haveF (ξ, J) := cbJc∑m=02m(|ξ|2mm∏k=1(2k − 1)(2J − 2k + 2)(2k)(2J − 2k + 1))(3.3.41)≈ c∞∑m=12m(|ξ|2m (2m− 1)!!(2m)!!)= c|ξ| dKd|ξ| . (3.3.42)Taking the change in the eigenvalue to be zero, we get thatc− x3 = cJ−1ξ d(lnK)dξ= cJ−1|ξ|21− |ξ|2 = J−1c(c1 − c4)2 + c224c1c4. (3.3.43)39102 103 104J10-410-310-210-1c−x3Figure 3.1: Difference between x3 at finite N (obtained numerically) and c (its large N asymptotics),as a function of N . The line represents equation (3.3.43), which has no free parameters and appearsto be an excellent match to the numerical data. In this figure, (a, b, c) = (1.5, 0.5, 3), c1 = 2, c2 = 5and c4 = 4. For these values, equation (3.3.43) implies that c− x3 = 2.71875/J .We have tested the correctness of this formula numerically,7 as can be seen in figure 3.1.Further, we have checked that[|α〉0]is a good approximation to the exact eigenvector. As canbe seen in figure 3.2, the magnitude of the difference decreases as N−1/2.Once we understand |α〉, we can ask about the leading correction to the exact eigenvector of Heff.To next order, the eigenvector has a form[|α〉+ |∆α〉|β〉], with corrections |β〉 and |∆α〉 that havemagnitudes of order no larger than N−1/2. Because we are working at a point where the normalvector points ‘up’, we have 〈α|β〉 = 0. However, generically 〈∆α|β〉 6= 0, so the actual normal vectorwill show a small deviation from this assumed direction. Finally, Ap ≈√||β||2 − |〈∆α|β〉|2.It is difficult to obtain a closed-form formula for |β〉, and even harder to obtain one for |∆α〉.We should proceed by finding a complete eigenbasis for the first part of Heff as written in equation7To facilitate numerical study, it is best to rewrite equation (3.2.1) in as a genuine eigenvalue equation. Considerthe operator σ3Heff. We can rewrite equation (3.2.1) as (−iσ2 ⊗ (X1 − x1) + iσ1 ⊗ (X2 − x2) + 12 ⊗X3) |Λ〉 = x3|Λ〉.Therefore, to find x3 on the emergent surface at a given x1 and x2, all we have to do is to solve an eigenvalue problem.It is important that the operator being diagonalized is no longer hermitian: most (or possibly all) of its eigenvalues arecomplex. Real eigenvalues (if any) correspond to points on the emergent surface. Since the dimension of the operatoris even, there must be an even number of real eigenvalues in non-degenerate cases. This naturally corresponds tosuch points on the emergent surface coming in pairs for a closed surface.40102 103 104J10-210-1∆Figure 3.2: Magnitude, ∆, of the difference between the approximate eigenvector and the exacteigenvector as obtained numerically, for the ellipsoid in figure 3.1. The straight line, shown to guidethe eye, is a best fit to the last few points and corresponds to ∆ = 1.12√J.(3.3.17), and then use standard perturbation theory to obtain the desired result. This is beyondthe scope of this chapter, so we will resort to less complete methods to obtain some insight into thestructure.The formal expression for |β〉 is|β〉 = (A†)−1(cL3J− x3 + p(2)3)|α〉 . (3.3.44)This expression is formal because A† might not have an inverse when acting on the above operator.However, we notice that since we already know x3, we are able to find, to leading order in N , thefirst non-zero coefficient of |β〉 (which is the coefficient of |J − 1〉). To do so, we take our alreadycomputed value of x3 and solve this equation:A†|β〉 = −(cL3J− x3)|α〉 . (3.3.45)Once we have the first coefficient, we can substitute it back into the above equation and solve for thenext coefficient. Repeating this will in principle yield nearly all components of |β〉 (with exceptionof the component with the most negative L3 eigenvalue).41Explicitly, we obtain that the coefficient of |J − 1〉 in |β〉 isc− x3√K√2Jc1 − c4 − ic2 . (3.3.46)The magnitude squared of this expression isc22J1detC|ξ|2(1− |ξ|2)1/2 . (3.3.47)We need this expression to be small (compared to 1), since we would like ‖β‖  ‖α‖. Thus, fornon-zero |ξ|, how large J needs to be for our analysis to be applicable depends, for example, onc. Numerical study confirms equation (3.3.47); further, it shows that the ratio of the expression inequation (3.3.47) and the total magnitude squared of |β〉 goes to a constant value at large N . Thus,‖β‖2 is proportional to c2 and decreases with large J like J−1.We will see in section 3.4 that corrections shown in equations (3.3.43) and (3.3.47) are large whenN is too small to describe the portion of a given surface with a high curvature.At the same order, we also get a correction to |α〉, |∆α〉. A formal expression, similar to the onefor |β〉 above,|∆α〉 = A−1((a+ ib)(L3J− 1))|α〉 , (3.3.48)does not have a well defined meaning as((a+ ib)(L3J − 1)) |α〉 generically has a significant com-ponent parallel to |α〉. It is not possible to solve for coefficients of |∆α〉 in the same way that wesolved for those of |β〉; we need a complete perturbation theory treatment. However, using the aboveexpression as a guide to structure at least, we see that the correction |∆α〉 is of order O(N−1/2),and that it would grow with a and b. While the coefficient c determines the local curvature of thesurface, the coefficients a and b control how fast the non-commutativity is changing, as we will seein section 3.3.6.As we already mentioned, |∆α〉 is not necessarily orthogonal to |β〉, so we will now have acorrection to the angle of the normal vector,ni ≈ (2<〈∆α|β〉, 2=〈∆α|β〉, 1) . (3.3.49)Numerical work confirms that the angle between the expected normal vector to the surface (whichhere points in the x3-direction) and the actual normal vector to the surface scales like N−1 andgrows linearly with the coefficients a and b. We will return to this point in section Polynomial maps from the sphereOur analysis of a generic polynomial surface will build on the analysis of an ellipsoid. Consider apoint of interest such that the normal at this point is pointing in the positive x3 direction. Let this42102 103 104J10-410-310-2c−x3Figure 3.3: The difference between the actual eigenvalue x3 and the classical (large N) position cfor a generic surface given by x1 = 1 + w1 + 0.5w3, x2 = 2w2, x3 = w3 + 0.2w1w2, at a point givenby (w1, w2, w3) = (1/2, 1/4,√11/4). The line shows equation (3.3.57).point lie at x1 = x2 = 0, setting y1 = y2 = 0. Without loss of generality, set y3 equal to zero aswell. This allows us to write Heff as a sum of two pieces as before:Heff(x3) =[0 A†A 0](3.3.50)+[cL3J − x3 + p(2)3 (a− ib)(L3J − 1)+ p(2)1 − ip(2)2(a+ ib)(L3J − 1)+ p(2)1 + ip(2)2 −(cL3J − x3) − p(2)3].p(2) are the polynomials introduced in section 3.3.3: to leading order, they can be written asp(2)k = dk,1(L+J)2+ dk,2(L−J)2+ dk,3L+L− + L−L+2J2(3.3.51)= ek,1(L1J)2+ ek,2(L2J)2+ ek,3L1L2 + L2L12J2(3.3.52)where ek,1 = dk,1 + dk,2 + dk,3, ek,2 = −dk,1− dk,2 + dk,3 and ek,3 = 2i(dk,1− dk,2). Second or higherorder polynomials containing at least one power of L3/J − 1 are either equivalent to polynomials inL1/J and L2/J (from L21 + L22 + L23 = N2 − 1), or subleading, as we will see in a moment.The vector defined in equation (3.3.18) together with |α〉 given in equation (3.3.19) is an ap-43proximate zero eigenvector of this more general Heff as well, as we have confirmed numerically.Generically, Ap decreases with large N like N−1/2.Analytically, we first compute the following quantities〈α|L−L+J2|α〉 ≈ 2J|ξ|21− |ξ|2 (3.3.53)〈α|L+L−J2|α〉 ≈ 2J11− |ξ|2 (3.3.54)〈α|L+L+J2|α〉 ≈ 2Jξ1− |ξ|2 (3.3.55)〈α|L−L−J2|α〉 ≈ 2Jξ¯1− |ξ|2 . (3.3.56)These imply that corrections to x3 due to the polynomials p(2)k in equation (3.3.52) are of orderJ−1, same as correction in equation (3.3.43). In fact, we can compute the new corrections to theeigenvalue x3 in this case:c− x3 = 1J(c|ξ|21− |ξ|2 −|1 + ξ|2e3,1 + |1− ξ|2e3,2 + i(ξ − ξ¯)e3,32(1− |ξ|2)). (3.3.57)Figure 3.3 shows comparison between this approximate result and the exact numerical values.The agreement is excellent.To summarize the size of the various higher order corrections, we notice that‖ (L3/J − 1)|α〉 ‖ ∼ O(N−1) (3.3.58)‖ (L1/J)|α〉 ‖ ∼ O(N−1/2) and∥∥ (L1/J)2|α〉 ∥∥ ∼ O(N−1) (3.3.59)‖ (L2/J)|α〉 ‖ ∼ O(N−1/2) and∥∥ (L2/J)2|α〉 ∥∥ ∼ O(N−1) . (3.3.60)To go further in our analysis, we could ask how introducing higher-order polynomials affects |β〉and |∆α〉 (and therefore Ap as well as the angle the actual normal vector makes with its expecteddirection), or more generally, what is the effect of all these terms on the exact eigenvector. Theanalysis parallels one at the end of the previous subsection: coefficients of the quadratic terms inp(2)3 enter in the same way that c does and coefficients of the quadratic terms in p(2)2 and p(2)3 enterin the same way that a and b do. Thus, again, having a larger curvature on the surface affects ‖β‖2while having the non-commutativity vary quickly affects ‖∆α‖2 (as we will see).As before, formulas for the first few coefficients of |β〉 can be computed recursively. The results aretoo complicated to be illustrative, however, they are qualitatively similar to those for the ellipsoid:‖β‖2 falls off like 1/J , grows with c2 and quadratically with the coefficients in p(2)3 and depends ina non-trivial way on |ξ|. In contrast to the ellipsoid case, it is possible for ‖β‖2 to be non-zero evenwith zero |ξ|.Finally, even higher order polynomials are proportionately more suppressed. For example terms44involving (L3/J − 1)2 are suppressed by N−2:〈α| (L3/J − 1)2 |α〉 ≈ 1J2|ξ|2(2 + |ξ|2)(1− |ξ|2)2 . (3.3.61)3.3.6 Local non-commutativityConsider −i[X1, X2], using the form in equation (3.3.12). We have−i[X1, X2] = (c1c4 − c2c3)(L3/J2) + terms linear in (L1/J2) and (L2/J2)+ terms with higher powers of Li . (3.3.62)From the formulas in section 3.3.5, the expectation value of this operator in the coherent state isjustθ12 = 〈α| − i[X1, X2]|α〉 = (c1c4 − c2c3)/J , (3.3.63)since the corrections to 〈α|L3/J |α〉 ≈ 1, as well as 〈α|L1/J2|α〉, 〈α|L2/J2|α〉 and those terms thatare higher order (in Lis), all lead to subleading contributions (of order 1/J2 or smaller). It isimportant to insist that c1c4 − c2c3 is non-zero, so the leading contribution above does not vanish.We can examine 〈α| − i[X1, X3]|α〉 and 〈α| − i[X2, X2]|α〉 in a similar way. In this case, onlythose sub-leading terms are non-zero and we obtain thatθi3 = 〈α| − i[Xi, X3]|α〉 ∼ 1/J2 for i = 1, 2. (3.3.64)Therefore, we have that θi3/θ12 is of order 1/J , which is well supported by our numerical data (seeFigure 3.4). We can then take θ = θ12. A more general, rotationally invariant equation isθ = 〈α| Θ |α〉 , where Θ :=√−∑i 6=j[Xi, Xj ]2 . (3.3.65)We have introduced a new operator, Θ, which will play an important role in the next section.Equation (3.3.63) has a simple geometric interpretation: the local non-commutativity on theround sphere is constant and equal to 1/J . A single non-commutative ‘cell’ with this area ismapped to an ellipse with area (det C)/J , which is just the non-commutativity in equation (3.3.63).In other words, the local non-commutativity is the volume form on the emergent surface divided bythe volume form on the sphere, times J−1.The local non-commutativity is not constant on the surface. An explicit computation on the45102 103 104J10-510-410-3θ 23/θ 12Figure 3.4: θ23/θ12 for the example in figure 3.3. This ratio appears to decrease like J−1.ellipsoid in equation (3.3.14) shows that its derivatives are∂θ∂x=b(c1c3 + c2c4)− a(c23 + c24)(c1c4 − c2c3)J and (3.3.66)∂θ∂y=a(c1c3 + c2c4)− b(c21 + c22)(c1c4 − c2c3)J . (3.3.67)If we include higher order polynomials, the appropriate coefficients in p(2)1 and p(2)2 enter in thesame way as a and b above. Thus, we see that having these coefficients larger makes the non-commutativity vary faster, as we have mentioned before.3.3.7 Coherent states overlaps, U(1) connection and Fµν on a D2-braneSince coherent states are associated with points, it is important that the overlap between coherentstates corresponding to well-separated points be small. Consider two points p and p′ on the emergentsurface which are within a distance of order 1/√N of each other. For large N ,8 the coefficientsci, a, b, c etc. . . that locally characterize the surface are approximately the same. However, thecorresponding pre-images of p and p′ on the unit sphere in w-space are sufficiently far apart thatthe basis in which equation (3.3.19) is written is completely different. Therefore, the approximate8The question of what constitutes a large enough N is discussed in section 3.4.460 1 2 3 4 5 6 7 8d/√θ0.|〈 α(0)|α(d)〉 |Figure 3.5: Magnitude of the overlap between the eigenstate corresponding to the point p at thenorth pole and the eigenstate corresponding to a point p’ a distance |d| away. The green H corre-sponds to points p’ with x2 = 0, while the blue N corresponds to p’ with x1 = 0. The dashed linecorresponds to equation (3.3.70). Plotted for an ellipsoid with c1=1, c2=0.75, c=12, with N=16,384.coherent state at the point p′ can be obtained from the coherent state at the point p by an SU(2)rotation (in the N -dimensional representation). Explicitly,|α′〉 = ei(−D2L1+D1L2) |α〉 , (3.3.68)where D1 and D2 are small displacements in w-space corresponding to moving from p to p′. Sincewe have positioned p at the north pole of the unit sphere, there is no displacement in the 3-direction.L1 and L2 can be written in terms of A and A† via equation (3.3.16), and we get that|α′〉 = e i2θ (dA+d¯A†) |α〉 , (3.3.69)where d = x′2− ix′1, with x′1, x′2 being the coordinates of point p′. To compute the overlap between|α〉 and |α′〉, we use the Baker-Campbell-Hausdorff formula to leading order, together with A|α〉 ≈ 0:〈α|α′〉 = 〈α| e− 18θ2 dd¯[A,A†] |α〉 ≈ e− |d|24θ , (3.3.70)since [A,A†] = 2θ(L3/J). As can be seen in figure 3.5, the actual coherent states have exactly thisexpected behaviour.47Further, we can look at the connection defined (to within a factor of 2) in equation (28) ofreference [22],2viAi = −ivi〈α(xi)|∂i|α(xi)〉 , (3.3.71)where vi is a tangent vector on the emergent surface. To evaluate it, we rewrite equation (3.3.69)in terms of the small displacements x1 and x2:|α′〉 = e iθ (−x2X1+x1X2) |α〉 . (3.3.72)Thus, the connection is just (A1, A2) = (−x1/2θ, x2/2θ) and the curvature is F12 = θ−1. This isexactly the expected result on an emergent D2-brane [14].3.3.8 Non-polynomial surfacesNot surprisingly, our general conclusions are applicable even when the maps from the sphere to thesurface of interest are not polynomial. As long as the maps are smooth enough to be approximatedby a Taylor polynomial, the largeN limit behaviours should be similar. Examples with many desiredproperties can be relatively easily ‘cooked up’. Here we consider two of conceptual relevance.Our first example using a non-polynomial map is designed to probe into the role of the parameterξ. To this end, we examinex1 = w3w1 +√1− w23 w2 , (3.3.73)x2 = −√1− w23 w1 + w3w2 . (3.3.74)x3 = w3 . (3.3.75)This example is designed produce a round sphere with a constant local non-commutativity θ by‘shearing’ the original sphere (to preserve the volume form). We have checked explicitly that θ isconstant over the surface and equal to 1/J in the large N limit. The parameter ξ, however, is notconstant, instead, we have ξ = −i sin(φ)/(2 − i sin(φ)). This shows that ξ does not play a role inthe large N limit of the surface: it can be changed by applying a volume preserving automorphismto the sphere. Another way to look at it is that the three matrices Xi defined by equations (3.3.73)-(3.3.75) can be obtained from Li/J by a conjugation (up to some ordering ambiguities). ξ can thusbe viewed as a basis-dependent quantity.Another interesting example is given byx1 =w1√w21 + w22 + µ2w23,x2 =w2√w21 + w22 + µ2w23, (3.3.76)x3 =µw3√w21 + w22 + µ2w23.480 5 10 15 20 25√Nθ′/√θ05101520NφFigure 3.6: Angle φ between the normal vector ~n computed using equation (3.2.7) and the non-commutativity vector ijkθjk, for the surface in equation (3.3.76) at a point given by x = 0.5, y = 0.The blue N corresponds to N=3000 and the red H to N=12 000; the agreement between plots atdifferent N shows that the plotted quantities scale with N in the expected way. On the horizontalaxis we have a derivative of the non-commutativity along the surface scaled by√θ, which increasesas µ is increased in equation (3.3.76).In this example, we again get a round sphere, but the local non-commutativity is no longer constant.As we would expect, the actual surface at finite N differs from a round sphere at order 1/N ; thiscorresponds to the normal vector deviating from the radial direction at the same order, as given byequation (3.3.49). Further, we can compute the non-commutativity vector ijkθjk. Our assertion isthat these two vectors should be nearly parallel. Figure 3.6 shows that, indeed, the angle betweenthese two vectors decreases as 1/N . This angle increases as the coefficient µ is increased, resultingin a more rapidly changing non-commutativity. Interestingly, Ap turns out to be subleading, oforder 1/N3/2 or smaller, instead of 1/N1/2, implying that |∆α〉 is nearly parallel to |β〉.The two examples in this subsection demonstrate that our approach works for surfaces which arenot given by polynomial maps from the sphere. This is not surprising, as our approach should workfor any surface which can be locally approximated by a polynomial map over the sphere. Relaxingthe polynomial condition allows for just about any smooth surface which is topologically equivalentto a sphere to be studied with our approach.493.4 Large N limit and the Poisson bracketIn the previous section, we have provided a series of examples increasing in generality and all sharingthe following common features: there existed a family of matrix triplets Xi labeled by their size N .Each such triplet give rise to a surface SN given by the locus of points where Heff(xi) had a zeroeigenvalue. The zero eigenvector of Heff at a point on a surface such that the normal to this surfacewas pointing in the x3 direction was, either exactly or approximately, of the form[|α〉0]. (3.4.1)Where the zero eigenvector was not exactly of this form, the corrections were small, of order N−1/2.More generally, since a rotation of the coordinate system can be effected by an SU(2) rotationof the σi matrices in Heff, the zero eigenvector at an arbitrary point p has the form|Λp〉 =[|α1〉|α2〉]=[a|αp〉b|αp〉]+ O(N−1/2)(3.4.2)where |a|2 + |b|2 = 1 and where |αp〉 is a unit N -dimensional vector.Given the two parts of a zero eigenvector of Heff, |α1〉 and |α2〉, at finite N , we compute |αp〉 asfollows: find the normal vector to the surface, ni = 〈Λp|σi|Λp〉. Then, find the SU(2) rotation thatbrings this vector to point in the positive x3 direction and apply it to Λp. Then, the top componentof |Λp〉 is |αp〉. Explicitly,|αp〉 = cos(θnˆ/2)eiφnˆ/2|α1〉 + sin(θnˆ/2)e−iφnˆ/2|α2〉 , (3.4.3)where θnˆ and φnˆ are the polar angles of the unit normal vector nˆ.Once the coherent state |αp〉 corresponding to a point is identified, we can define a correspondencebetween functions on the large-N surface f and operators (N ×N matrices) Mf throughf(τ) = 〈αp|Mf |αp〉 , (3.4.4)where τ = (τ1, τ2) is a coordinate of some point p on the surface.The function s : Mf → f is usually called the symbol map; using a coherent state to define thesymbol is an approach due to Berezin [38]. The implied non-commutative star product is(f ? g)(τ) := 〈αp|Mf Mg|αp〉 . (3.4.5)The star product is not unique, ie it is not fixed by the surface and the non-commutativityparameter θ alone. There are many different triplets of matrices that give the same surface andnon-commutativity; different triplets would lead to different star products. Only the leading order50of the commutator f ?g−g ?f ≈ θ is universal. For example, the details of the star product dependon ξ which we know to be arbitrary. However, the star product implies, in the large N limit, aunique antisymmetric bracket,{f, g} := N (f ? g − g ? f) . (3.4.6)We would like this bracket to give us a Poisson structure on our emergent surface. It is naturallyskew-symmetric and satisfies the Jacobi identity, so it is a Lie bracket. To be a Poisson bracket, italso needs to satisfy the Leibniz Rule:{fg, h} = f{g, h}+ g{f, h} . (3.4.7)(Notice that these are ordinary multiplications now, not star-products.)Instead of directly proving that the Leibniz Rule holds, we will show that our definition of a starproduct is equivalent to{f, g} = 1ρab ∂af ∂bg (3.4.8)for some function ρ on the surface. In particular, we will haveρ =√det gNθ, (3.4.9)where g is the pullback metric on the non-commutative surface and θ is the local non-commutativityparameter defined in subsection 3.3.6.Let’s follow our previous approach, and consider not onlyXi to be polynomials in L1/J , L2/J andL3/J−1, but also consider operators that are polynomials in Xi (and therefore polynomials in L1/J ,L2/J and L3/J − 1). The degrees and coefficients of all the polynomials are fixed while N → ∞.First, consider the expectation value 〈αp|M |αp〉 of some such operator M = m(X1, X2, X3) in acoherent state, where m(·, ·, ·) is a polynomial function. We can compute 〈αp|M |αp〉 at a pointp where the normal points straight up by first writing M as a polynomial in L1/J , L2/J , and(L3−1)/J . Then, from equations (3.3.58), (3.3.59) and (3.3.60), we see that the leading order piece(which stays finite as N →∞) is simply the constant term9. Thus,〈αp|M |αp〉 = m(y1, y2, y3) , (3.4.10)where yi are the coordinates of the surface at point p as defined in equations (3.3.12).Now that we have shown that the expectation value in a coherent state at a point of any poly-nomial (in Xi) operator is exactly what we would expect, let’s think about the expectation value ofthe commutator of two such operators M1 and M2. Consider then two polynomials, m1 and m2 inx1, x2 and x3, and the corresponding operators M1 = m1(X1, X2, X3) and M2 = m2(X1, X2, X3).We have already argued that θ12 is much larger than θ13 and θ23. A similar argument extended to9Any ambiguities due to the fact that L21 + L22 + L23 = N2 − 1 are subleading in N.51functions of Xi shows that, as long as Xis are of the form (3.3.12), we have〈αp| − i[M1,M2] |αp〉 = θ12(∂m1(y1, y2, y3)∂y1∂m2(y1, y2, y3)∂y2− ∂m1(y1, y2, y3)∂y2∂m2(y1, y2, y3)∂y1).(3.4.11)Thus, for the two functions on the non-commutative surface given as restrictions of the polynomialsma: fa(σ) = ma(xi(σ)), the bracket is{f1, f2} = N〈αp| [M1,M2] |αp〉 (3.4.12)= Nθ(∂σa∂x1∂σb∂x2− ∂σa∂x2∂σb∂x1)∂f1∂σa∂f2∂σb= Nθab√det g∂f1∂σa∂f2∂σb,in agreement with equations (3.4.8) and (3.4.9).To summarize, we have proven that our emergent surface is equipped with natural Poisson bracketwhich satisfies the correspondence principle{·, ·} ↔ − iN [·, ·] . (3.4.13)Essential for our argument to work was the non-commutativity vector ijkθjk being nearly parallelto the normal vector ni, as shown in Figure 3.6. If this was not the case, the bracket we definedwould fail to be a Poisson bracket.For the remainder of this section, we will answer the following question: given a surface embeddedin three dimensions and a Poisson structure on this surface, does there exist a matrix descriptionthat approximates this surface?Our construction gives a positive answer to this question, and provides restrictions on the surfaceand on Nθ for the approximation to be good. We focus on Nθ (rather than θ itself) as this is afinite quantity in the large N limit and determines the Poisson structure through equation (3.4.9).Given a surface and a function Nθ on this surface, we can always define a map from the unit sphereto this surface such that the ratio of the volume form on the surface to the volume form on thesphere is Nθ (see equation (3.3.13)). In fact, we can find many such functions. Which we pick willaffect ξ and the higher orders of the star product, but not the overall non-commutative structure.Note, however, that it is not possible to set ξ to zero everywhere for a generic non-commutativesurface. ξ is zero if the metric on the emergent surface is proportional to the metric on the sphere,while the coefficient of this proportionality must be the non-commutativity θ, which is fixed. Thesetwo requirements would fix (up to diffeomormisms) the metric on the emergent surface, which isalready fixed by the embedding. To view this in a different way, the freedom in choosing a mapfrom the sphere to the emergent surface is the freedom to pick two functions on the sphere. One ofthese functions is fixed by requiring a particular non-commutativity θ. The remaining function canbe used to change ξ. However, ξ is a complex function, so requiring it to vanish over-constrains theproblem.52Given a map from the sphere to the desired surface, we need only replace the rectilinear coordi-nates on the sphere with some SU(2) generators Li and we obtain a triplet of matrices Xi which leadus to the appropriate non-commutative structure. Here, again, there is ambiguity in the orderingof the operators. Its effects are suppressed by powers of 1/N and it affects higher order terms inthe star product (but not the leading order term).For this construction to work, the surface we start with must be sufficiently smooth. Alterna-tively, we could say that we need to pick an irrep of SU(2) large enough to accommodate a rapidlyvarying surface. Two conditions seem necessary: that the curvature radii of the surface at any pointbe much larger than the diameter of a non-commutative ‘cell’ (Rcurvature √θ ∼ N−1/2) andthat θ change slowly. Let θ′ be a derivative of θ in some tangent direction. Then, the change innon-commutativity over a single cell (which has an approximate diameter of√θ),√θθ′, should besmall when compared with θ itself: θ′/√θ  1 (θ′/√θ ∼ N−1/2). As we have already discussed, inequation (3.3.50)—which was the basis for our perturbative definition of a general surface near somepoint—the coefficients in the two diagonal terms (such as c) control the curvature of the surfacewhile the coefficients of the off-diagonal terms (such as a and b) control θ′/θ (see equations (3.3.66)and (3.3.67)). Further, as we have discussed, large ‘curvature coefficients’ lead to large |β〉 whilelarge ‘theta variability coefficients’ lead to large |∆α〉. The larger these coefficients are, the largerN must be to compensate, or higher order terms would spoil the correspondence with the classicallimit we have built up. Generally speaking, the factorization of eigenstate property in equation(3.4.2) fails when curvatures are too large at a given N (since |β〉 becomes large). On the otherhand, when the non-commutativity varies too quickly, the Poisson brackets involving it (such as{Nθ, f}) will turn out to be too large.3.5 Area and minimal area surfacesIn equation (3.3.65), we introduced an operator whose expectation value in a coherent state is thelocal non-commutativity θ. The non-commutativity θ has units of length-squared, and it can beinterpreted as the area of a single non-commutative ‘cell’. This is similar to thinking of phase spaceas made up of elementary cells whose area is ~. In string theory, where a non-commutative surfaceis made up of lower dimensional D-branes ‘dissolved’ in the surface, we can think of θ as the areaoccupied by a single D-brane, or, equivalently, the inverse of the D-brane density. If we divide thesurface into N non-commutative cells, adding up the areas of all these cells we should get the totalarea of the surface. This is in fact borne out here, as the operator Θ introduced in equation (3.3.65)has a second role: its trace seems to correspond to the area of the surface10A = 2pi Tr Θ = 2pi Tr√−∑i,j[Xi, Xj ]2 . (3.5.1)10Factor of 2pi can be arrived at by considering the round sphere. Since our matrices Xi are the SU(2) generatorsscaled by J, the more usual factor of 4pi/N is multiplied by J ≈ N/2.53102 103J10-310-2Relative errorFigure 3.7: Relative error in the non-commutative area as given in equation (3.5.1) compared tothe classical area, for an ellipsoid with major axes 6, 3 and 1. The error falls off with J like J−1; abest fit line, 1.02/J , is shown to guide the eye.Numerical evidence that this formula holds in is shown in figure 3.7.Consider now minimal area surfaces. If we parametrize our emergent surface with coordinatesσa and define the pullback metric on this surface:gab =3∑i=1∂xi∂σa∂xi∂σb, (3.5.2)(locally) minimal surfaces are solutions to the equations∆xk(σa) = 0 , k = 1 . . . 3 , (3.5.3)where the Laplacian is, as usual∆ =1√g∂∂σa√ggab∂∂σb, (3.5.4)and where g is the determinant of the metric gab.It is easy to check that these minimal surface equations can be written in terms of the Poisson54bracket (3.4.8) as113∑i=1{xi, {xi, xk}} − 123∑i=1ρ2g{xi,gρ2}{xi, xk} = 0 . (3.5.5)Let’s now rewrite this equation in terms of θ (using equation (3.4.9)):3∑i=1{xi, {xi, xk}} − 123∑i=1θ−2{xi, θ2} {xi, xk} = (3.5.6)3∑i=1{xi, {xi, xk}}+3∑i=1θ{xi, θ−1} {xi, xk} = 0 ,or, in a more suggestive form (removing an overall factor of θ),3∑i=1{xi, θ−1{xi, xk}} = 0 . (3.5.7)This should be compared with the variation of our expression for the area of the non-commutativesurface (3.5.1):∂A∂X1=12( [X2,Θ−1[X2, X1] + [X2, X1]Θ−1]+ (2→ 3) ) = 0 . (3.5.8)Taking an expectation value of equation (3.5.8) w.r.t. a coherent state, we obtain equation (3.5.7),confirming that the area of the non-commutative surface is indeed given by equation (3.5.1).Notice that this equation differs from that for a static configuration in a generic matrix model(such as BFSS or IKKT), which is[Xi, [Xi, Xk]] = 0 . (3.5.9)This is because the Lagrangian for these matrix models contain a term of the form [Xi, Xj ]2 whichis the square of our operator Θ. When considering minimum area surfaces in matrix models, whenthe non-commutativity varies over the surface, the appropriate equation is not (3.5.9), but (3.5.8),or more generallyΘ−1[Xi, [Xi, Xk]] + [Xi, [Xi, Xk]]Θ−1 + [Xi,Θ−1][Xi, Xk] + [Xi, Xk][Xi,Θ−1] = 0 , (3.5.10)which, in the large N limit where ordering issues can be ignored, can be simplified to[Xi, [Xi, Xk]] + Θ[Xi,Θ−1][Xi, Xk] = 0 (3.5.11)11This approach was used to study matrix models for minimal area surfaces in reference [83].55or[Xi, [Xi, Xk]] − 12Θ−2[Xi,Θ2][Xi, Xk] = 0 . (3.5.12)This last equation matches the original equation (3.5.5). It is important to notice that the secondterm in the above equation (3.5.12) has the same N-scaling as the first term: both are proportionalto N−2. Thus, this term cannot be neglected even in the large N limit.To gain more insight into the formula for the area of the surface, we can examine the formulafor the area in terms of the Poisson bracket:A =∫d2σ√gNθ√∑i,j{xi, xj} →∫d2σ√gθ√−[Xi, Xj ]2 . (3.5.13)The formula in equation (3.5.13) is essentially the bosonic part of the Nambu-Goto action for astring worldsheet. This action is classically equivalent to the Schild action [57], whose quantizationvia matrix regularization gives the IKKT model [56]. Equivalence of these two actions is provenby the standard method involving an auxiliary field the inclusion of which removes the square rootfrom the action [58] (for a review, see reference [84]). In the case of the correspondence betweenthe Nambu-Goto and the Polyakov action, this auxiliary field is the worldsheet metric. Here, itsrole seems to be linked to the local non-commutativity θ. This is not surprising: if the matrixmodel is to be viewed as a quantization of the surface, we should be free to pick any local non-commutativity we chose, so it can play the role of an auxiliary field. This point of view providesa physical interpretation to the quantum equivalence of the IKKT and the non-abelian Born-Infeldmodel.Finally, our computation allows us to write down the non-commutative Laplacian on our emergentsurface; it is, ignoring higher 1/N -corrections∆ = Θ−2[Xi, [Xi, · ]] − 12Θ−4[Xi,Θ2][Xi, · ] . (3.5.14)This equation could be the starting point for a study of the effects of varying non-commutativityon non-commutative field theory.3.6 The torusOur construction has a natural extension to a toroidal surface embedded in flat three space. Justas surfaces topologically equivalent to a sphere were build by considering maps from the non-commutative sphere algebra, to make a torus we use maps from the appropriate algebra.56Consider a surface given byx1 = (R+ r cosu) cos v , (3.6.1)x2 = (R+ r cosu) sin v , (3.6.2)x3 = r sinu , (3.6.3)where u, v ∈ [0, 2pi] and r < R. Now, consider the standard clock-and-shift matrices U and V thatare usually used to define the non-commutative two-torus:UV = e2pii/NV U , (3.6.4)Ukl = δkle2pii(k/N) , (3.6.5)Vkl = δkmodN ,(l+1)modN . (3.6.6)In the non-commutative torus, operators of the form UnV m are associated with functions on thetorus of the form einueinv. To define the non-commutative torus embedded in R3 we thus simplysubstitute eiu → U and eiv → V in equations (3.6.1)-(3.6.3), symmetrizing when necessary to obtainhermitian matrices. Numerical analysis shows that the resulting toroidal surface is smooth and hasthe appropriate large N limit (with Ap decreasing for large N as N−1/2, the surface approachingthe classical shape and the area of the surface well approximated by equation (3.5.1)).Once we have obtained this particular toroidal surface, any other surface with this topology(including surfaces with the same shape but different local non-commutativity, for example uniformone) can be obtained by smooth maps in a way that parallels our discussion of spherical surfaces.It would be interesting to consider a deformation which connects the torus and the sphere and toexamine what happens at the point of topological transition in detail.3.7 Open questions and future workThere are many questions which our work does not address.For example, one can ask if equation (3.5.1) can be proven analytically, starting with the definitionof the surface fromHeff. A reasonable start for such a proof might be equation (3.5.13). If we assumethat1NTr · = 12pi∫d2σ√gNθ〈α(σ)| · |α(σ)〉 , (3.7.1)we recover equation (3.5.1). Equation (3.7.1) is equivalent to12pi∫d2σ√gθ|α(σ)〉〈α(σ)| = 1N . (3.7.2)Above equation implies a relationship between the trace and the integral of the non-commutative57surface1NTr ↔ 12pi∫d2σ√gNθ. (3.7.3)A completeness relationship such as (3.7.2) is necessary for the symbol map from operators tofunctions on the emergent surface to have a unique inverse, which in turn is necessary for thedefinition of the star product to make sense. In principle, it should be possible to prove such acompleteness relationship starting with equation (3.1.1).In subsection 3.3.7, we briefly addressed the question of the U(1) connection on the emergentD2-brane. Extending this approach should allow us to prove the equivalence of the non-abelianeffective action for D0-branes and the abelian effective action for a D2-brane. More simply, itshould be possible to show the equivalence of the BPS conditions in these two scenarios.It would be interesting to see how our set up could be extended to surfaces which are nottopologically equivalent to a sphere or a torus. It should be possible, for example, to find matrixtriplets Xi which correspond to emergent surfaces with a larger number of handles—and for whichthe large N limit we describe holds. One could check, for example, whether the non-commutativesurfaces given in reference [82] have a large N limit in the sense in which we define it here. Further,it would be interesting to see how our toroidal construction in section 3.6 is related to that inreference [82].Finally, there are many generalizations of equation (3.1.1) that would be interesting to explore,including higher dimensional generalizations in Chapter 4 (both of the embedding space and theemergent surface) and those to curved embedding space. One could also consider Lorentzian sig-nature models, which would be useful in the context of recent progress in cosmology arising frommatrix models, as in reference [21].58Chapter 4Matrix embeddings on flat R2n+14.1 Introduction and conjectureThe appearance of matrix coordinates, where the positions of N identical objects in d dimensions aredescribed by d N ×N matrices instead of N d-vectors, is common in string theory. The geometricinterpretation of non-commuting matrix coordinates often involves an emergent higher dimensionalobject. The exact shape and other properties of this emergent object can be hard to study; outsideof highly symmetric surfaces such as spheres, only some approximate methods (such as diagonalizingthe matrices one at a times) are usually employed. In reference [22], a method for determining asurface embedded in R3 and associated with any three matrices was given, providing a concretesolution to this problem. In Chapter 3, the geometry of this surface was examined in detail, provingthe correspondence principle between matrix commutators and a Poisson structure on the emergentsurface.It is natural to ask about generalizing these results to higher dimensions. Higher dimensional non-commutative spaces posses a much richer phenomenology than non-commutative surfaces do and anexplicit embedding into flat space would make their study easier and more concrete. Below, in equa-tion (4.1.1), we conjecture an embedding operator which makes this possible for even-dimensionalnon-commutative hypersurfaces embedded in a odd-dimensional flat space.The effective Hamiltonian plays a role of an ‘embedding operator’: it specifies how the emergentsurface given by three matrices Xi should be embedded in flat R3. Since equation (3.1.1) wasobtained from an orbifold construction, with Pauli matrices arising from a dimensional reductionof Dirac Γ matrices from 9 dimensions to 3, a natural guess for the generalization of the embeddingoperator to arbitrary odd dimensions isEd(xi) =d∑i=1γi ⊗ (Xi − xi) , (4.1.1)where γi are the (Euclidean) Dirac matrices in d dimensions, which form a representation of the59Clifford algebra{γi, γj} = 2δij . (4.1.2)We have introduced a new symbol, Ed, to denote the embedding operator in Rd. For d = 9 thisoperator has been used in reference [85] to study thermal configurations in the BFSS model. SimilarDirac operators have been used in reference [86] to define the location of D-brane intersections andthe resulting emergent gravity.As we will see, our conjectured embedding operator ‘knows’ a lot about non-commutative geom-etry. For example, a non-commutative sphere S2d with SO(2d+1) symmetry cannot locally (nearsome point p) look like the standard flat non-commutative space, since the latter is never fullyrotationally symmetric, while S2d should retain SO(2d) symmetry around point p. Examining thekernel of the embedding operator Ed for a non-commutative four-sphere we find an auxiliary spinspace whose presence restores SO(4) invariance, resolving the puzzle.It would be very interesting to obtain formula (4.1.1) from string theory considerations. Ford = 5 and d = 7, the computation might proceed along the lines of reference [22], using an orbifold.For d = 9, another method might be more applicable (see the discussion in reference [22]).The remainder of this chapter is organized as follows. In the next section, we set conventionsand observe that once Ed is known in some odd dimension d, it is possible to obtain the embeddingoperators in all lower dimensions by simply setting some of the matrices to zero, two at a time. Insection 4.3, we discuss flat non-commutative space and generalize most of our results from reference[1] to higher dimensions. In section 4.4, we study the non-commutative four-sphere embedded inR5, in particular obtaining a flat non-commutative space with SO(4) rotational symmetry as anapproximation to the sphere on a small patch. In section 4.5, we discuss further examples of fourdimensional non-commutative surfaces. Finally, in section 4.6 we try to study even dimensions bysetting just one of the matrices to zero. That this naive guess fails to work can be demonstrated byconsidering the non-commutative three-sphere, S3.4.2 Conventions and a recursive property of EdOur embedding operators have the property that once Ed is known in some odd dimension d, it ispossible to obtain the embedding operators in all lower dimensions recursively. The easiest way tosee that our family of embedding operators has this property is to use an iterative definition of theγ matrices as follows.1 In d = 1, we trivially take γ1 to be the 1 × 1 unit matrix. Then, denoting1We follow here, reference [87].60the γ matrices in d− 2 dimensions with γ˜i, we have in d dimensions thatγi = σ3 ⊗ γ˜i , i = 1, . . . , d− 2 , (4.2.1)γd−1 = σ1 ⊗ 1 , (4.2.2)γd = σ2 ⊗ 1 . (4.2.3)The dimension of the γ matrices is thus 2n = 2(d−1)/2. For d = 3, we obtain a permutation of thePauli matrices: γ1 = σ3, γ2 = σ1, γ3 = σ2.Now, in some odd number of dimensions d set the last two matrices Xd−1 and Xd to zero. Wecan then reduce Ed to Ed−2: if Xd−1 = Xd = 0, thenEd(x1, . . . , xd) =d−2∑i=1(σ3 ⊗ γ˜i)⊗ (Xi − xi)− (σ1 ⊗ 1)⊗ (xd−1)− (σ2 ⊗ 1)⊗ (xd). (4.2.4)One can show that for the above operator to have a zero eigenvector, we must necessarily havexd−1 = xd = 0. Then, the operator above can be reduced toσ3 ⊗ Ed−2(x1, . . . , xd−2) . (4.2.5)Thus, once a construction of Ed is known in some odd number of dimensions, it is easy to constructall the smaller odd dimensional cases. In section 4.6 we will discuss our attempt to obtain anembedding operator in an even number of dimensions by setting just one of the matrices to zero.To write the γ matrices in an explicit form it is convenient to introduce the following notation:σn(c1, . . . , cn) := σc1 ⊗ . . . ⊗ σcn , (4.2.6)where the coefficients ci take integer values from 0 to 3 and where we define σ0 = 1. In this notation,the recursive definition of γ matrices implies thatγ1 = σn(3, 3, 3, . . . , 3, 3, 3)γ2 = σn(3, 3, 3, . . . , 3, 3, 1)γ3 = σn(3, 3, 3, . . . , 3, 3, 2)γ4 = σn(3, 3, 3, . . . , 3, 1, 0)γ5 = σn(3, 3, 3, . . . , 3, 2, 0)...γd−1 = σn(1, 0, 0, . . . , 0, 0, 0)γd = σn(2, 0, 0, . . . , 0, 0, 0)61To complete our conventions, we make the following choice for the Pauli matrices:σ1 =[0 11 0], σ2 =[0 −ii 0], σ3 =[1 00 −1], σ− =[0 01 0], σ+ =[0 10 0]. (4.2.7)4.3 Non-commutative R2nAs the first example, set X1 = 0 and consider the other d − 1 matrices to have a commutationrelation[Xi, Xj ] = iθij for i, j = 2, . . . , d . (4.3.1)This, of course, is simply flat non-commutative space, extending in dimensions 2 through d (assum-ing θ has full rank). θ is an antisymmetric even dimensional matrix which can be, by an orthogonalchange of basis and therefore without loss of generality, brought into the block-diagonal formθ = diag([0 θ1−θ1 0], . . . ,[0 θn−θn 0]). (4.3.2)We define Aa = X2a + iX2a+1 for a = 1, . . . , n. Aa and A†a are the lowering and raising operatorsof a harmonic oscillator with [Aa, A†a] = 2θa. The lowering operators Aa have eigenstates |α〉a (thecoherent states), corresponding to every complex number α: Aa|α〉a = α|α〉a. Ed can be written asn∑a=1(Λa− ⊗ (Aa − αa) + Λa+ ⊗ (A†a − α¯a)), (4.3.3)whereΛa± = σn( 3, . . . , 3︸ ︷︷ ︸n−a times, ±, 0, . . . , 0︸ ︷︷ ︸a−1 times) = γ2a ± iγ2a+i . (4.3.4)In this form, it is easy to see that|Λ(α)〉 =(n⊗a=1[10])⊗(n⊗a=1|α〉a)(4.3.5)is a zero eigenvector for Ed at a point given by x1 = 0 and x2a + ix2a+1 = αa. Thus there is a zeroeigenvector for every point on the co-dimension one hypersurface given by x1 = 0. The first factorin the above zero eigenvector is simply one of the highest weight vectors of the Clifford algebraselected by the particular form of raising operators Λ+ which we are using. We will denote it withVd:Vd :=(n⊗a=1[10]). (4.3.6)Vd is an eigenvector of γ1 with eigenvalue 1 and has the property that Λa+Vd = 0 for all a.62We can expect many non-commutative spaces to have the property that the embedding operatorhas a single zero eigenvector at a given point on the emergent surface. Those spaces should, locally,look like non-commutative flat space given by equation (4.3.1). We will see in the next section that,for d > 3, non-degenerate non-commutative spaces exist whose embedding operators have multiplezero eigenvectors at a point. However, for those that don’t, our work [1] on emergent surfaces inthe large N limit can easily be generalized to higher dimensions. Similar results have been obtainedbefore in reference [70] (see also reference [80] and the references therein).In the rest of this section, we state the salient results and conjectures.Assume, then, that the zero eigenvector |Λp〉 of the embedding operator is unique at every pointp of the emergent surface. The normal vector to this surface at point p is given by2ni = 〈Λp|γi ⊗ 1|Λp〉 . (4.3.7)For simplicity, we now rotate our surface so that the normal vector at the point of interest pointsin the x1 direction. We conjecture that the eigenvector is equal to, approximately, a product of anappropriate highest weight state V θd and a N -dimensional vector:|Λ〉 = V θd ⊗ |α〉 + corrections that vanish for N → 0 . (4.3.8)Further, we can define a local non-commutativity matrix θij at point p byθij = 〈α| − i[Xi, Xj ]|α〉 , for i, j = 2, . . . d. (4.3.9)θij is an antisymmetric two-form on the emergent surface; it defines a Poisson bracket of twofunctions f and h:{f, h} := Nθab√det g∂af∂bh , (4.3.10)where gab is the pullback of the flat metric on Rd to the d − 1 dimensional emergent space. Fromthis Poisson bracket, we divide the d − 1 directions x2, . . . , xd into raising and lowering operatorsjust like we did above. In particular, we have a new set of lowering and raising operators on thespinor space, Λθ,a± (defined, in a particular basis, in equation (4.3.4)). The highest weight state inequation (4.3.8) has Λθ,a+ V θd = 0.The N -dimensional state |α〉 should be interpreted as a coherent state associated with the pointp. Since (Ed)2|Λ〉 = 0, we have〈Λp|1⊗∑i(Xi − xi)2|Λp〉 = −12〈Λp|∑i 6=j(γiγj)⊗ [Xj , Xk]) |Λp〉 . (4.3.11)2The arguments for this and other statements below are basically identical to that given in reference [1] for d = 3.63Substituting the factorization condition (4.3.8), we obtain〈α|∑i(Xi − xi)2|α〉 = −12〈α|∑i 6=j(nij [Xi, Xj ]) |α〉 , (4.3.12)where the two-form nij is defined below, in equation (4.3.15). However, since our non-commutativespace is a direct product of n copies of two dimensional non-commutative space, a better way tostudy the properties of the coherent state is work in a basis where the non-commutativity is givenby equation (4.3.2) and to write |α〉 as a product of n coherent states |α〉 = |α〉1 ⊗ . . .⊗ |α〉n.Once we have coherent states |αp〉 corresponding to every point p on the surface, we can associateany N × N matrix M with functions on the surface, via M → 〈αp|M |αp〉. This is the Berezinapproach to non-commutative geometry [38]. It gives a natural map between commutators ofoperators and an antisymmetric Lie bracket on the surface. This bracket turns out to be equal tothe Poisson bracket defined in equation (4.3.10) as long as, in addition to the factorization condition(4.3.8), we also have that〈α| − i[Xj , X1]|α〉 , for i, j = 2, . . . d, (4.3.13)is much smaller than ‖θij‖ for N →∞.It is useful to define two antisymmetric two-forms on Rd:θˆij = 〈α| − i[Xi, Xj ]|α〉 , for i, j = 1, . . . d. (4.3.14)andnij =12〈V θd |i[γi, γj ]|V θd 〉 for i 6= j . (4.3.15)It is easy to see that n1k = 0. In the basis in which θij is given by equation (4.3.2), we havenij = diag(0,[0 1−1 0], . . . ,[0 1−1 0]). (4.3.16)The necessary condition for the correspondence principle to hold can then be stated more covariantlyasnijnklθˆik = θjl + corrections that vanish for N → 0 . (4.3.17)It follows that the vector i1,i2,...,id θˆi1,i2 . . . θˆid−2,id−1 should be nearly parallel to the normal vectorni. We conjecture that this vector is related to the total volume of the surface viaVolumed−1(emergent surface) = C Tr√∑id(i1,i2,...,id [Xi1 , Xi2 ] . . . [Xid−2 , Xid−1 ])2. (4.3.18)C in the above is some numerical coefficient which does not depend on N (for d = 3, this coefficientwas 2pi).64When interpreting the emergent surface as a higher-dimensional D-brane emerging from D0-branes via the dielectric effect [16], the two form θˆij and its pullback to the worldvolume of theD-brane, θij , will enter into the non-abelian BI and CS actions as expected. Finally, an emergentD-brane should have a U(1) connection living on its worldvolume; following reference [22], we candefine it as2viAi = −ivi〈α(xi)|∂i|α(xi)〉 , (4.3.19)where vi is a tangent vector on the emergent surface. Working with a coherent state in a factorizedform, we obtain that associated curvature is ∂[iAj] = (θ−1)ij , as expected.4.4 Even dimensional spheres S2n and non-commutative space withSO(2n) invarianceThe non-commutative four sphere can be constructed as in reference [88] (see also reference [89]).The starting point is a representation of the Clifford algebra in four dimensions: the γ matricesof section 4.1. The matrices in this representation act on vectors in a four-dimensional spinorrepresentation. Consider then an irreducible representation of Spin(5) given by the completelysymmetric tensor product of k copies of this irrep. To each γ, associate a matrix Xi that acts onthis tensor product as followsXi =1k(γi ⊗ 1⊗ . . .⊗ 1 + 1⊗ γi ⊗ . . .⊗ 1 + . . . + 1⊗ 1⊗ . . .⊗ γi)sym (4.4.1)The claim is that these five position matrices represent a four-sphere of radius one. Their dimensionis N = (k + 1)(k + 2)(k + 3)/6.In the four dimensional spinor irrep, consider the vector V4, and take its image under the kthsymmetric tensor product map, (V4⊗ . . .⊗V4)sym := (V4)⊗k. Because γ1V4 = V4 and (γ2 + iγ3)V4 =(γ4 + iγ5)V4 = 0, the matrices Xi above have a simple action on this vector,X1 · (V4)⊗k = (V4)⊗k , (4.4.2)(X2 + iX3) · (V4)⊗k = 0 , (4.4.3)(X4 + iX5) · (V4)⊗k = 0 . (4.4.4)Consider the point (1, 0, 0, 0, 0) in R5, which we hope lies on the emergent sphere. We need theembedding operator E5 at this point to have a zero eigenvector. We rewrite E5 at this point asγ1 ⊗ (X1 − 1) +2∑a=1(Λa− ⊗X+a + Λa+ ⊗X−a), (4.4.5)whereX±a = X2a ± iX2a+1 . (4.4.6)65Now, consider a vector Λ = V4 ⊗ (V4)⊗k. It is easy to see that this is a zero eigenvector of E5 asgiven in equation (4.4.5). Since the γ matrices form a fundamental (or standard) representationof SO(5), we recover the entire spherical surface with radius 1 by symmetry. However, using themethodology from section 4.3, rotational symmetry appears lost, as 〈(V4)⊗k|[X2, X3]|(V4)⊗k〉 =〈(V4)⊗k|[X4, X5]|(V4)⊗k〉 = 1/k and the other four commutators vanish. Since we know that thenon-commutative sphere has SO(5) symmetry and therefore SO(4) symmetry once a point on thesphere is fixed, the non-commutativity on the 4-sphere must be of a different kind than that insection 4.3.In fact, Λ = V4 ⊗ (V4)⊗k is not the only zero eigenvector of the embedding operator in equation(4.4.5). In contrast to the flat non-commutative space above, here both the raising and the loweroperatorsX±a have zero eigenvectors. Let V˜4 = Λ1−Λ2−V4 be the spinor with γ1V˜4 = V˜4 and Λa−V˜4 = 0.Then, consider an arbitrary unit spinor W in the span of {V4, V˜4}, W = µV4 − νV˜4, |µ|2 + |ν|2 = 1,with γ1W = W , (µΛ1+− νΛ2−)W = 0 and (µΛ2+ + νΛ1−)W = 0. Rewrite the embedding operator inequation (4.4.5) asγ1 ⊗ (X1 − 1) + (4.4.7)(µΛ1+ − νΛ2−)⊗ (µX−1 − νX+2 ) + (νΛ1+ + µΛ2−)⊗ (νX−1 + µX+2 ) +(µΛ2+ − νΛ1−)⊗ (µX−2 − νX+1 ) + (νΛ2+ + µΛ1−)⊗ (νX−2 + µX+1 ) .This demonstrates explicitly thatΛ˜ = W ⊗ (W )⊗k (4.4.8)is also a zero eigenvector of the embedding operator in equation (4.4.5). The kernel of the embeddingoperator is a (k+ 2)-dimensional space, while the space of the associated coherent states is (k+ 1)-dimensional.3 Its presence has a natural interpretation: it is the auxiliary space necessary to ensurethat the emergent non-commutative space has SO(4) symmetry. (Notice that the non-commutativeflat space we defined in the previous section does not have full rotational symmetry even when weset all θa equal to each other.)To see how rotational invariance is restored, first notice that the SO(4) symmetry we wish tosee restored is generated by the six commutators [Xi, Xj ]. For k = 1, we write these commutatorsexplicitly:L1 := −i[X2, X3] = σ2(0, 3) , L2 := −i[X2, X4] = σ2(2, 1) , L3 := −i[X3, X4] = σ2(2, 2) ,K1 := −i[X4, X5] = σ2(3, 0) , K2 := −i[X5, X3] = σ2(1, 2) , K3 := −i[X2, X5] = −σ2(1, 1) .3First, let’s understand why the vectors (W )⊗m span a (m + 1)-dimensional space (ie, why Symm(span{V4, V˜4})is (m + 1)-dimensional), by drawing a parallel with representations of SU(2). The fundamental irrep of SU(2) isof course 2-dimensional, and all higher irreps correspond to completely symmetric tensor powers of the fundamentalrepresentation. Thus we know that the dimension of SymmS where S is any two dimensional vector space is m+ 1.Thus, the kernel has dimension k+2 (because it corresponds toW⊗(k+1)), but there are only k+1 linearly independentN -dimensional coherent states once the first term in the product is stripped off.66For larger k, we just consider these operators acting on the symmetric kth tensor power of the four-dimensional irrep of Spin(5). Notice than when one of these six generators acts on any vector in thekernel of the embedding operator, we get another vector in the kernel. Thus, we get a representationof the algebra so(4). To see which representation it is, consider two mutually commuting sets ofgenerators, Li ±Ki. Their commutation relationships are[(Li ±Ki), (Lj ±Kj)] = 2iijk(Lk ±Kk) and [(Li ±Ki), (Lj ∓Kj)] = 0 , (4.4.9)which is nothing more but the standard fact that SO(4) ∼ SU(2)×SU(2). By explicit computation,we see that when acting on the kernel of the embedding operator, Li−Ki vanish, while the action ofLi+Ki is that of a (k+1)-dimensional irreducible representation of su(2). Thus, the zero eigenvectorsof the embedding operator form the (k/2, 0) irrep of SU(2) × SU(2). For example, for k = 1 wehave, explicitly in the {V4, V˜4} basis−i[X2, X3] = σ2(0, 3)→ m23 :=[1 00 −1],−i[X3, X4] = σ2(2, 2)→ m34 :=[0 −1−1 0], (4.4.10)−i[X2, X4] = σ2(2, 1)→ m24 :=[0 i−i 0].So far, we have focused on the point (1, 0, 0, 0, 0). However, when other points close enough tothis one are considered, the commutators [Xi, Xj ] for i, j = 2, . . . , 5 are nearly constant. Consider,for example, a zero eigenvector of the embedding operator E5 at a point (1, βˆ, 0, 0, 0), with β  1.Let’s use a basis for the four dimensional spinor representation given by σ2(3, 0)|s1, s2〉 = s1|s1, s2〉and σ2(0, 3)|s1, s2〉 = s2|s1, s2〉, where si = ±1. In this notation, V4 = |++〉 and V˜4 = |−−〉. Forclarity, pick an eigenvector of the embedding operator at point (1, βˆ, 0, 0, 0) of the form(|++〉 + β|+−〉+ . . .)⊗k = (4.4.11)|++〉⊗k + kβ(|+−〉 ⊗ |++〉⊗(k−1))sym+12k(k − 1)β2(|+−〉 ⊗ |+−〉 ⊗ |++〉⊗(k−2))sym + . . .where β is proportional to βˆ. (|+−〉⊗|++〉⊗(k−1))sym has length 1/√k, (|+−〉⊗|+−〉⊗|++〉⊗(k−2))symhas length approximately 1/√k(k − 1)/2, etc...,Thus, √kβ is of order 1, this vector’s overlap with(|++〉)⊗k decreases sharply to zero45. That a smooth sphere is recovered in the large k limit tell4From our work [1], we would expect this overlap to have Gaussian fall-off.5Thus, the radius of a non-commutative ‘cell’ is 1/√k and its 4-volume is 1/k2. In a sphere of radius 1, wethen have approximately k2 such ‘cells’. Each corresponds to a k + 2 dimensional kernel of the embedding operator,so the total dimensionality of the matrices needs to be approximately k3, in agreement with the exact formulaN = (k + 1)(k + 2)(k + 3)/667us that there is a range of values for β (or βˆ) where the vector above is close to being linearlyindependent of (|++〉)⊗k but where terms with powers of β greater than some p k can be ignored.In this range, the matrix elements of [Xi, Xj ] when acting on the kernel of the embedding operatorare approximately independent of β. As an example, consider that〈++|⊗k − i[X3, X4] (|+−〉 ⊗ |++〉⊗(k−1))sym (4.4.12)is of order 1/k, because the above overlap is only non-zero when the non-trivial operator in−i[X3, X4] = 1k(σ2(2, 2)⊗1⊗. . .⊗1 + 1⊗σ2(2, 2)⊗. . .⊗1 + . . . + 1⊗1⊗. . .⊗σ2(2, 2))sym (4.4.13)‘finds’ |+−〉 when acting on (|+−〉 ⊗ |++〉⊗(k−1))sym.Thus, for points near (1, 0, 0, 0, 0), the relevant commutators, when acting on the kernel of theembedding operator, are nearly constant (with 1/k corrections) and we get the following approxi-mate non-commutative algebra[Xi, Xj ] =ikmij , (4.4.14)where mij are (k+1)× (k+1) matrices in the (k/2, 0) irreducible representation of SU(2)×SU(2).m23, m24 and m34 are defined in equation (4.4.10), while m25 = m34, m35 = m24 and m45 = m23.The factor 1/k comes from normalization of Xi in equation (4.4.1). This non-commutativity algebrais similar to spin non-commutativity with SO(3) symmetry in three spacial dimensions in reference[90] (see also references therein).SO(4) is restored in equation (4.4.14) because the action of SO(4) on X2, X3, X4, X5 can be‘undone’ by a similarity transformation on matrices mij . Since SU(2) × SU(2) is a double coverof SO(4), a rotation in R4 that goes ‘all the way around’ (ie, is trivial in SO(4)) corresponds to anon-trivial element of SU(2) × SU(2), namely (−1) ⊗ (−1). In the (1/2, 0) irrep (and all (k/2, 0)irreps for k odd) this corresponds to multiplying all the vectors in the kernel of the embeddingoperator by −1. Such a change of basis has no effect on the matrix elements of [Xi, Xj ], or on mij .For (k/2, 0) irreps with k even, (−1)⊗ (−1) is trivial.Another observation concerns orientability: a non-commutative 4-space with opposite orientationto the one we have considered is found at the other pole of the sphere, near the point (−1, 0, 0, 0, 0).This can be obtained by taking V4 → V˜4 and V˜4 → V4. As such a map is not an element of SU(2),it has a non-trivial effect on the matrices mijThat the space of coherent states has dimension k+1 fits well with string theory: in reference [89]it was found that the correct interpretation of the four-sphere is that of a D4-brane stack with koverlapping branes6. Further, we notice that if we make a definition of a connection similar to thatin equation (4.3.19), we will obtain a U(k + 1) gauge field, consistent with the interpretation of astack of k + 1 emergent D-branes. Finally, substituting our solution into equation (4.3.18) we get6Up to corrections of order 1/k, which explains the discrepancy between k and k + 1.68an answer of the form (numericalcoefficient) · k + O(1/k) corrections, again confirming that whatwe have obtained is a sphere of radius one, wrapped k (or k+ 1) times. This wrapping seems to benecessary to recover full rotational symmetry.The string theory representation raises the following puzzle: is it possible to make a singleemergent spherical D4-brane? In reference [89] this puzzle was phrased differently: is it possible toseparate the k+ 1 branes making up the stack and give them different radii? We take a partial steptowards a positive answer in the next section by giving up local SO(4) invariance.The generalization to from the four sphere to higher even-dimensional spheres, S2k is straight-forward. These spheres are constructed in the same way as the four sphere, S4, by simply usingthe higher dimensional γ matrices (see, for example, reference [91] for a review). SO(2k) sym-metry around a point on S2k will be restored in much the same way that SO(4) symmetry wasrestored around a point on S4, leading to higher dimensional versions of the non-commutative al-gebra (4.4.14). Even-dimensional non-commutative spheres have a rich phenomenology (see forexample [92]), which it would be interesting to explore from the point of view of our embeddingoperator.4.5 More examples in d = 5In this section, we consider two relatively simple co-dimension one hypersurfaces in R5, one of whichhas the topology and the symmetries of (S2 × S2)/Z2, and the other is a round S4 whose SO(5)symmetry is broken by non-commutativity.To embed (S2 × S2)/Z2 in R5, we consider the equation(1− x22 − x23)(1− x24 − x25) = x21 . (4.5.1)The non-commutative version of this hypersurface is given byX1 = J(1)3 ⊗ J (2)3 , (4.5.2)X2 = 1⊗ J (2)1 ,X3 = 1⊗ J (2)2 ,X4 = J(1)1 ⊗ 1 ,X5 = J(1)2 ⊗ 1 ,where the matrices J (a)i = L(a)i /ja, while L(a)i form two irreducible representations of su(2): [L(a)i , L(a)j ] =iijkL(a)k , each with spin ja, a = 1, 2. It is easy to see that, in the large spin limit, these matricessatisfy equation (4.5.1).At the point (1, 0, 0, 0, 0), the corresponding embedding operator has two zero eigenvectors,V4 ⊗ (|j1〉 ⊗ |j2〉) := V4 ⊗ |α1〉 and V˜4 ⊗ (| − j1〉 ⊗ | − j2〉) := V4 ⊗ |α2〉, where J (a)3 |m〉a = m|m〉a.69The local non-commutativity at this point is〈α1| − i[X2, X3]|α1〉 = −〈α2| − i[X2, X3]|α2〉 = 1j1, (4.5.3)〈α1| − i[X4, X5]|α1〉 = −〈α2| − i[X4, X5]|α2〉 = 1j2(4.5.4)with the expectation values of the other commutators vanishing, and with all cross-terms between|α1〉 and |α2〉 vanishing as well for ji > 1/2.The set of matrices (4.5.2) has the expected SO(3)×SO(3) symmetry: an action of the symmetrygroup on the lower indices of J (a)i is equivalent to a conjugation. However SO(3) × SO(3) is not asubgroup of SO(5), so different points on the emergent surface are not equivalent and we cannotuse symmetry to study zero eigenvectors of the embedding operator. Instead, we must resort tonumerical analysis. Preliminary numerical study at various small spins (at most 2) shows that theemergent surface gets closer to that in equation (4.5.1) for larger matrices, and that the embeddingoperator has two zero eigenvectors everywhere on the emergent surface. This would imply that theemergent surface locally looks like a direct sum of two non-commutative spaces described in section4.3. It is possible that there are some points of enhanced symmetry, though we did not find any.That we get two copies of non-commutative flat space locally is consistent with S2 × S2 being adouble-cover of the surface in equation (4.5.1).A different non-commutative surface is given byX1 = J(1)3 ⊗ J (2)3 , (4.5.5)X2 = J(1)3 ⊗ J (2)1 ,X3 = J(1)3 ⊗ J (2)2 ,X4 = J(1)1 ⊗ 1 ,X5 = J(1)2 ⊗ 1 .These five matrices satisfy, in the large spin limit, the equation∑iX2i = 1. Again, at the point(1, 0, 0, 0, 0), V4 ⊗ (|j1〉 ⊗ |j2〉) and V˜4 ⊗ (| − j1〉 ⊗ | − j2〉) are zero eigenvectors of the correspondingembedding operator. At this point, the non-commutativity is the same as in the previous example.Since SO(5) symmetry here is broken to SO(3) × SO(2), to study the whole surface, we resort tonumerical analysis, which shows that the embedding operator has two eigenvectors at nearly allpoints on the sphere∑i x2i = 1, except on the circle x1 = x2 = x3, where the degeneracy is 2j2 + 2.We can explain the enhanced degeneracy on the circle as follows: On this circle, let’s take (withoutloss of generality) the point (0,0,0,1,0). The operator Λ2−⊗1+1⊗L(2)+ commutes with E5(0, 0, 0, 1, 0)and generates a basis for its kernel when acting on (|σ1,+1〉 ⊗ |σ3,−1〉)⊗ (|L(1)1 ,+j1〉 ⊗ |L(2)3 ,−j2〉)where the notation |L, l〉 means an eigenvector of operator L with eigenvalue l. Our interpretationis that this corresponds to a stack of two non-commutative spherical surfaces which ‘merge’ on70the circle x1 = x2 = x3 where, perhaps, the full SO(4) symmetry is locally restored. Away fromthis circle, non-commutativity breaks SO(4) symmetry while the surface is still a round sphereindependent of matrix size.These two examples illustrate the rich non-commutative phenomenology that can be studiedusing our embedding operators.4.6 Even dimensionsIn this section, we try to use dimensional reduction of our embedding operator Ed to obtain anembedding operator in even dimensions. However, we find that this naive attempt does not producean embedding operator compatible with the usual construction of the non-commutative three sphereS3. Therefore, we leave even dimensional spaces for future work.To obtain a guess for the embedding operator in even dimensions, simply assume that Xd = 0in equation (4.1.1):Ed(x1, . . . , xd) =d−2∑i=1(σ3 ⊗ γ˜i)⊗ (Xi − xi) − (σ1 ⊗ 1)⊗ (Xd−1 − xd−1) − (σ2 ⊗ 1)⊗ (xd)(4.6.1)It is possible to show that this operator has an eigenvector with eigenvalue zero only if xd = 0 andif another operator, which we would like to identify with Ed−1, has an eigenvector with eigenvaluezero. This would lead us to propose thatEd−1 =d−2∑i=1γi ⊗ (Xi − xi) + i1⊗ (Xd−1 − xd−1) , (4.6.2)where the γ matrices are those for dimension d − 1, is a suitable embedding operator in an evendimension d − 1 = 2n. The last term can, equivalently, have a minus sign in front of it. Noticethat the above embedding operator is not hermitian: This is inconvenient but seems unavoidable.A potentially interesting observation is that if we take the last dimension, d, to be time, then thematrix Xd would be anti-hermitian and Ed itself would be hermitian.The most natural place to test this embedding operator is to take d−1 = 4 and try the matricescorresponding to a non-commutative S3 (see, for example, reference [91, 93]). The correspondingembedding operator does not seem to have any eigenvectors away from the origin (0, 0, 0, 0). Inparticular, for the two smallest presentations of S3, withN = 4 andN = 12, when the correspondingembedding operator is evaluated at a point (x, y, z, w), its determinant is r6(r2 + 8) for N = 4 andr16(r2 + 6)(r2 + 4)3 for N = 12. We have normalized our matrices so that their largest eigenvalueis 1, and r2 = x2 + y2 + z2 +w2. Clearly, the embedding operator has zero eigenvectors only at theorigin r = 0. Thus, (4.6.2) does not seem to be the correct operator.To understand why the embedding operator in equation (4.6.2) does not have the right properties,71it is useful to look at equation (4.3.12). Let the Xi be a series of representations of some Lie algebra(such as su(2) for the two-sphere), so scaled that eigenvalues have a fixed range. Due to this scaling,the commutators on the right hand side of equation (4.3.12) get smaller as the matrices grow.This, in turn, guarantees that the width of the coherent state, on the left hand side of equation(4.3.12), approaches zero as the matrices get large. However, when the embedding operator inequation (4.6.2) is squared, the off-diagonal terms fail to arrange themselves into commutators, andwe cannot make any conclusions about the size of the coherent state. In work [23], the existenceof coherent states whose width approaches zero as the matrices grow large was used to define anemergent surface in any number of dimensions at infinite N (but not at finite N , in contrast toour work). We suspect that the corrent embedding operator must lead to an equation similar instructure to (4.3.12), and this is why (4.6.2) fails.72Chapter 5Noncommutative layered spherical shellsS2dso(2d+2)5.1 IntroductionThe matrix configuration studied in this chapter is known as the "generalized fuzzy sphere", seereferences [27, 77,94,95]. The generalized fuzzy sphere SDso(D+2) embedded in RD+1 has the matrixcoordinates{Xi} = {Σi D+2 ∈ so(D + 2)|i = 1, . . . , D + 1}. (5.1.1)Here Σµν satisfies the algebra1ı[Σµν ,Σσρ] = σνσΣµρ + σµρΣνσ − σνρΣµσ − σµσΣνρ. (5.1.2){Xi} are picked from so(D+2) uniquely up to conjugacy concerning full non-commutativity [Xi, Xj ] 6=0 ∀i 6= j. If we try to construct a D-dimensional sphere from other algebras, it is also clear thatso(n < D+2) does not have enough non-commutative generators, and so(m > D+2) can be writteninto the subalgebra so(D+2) in its reducible representation.As a solution of an emergent universe in the regulated bosonic IKKT matrix model, i.e., massiveYM matrix model, SDso(D+2) is studied in references [25,26,62,77]. By choosing a different signaturein so(D, 2), the sphere SDso(D+2) becomes the fuzzy hyperboloid SDso(D,2), and S4so(4,2) representsan expanding 3 + 1 universe with a Big Bang [26, 67]. Also, one can potentially introduce thestandard model in the so(4,2) universeis due to its large symmetry, where the fields in the matrixmodel are fluctuations around the matrix configuration. The fields on S4so(6) are studied in references[27,94,95], and in particular, reference [94] obtained the linearized Einstein equations by consideringthe emergent graviton fields.We consider the SO(D + 1) symmetric bosonic IKKT matrix model with a mass term, which is1In this chapter, we denote√−1 by ı, so ı is not confused with the index i.73treated as the IR regulator asSM = Tr(−14[Xµ, Xν ][Xµ, Xν ]− λ2(XµXµ − L2)), (5.1.3)where the dummy indices are summed with respect to the Euclidean signature δµν . The variationδSM/δXi gives the equation of motionδµν [Xµ, [Xν , Xκ]]− λXκ = 0, (5.1.4)which is satisfied by takingXµ =√λDΣµ D+2. (5.1.5)Here λ must be positive for Xµ to be Hermitian. This model has a scale L as the infrared cutoff,which restricts1NTr (XµXµ) = L2. (5.1.6)We can also view SM as the dimensionally reduced SU(N) super Yang-Mills model with a negativemass term −λ2XµXµ. The negative mass implies an instability, and negative eigenvalues appearconsequently in the second-order variation:δ2SMδXjδXj=∑i 6=j(Xi ⊗ 1N − 1N ⊗Xi)2 − λ1N2 . (5.1.7)The first term on the right-hand side contains a zero eigenvalue, so eq.(5.1.7) has a negative eigen-value when λ > 0, which is the condition of X being Hermitian, and therefore SDso(D+2) with theinfrared cut is not classically stable.Although SDso(D+2) is not a classical solution with a positive mass term, it can still be stabilized viaquantum effects. The one-loop effective action around the non-commutative sphere in the massiveIKKT model is given in reference [77], and one finds the supersymmetric breaking effect with apositive mass term, in which the SUSY breaking suppresses the bosonic fluctuations and leaves thefermionic fluctuations unaffected. Without the quantum effect, SDso(D+2) has zero radius becauseof the positive mass. Since the bosonic strings bounding the D0-branes are suppressed, the SUSYbreaking leads to expansion of the sphere until the balance is reached at larger radius. Eventuallythe fuzzy sphere with quantum fluctuations is stabilized at a physical radius, which depends on thegiven negative value of λ, see reference [77].The main object examined in this chapter is the simplest two-sphere S2so(4) constructed by so(4)generators. The collective coordinates are written as the product of two spin representations for74sul(2)⊗ sur(2), which is the double cover of so(4). Explicitly, we list the so(4) generators asFor i, j, k = 1, 2, 3,{Σij = ijk(Jkl ⊗ 1 + 1⊗ Jkr),Σi4 = Jil ⊗ 1− 1⊗ J ir.(5.1.8)To specify S2so(4) in the (Jl, Jr) representation, we change the notation to S2Jl,Jrfor short in thefollowing. The convenient basis consists of the orthonormal states|jl,ml〉 ⊗ |jr,mr〉 for mi = −ji, . . . , ji, (5.1.9)such that Σ12 and Σ34 are both diagonalized. su(2) generators acting on these states giveJ3i |ji,mi〉 = mi |ji,mi〉 , (5.1.10)J±i |ji,mi〉 =√(ji ∓mi)(ji ±mi + 1) |ji,mi ± 1〉 . (5.1.11)The collective coordinates of S2Jl,Jr are selected as{Xso(4)} = {Σ14,Σ24,Σ34}. (5.1.12)It is straightforward to computeTrΣ2µν =N3[Jl(Jl + 1) + Jr(Jr + 1)], where N = (2Jl + 1)(2Jr + 1), (5.1.13)and the constraint (5.1.6) gives the Lagrangian multiplier λ asλ =2L2Jl(Jl + 1) + Jr(Jr + 1). (5.1.14)Let us first try guessing what S2Jl,Jr could look like from Kirillov’s coadjoint method [32], seesection 2.1 for a brief review, and we can also gain some faith in S2Jl,Jr that it can be quantizedproperly. The coadjoint orbit means the orbit of a Lie group G acting on its coadjoint algebra(dual of the adjoint algebra) with the orbit forming a nice symplectic manifold [33]. Thinking ofthe coadjoint orbit as the phase space of a mechanical system, it can be quantized by a unitaryirreducible representation of the Lie group G. Consider the double cover SU(2)× SU(2) of SO(4),the coadjoint representation is equivalent to the adjoint reppresentation for any compact Lie group,so the method says that the coadjoint orbit can be quantized by the representation of SU(2)×SU(2).Furthermore, the partial coadjoint orbit2,eıφijΣijΣk4e−ıφijΣij = RlkΣl4, (5.1.15)2This is due to the fact that SO(d) generators are rotated like Lorentz bi-vectors in the action,eıφµνΣµνΣαβe−ıφµνΣµν = RµαRνβΣµν in which Rµν = exp(φµν ).75suggests {Σk4} is a SO(3) symmetric sub-region of the quantized coadjoint orbit. Since {Σij} inthe direct sum of SO(3) has the geometry of a stack of separated spheres, it is reasonable to expectthe sub-region {Σk4} to have a structure like layers of concentric spheres.The first method we applied on S2Jl,Jr constructs an effective Hamiltonian reflecting the leastlength of a string connected from an extra point probe to the collective D-branes, and the emergentsurface is mapped by scanning over the embedding background, see reference [22]. The effectiveHamiltonian takes the Dirac term in the BFSS model, since the Dirac mass reflects the length ofthe fermionic string. Considering D-branes distributed in a probability wavelet, the Dirac-probemethod searches for the zero Dirac mass when the probe is at the peak of probability, but we haveno knowledge of the region slightly away from the peak. A problem pointed out in reference [2]is that this method forbids the emergence of odd-dimensional spheres. The effective Hamiltonianin the even-dimensional background has one gamma matrix as the identity matrix3, see eq.(4.6.2),and the symmetry of the surface eigen-equation reduces to so(2d + 1) in the 2d + 2-dimensionalbackground, which is not enough for the existence of S2d+1.The second method we applied on S2Jl,Jr takes the matrix Laplacian acting on the connectingstring to be the effective Hamiltonian, similar to the Dirac-probe method, see reference [24]. Thespectrum reflects the displacement energy separating the probe from the target D-brane. When theprobe coincides with the target, one expects the displacement energy to be locally minimized. TheLaplace-probe method could have a problem of being ’too fuzzy’ [24], e.g., the point probe couldmix up two close emergent surfaces into one surface by failing to distinguish two local minima ofthe displacement energy. The tangled internal structure of S2Jl,Jr is too fuzzy for the Laplace probeand results in a different outcome comparing to the Dirac probe.To resolve the conflict, we applied the third method depicting the D-brane by the supergravitycharge distribution, where D-branes are considered as the source of the supergravity field in thelow energy regime, see references [46, 96]. This supergravity method developed in early researchon the matrix model is known to be invalid when it is applied to matrix configurations with finitedimensions. Reference [97] examined the fuzzy sphere with this method and found that the su-pergravity charge does not localize at fixed radius unless taking the infinite N limit of the matrixdimension. Unfortunately, we found that the third method applied on S2Jl,Jr is self-contradictoryand inconsistent with the other two probe methods.Conclusively, we showed that S2Jl,Jr is a thick non-commutative spherical shell4 consistently inthree methods if the modification according to the local symmetry is applied. Because S2Jl,Jr istoo fuzzy for its tangling layers, we probed a patch isolated with reference to the local azimuthalsymmetry by projecting modes with the same azimuthal angular momentum. S2Jl,Jr has stackingcoherent shells like an onion and incoherent margins between. If we label the conventional fuzzy3Irreducible representations (irreps) of the Clifford algebra cl(d+2) are direct sum of two irreps of cl(d+1) plus thechirality matrix σ3 ⊗ 1, and the embedding operator can be factorized into two pieces, each with a reduced gammamatrix ±1.4From the geometric quantization point of view, reference [27] obtained the geometry of S4so(6) in the thin limit.76sphere of S2so(3) in the spin-J representation by S2J , the coherent shells of S2Jl,Jrare quite similar to astack of fuzzy spheres ⊕Jl+Jrj=|Jl−Jr|S2j , and the decoherent margins reflect the distinction that the D0constituents of S2Jl,Jr spread across layers instead of being distributed at fixed radii like ⊕J+Kj=J−KS2j .The internal structure of S2Jl,Jr is tangled, complicated, and yet fruitful for future exploration.5.2 Embedding operatorIn the embedding background R3, the Dirac typed embedding operator [22] isE3(~x) ≡3∑i=1σi ⊗ (Xˆi − xi · 1N ), (5.2.1)where σ are Pauli matrices, and the emergent manifold is described by the surface equationdet (E3(~x)) = 0. Ed(~x)’s eigenvectors with the zero eigenvalues play the role of the general co-herent states [39] in Berezin symbol map [38]. We are interested in solving the general coherentstate |Λ(~x)〉 for the surface equation Ed(~x)|Λ(~x)〉 = 0.We now apply the embedding operator on S2Jl,Jr embedded in R3. Without lost of generality,we can restrict the probe on the x3-axis because of the spherical symmetry, and the surface eigen-equation at ~x = (0, 0, z) has a concise form,z|Λ〉 = (σ3σ1 ⊗ Σ14 + σ3σ2 ⊗ Σ24 + 1⊗ Σ34) |Λ〉=(J l3 ⊗ 1− 1⊗ Jr3 J l− ⊗ 1− 1⊗ Jr−−J l+ ⊗ 1 + 1⊗ Jr+ J l3 ⊗ 1− 1⊗ Jr3)|Λ〉. (5.2.2)It is straightforward to spot a zero state at the North pole (x = 0, y = 0, z = Jl−Jr) as the highestweight state,|Λ(x = 0, y = 0, z = Jl − Jr)〉 =(|Jl, Jl〉 ⊗ |Jr, Jr〉0). (5.2.3)However, the highest weight state is not the only eigenstate with the zero eigenvalue. Along theradial direction, we can find total 2Jr + 1 zerostates, and these are the only eigenstates with realeigenvalues z in (5.2.2) when Jl > Jr.When Jl = Jr, there are total 4Jr+2 degenerate zerostates with the zero eigenvalue z = 0, whichis the only real eigenvalue in the surface eigen-equation. Zero radius implies these degenerate statesare SO(3) invariant. The highest weight state is a known zerostate, so we can generate anotherzerostate as|Λ(θ, ~n)〉 = exp(ıθ~n · ~σ2)⊗ exp(ıθijkΣijnk)︸ ︷︷ ︸RSO(3)(|Jl, Jl〉 ⊗ |Jr, Jr〉0)(5.2.4)77from the rotation RSO(3) around the unit vector ~n by the angle θ. RSO(3) is in the representationspace of SU 12⊗ SUJl ⊗ SUJr , which can be decomposed as12⊗ Jl ⊗ Jr = ⊕Jrk=0Jl + Jr − k ±12. (5.2.5)The highest weight state is clearly in the spin Jl + Jr + 12 representation, which has dimensions2(Jl + Jr) + 2, so we can identify the 4J + 2 zerostates as the rotated highest weight state whenJl = Jr = J . Since S2Jl,Jr collapse when Jl = Jr, we only consider the case of Jl > Jr for the restparts.The embedding operator of S2Jl,Jr in the representation space of SU 12 ⊗ SUJl ⊗ SUJr is mostconveniently expressed in the representation diagonalizing the total azimuthal angular momentumJ totz =12σ3 ⊗ 1⊗ 1 + 1⊗ Jzl ⊗ 1 + 1⊗ 1⊗ Jzr (5.2.6)because of the azimuthal symmetry from restricting the point probe at the North pole. The az-imuthal angular momentum has values Jl + Jr − k ± 12 for k = 0, . . . , 2Jr. In the sub-Hilbert spacefor the collective coordinates, i.e. the representation space of SUJl ⊗ SUJr , we denote a state in thesubspace with the degenerate eigenvalue Jl + Jr − k with respect to Σ12 = Jzl ⊗ 1 + 1⊗ Jzr by|k,~a〉 =k∑i=0ai+1 |Jl, Jl − k + i〉 ⊗ |Jr, Jr − i〉 for k = 0, . . . , 2Jr. (5.2.7)We are interested in J totz ’s eigenstate|Λk〉 =(|k,~a〉|k − 1,~b〉)(5.2.8)such that J totz |Λk〉 =(Jl + Jr − k + 12) |Λk〉. For the sake of the azimuthal symmetry, we find thatthe zerostates of the embedding operator are the eigenstates |Λk〉 of J totz because(σ3σ1 ⊗ Σ14 + σ3σ2 ⊗ Σ24 + 1⊗ Σ34) |Λk〉 = zk|Λk〉 for k = 0, . . . , 2Jr. (5.2.9)We emthasize that {|Λk〉|∀k = 0, . . . , 2Jr} are the only eigenstates with real eigenvalues.In the eq.(5.2.9), the coefficients ~a and ~b are functions of z satisfying recursive formulasa0 = 0, a1 = 1,LnRn+1∆n−1 an+1 =(∆n − 2 + L2n∆n−1 +R2n∆n−3)an − Ln−1Rn∆n−3 an−1for n = 1, . . . , k + 1,(5.2.10)78andbm =Lmam −Rm+1am+1∆m − 1 for m = 1, . . . , k, (5.2.11)whereLn =√Jl(Jl + 1)− (Jl − k + n− 1)(Jl − k + n),Rn =√Jr(Jr + 1)− (Jr − n+ 1)(Jr − n+ 2),∆n = Jl − Jr + 2n− k − z.(5.2.12)Recursively building up coefficients an(z) and bn(z), one can solve zk as the only one real root in thepolynomial derived from (5.2.9). Unfortunately, the root zk does not have a closed form expression,and we would like to emphasize our two claims, 1. there are only 2Jr + 1 zerostates with a realeigenvalues in (5.2.2) and 2. only one real root zk exists, are based on the observation with exactcomputations for a given finite N.We find approximatelyRk ≈ Rapp = Jl − Jr1− kJl+Jrfor k = 0, . . . , 2Jr (5.2.13)based on our numerical data, see fig 5.1 and 5.2. The approximation generally fits the data wellfor the radii error Rapp−RkRk ≈ 1Jl+Jr , and it fits the inner shell better than the outer shell. We canvisualize the emergent radii better in two cases of the large N limit, one is the thin layers limitJl  Jr, and the other is the thick layers limit Jl ∝ Jr  1. In the thin layers limit, the radius ofeach coherent shell is approximatelyJl − Jr + k +O(1Jl). (5.2.14)In the thick layers limit, radii of inner shells at Jr  k are approximatelyJl − Jr + kJl − JrJl + Jr+O(1Jr), (5.2.15)and radii of outer shells at k  1 are approximatelyJl − Jr1− kJl+Jr(5.2.16)until the outermost shell at R2Jr ≈ Jl + Jr. Under both limits, we have reciprocal radii roughlydistributed from (Jl + Jr)−1 to (Jl − Jr)−1 evenly.2Jr + 1 real eigenvalues z indicate 2Jr + 1 emergent spherical shells because the eigenstatesrotated spherically satisfy the surface equation as well. We called these shells by coherent shellsto reflect that every point on the shell has a corresponding coherent state. Coherent shells are fuzzyand intercept with each other as shown in fig 5.3, the variance of z expectation 〈(Σ34 − z)2〉 coversthe gap between different layers of spheres, and the coherent states on different layers overlap one790 10 20 30 40 50 60 70 80k0.650.70.750.80.850.90.951Jl−JrRJl−JrRkJl−JrRapp-0.0041503*k+0.999940 10 20 30 40 50 60 70 80k10-510-410-310-2Rapp−RkRkFigure 5.1: The upper figure plots the reciprocal radius versus the shell number k in irrep (Jl, Jr) =(200, 40). We expect the emergent radius distributed from R0 = 160 to roughly R80 ≈ 240 with 81layers. 20 samples of Rk solved from the eigen-equation are presented in plus signs and Rapp in crosssigns. The dot line is the linear fit with respect to Rk, and it is fairly close to the approximation1− k240 . In the lower plot, we see the radial error (Rapp −Rk)/Rk increases toward outer shells.800 20 40 60 80 100k00.−JrRJl−JrRkJl−JrRapp-0.0098521*k+10 20 40 60 80 100k10-410-310-210-1100Rapp−RkRkFigure 5.2: The upper figure plots the reciprocal radius versus the shell number k in irrep (Jl, Jr) =(51, 50). We expect the emergent radius distributed from R0 = 1 to roughly R100 ≈ 101 with 101layers. 20 samples of Rk solved from the eigen-equation are presented in plus signs and Rapp in crosssigns. The dot line is the linear fit with respect to Rk, and it is fairly close to the approximation1 − k101 . In the lower plot, we see the radial error (Rapp − Rk)/Rk increases much dramaticallytoward outer shells comparing to fig 5.1.81another at different angles, shown in (5.4.21). To investigate the space between the layers, we definethe off-shell coherent state|r(k, k + 1)〉 = cos(ξ)|Λk〉+ sin(ξ)|Λk+1〉 (5.2.17)by interpolation between two adjacent coherent states in the radial direction, where the coefficientξ is resolved fromcos2(ξ)zk + sin2(ξ)zk+1 = z for z ∈ [zk, zk+1], (5.2.18)which guarantees 〈r(k, k + 1)|Σ34|r(k, k + 1)〉 = z. 〈r(k, k + 1)|Σ14|r(k, k + 1)〉 = 〈r(k, k +1)|Σ24|r(k, k + 1)〉 = 0 follows automatically. Thus, we have a smooth symbol map within theincoherent margins between coherent shells.0 10 20 30 40 50 60 70 80k160170180190200210220230240250radiusFigure 5.3: Radii versus shell number k in irrep (Jl, Jr) = (200, 40). The cross sign indicates theemergent radius, and the error bar refers to the variances 〈Λk|(Σ34 − zk)2|Λk〉. We find that thevariance as the thickness of most shells covers incoherent margins, and coherent shells interceptwith each other.To analyze S2Jl,Jr in the large N limit, we need an analytic approximation beyond the numericalobservation. This can be done for both the eigenvalues and the eigenvectors in the thin layers limitJl  Jr. The surface eigen-equation(5.2.9) can be separated into two parts as(J l3 + Jr3 Jl− − Jr−−J l+ + Jr+ −J l3 − Jr3)︸ ︷︷ ︸S0−(2Jr3 00 −2Jr3)︸ ︷︷ ︸S1 |Λk〉 = zk|Λk〉. (5.2.19)82By normalizing the radius to 1, the small factor JrJl−Jr+k out of S1 indicates that S1 is a small pertur-bation compared with S0, so zk and the corresponding eigenstate can be computed approximatelywith the standard perturbation theory.On the k-th shell, we denote the zeroth-order zerostate|Λ0k〉 =(|k,~c〉0)such that S0|Λ0k〉 = (Jl + Jr − k)|Λ0k〉. (5.2.20)It is straightforward to derive|k,~c〉 = 1Ak∑i=0√√√√( ki)F (i)|Jl − k + i, Jr − i〉, (5.2.21)whereF (i) =(2Jl − k + i)!(2Jl − k)!(2Jr − i)!2Jr!and A =√√√√ k∑i=0(ki)F (i). (5.2.22)Referring to a lemma of Gamma function, limn→∞(n+α)!n!nα = 1, we havelimJl−k→∞Jr→∞F (i) = limJl−k→∞Jr→∞(2Jl − k + i)!(2Jl − k)!(2Jr − i)!2Jr!=(2Jl − k2Jr)i. (5.2.23)Hence, we can recognize the coefficients in the large N limit aslimJl−k→∞Jr→∞|k,~c〉 =k∑i=0√√√√( ki)P i(1− P )k−i|Jl − k + i, Jr − i〉 (5.2.24)for P = Jl−k/2Jl+Jr−k/2 .Taking the resolved coefficients as the probability amplitudes, the probability distribution isequivalent to the binomial distribution such that we have the probability P to subtract from mrand the probability 1−P to subtract from ml, and(ki)P i(1−P )k−i is the chance ending up at(ml = Jl − (k − i),mr = Jr − i) over k times subtraction from (ml = Jl,mr = Jr). This binomialdistribution has the mode b(k + 1)P c with the variance σ2 = kP (1 − P ). Accordingly, |k,~c〉 isdominated by the peak state|Jl − k + im, Jr − im〉 for im = b(k + 1)P c . (5.2.25)83With resolved ~c, we find the first-order approximate emergent radius in the large N limit aslimJl−k→∞Jr→∞〈Λ0k|S0 + S1|Λ0k〉= Jl − Jr − k + 2k∑i=0(ki)iP i(1− P )k−i= Jl − Jr + Jl−Jr−k/2Jl+Jr−k/2k.(5.2.26)To compare with our observationJl − Jr1− kJl+Jr= (Jl − Jr)∞∑n=0(kJl + Jr)n(5.2.27)we haveJl − Jr + Jl−Jr−k/2Jl+Jr−k/2k= Jl − Jr + (Jl − Jr) kJl+Jr − Jr(kJl+Jr)2+O((kJl+Jr)3),(5.2.28)and two expansions agree with each up to the first-order of kJl+Jr .The perturbation method also provides an excellent estimation that when we split the 2N ze-rostate into the upper and the lower half parts, the norm of the upper part be about Jl/Jr timeslarger than the norm of the lower part, so the proper coherent state in the subspace of collectivecoordinates is the upper half of the zerostate from the embedding operator.5.3 Laplace operatorLaplace operator in reference [24] is defined asL =∑i(Xˆi − xi)2. (5.3.1)The coherent state used in the symbol map is taken to be the ground state of the Laplace operatorat the emergent point. Physically, the expectation of the Laplace operator measures the varianceof emergent point. While the eigenvalues are functions of ~x, the emergent surface point is locatedat where the ground energy reaches its global minimum along the line of the probe passing throughthe emergent surface. If there are multiple n layers, we need n coherent states from n eigenstatesof L, and each layer is located at where the corresponding eigenvalue reaches its local minimum.It is unsolved that how many eigenstates shall be considered for the given collective coordinates{X} before knowing the number of layers. Another difficulty is to track the minimum emergentenergy of each layer along the probe’s trajectory ~x while we do not know how to correspond layersto eigenstates at different probe locations. For example, two eigenvalues of L at different loci canbe the emergent energies of two separated layers, and the local minimum of the lowest eigenvalue84does not reflect the emergent of a layer if this eigenvalue corresponds to another layer. As shownin the following, only the ground state is not enough to describe S2Jl,Jr , and we find the eigenstatescorresponding to different shells at different radii.Applying on S2Jl,Jr , the Laplace operator at ~x = (0, 0, z) is given byL(z) = Σ214 + Σ224 + (Σ34 − z)2= 2 (Jl(Jl + 1) + Jr(Jr + 1)) + z2 − 2zΣ34 − Lˆ2,(5.3.2)where Lˆ2 = Σ223 + Σ231 + Σ212. Numerically, it is straightforward to compute the lowest eigenvalueof L(z) for any given z as shown in fig.5.4. Rather than wavy curve showing layers of S2Jl,Jr , thelowest eigenvalues distribute like a deep well with bottom from Jl−Jr to Jl +Jr roughly, and thereare no clear hierarchy between the second lowest eigenvalue to the lowest one to be the indicationof emergent surface introduced in reference [24].20 40 60 80 100 120 140 160 180radius0200400600800100012001400160018002000eigvallowest eigval2nd lowest eigvalFigure 5.4: This data took so(4) generators in irrep (Jl, Jr) = (100, 40) for the lowest two eigenvaluesof the Laplace operator at different given radii.Because it is hard to identify the emergent energy of each layer, we need to analyze more eigen-states of the Laplace operator at one fixed location. The goal is to associate the emergent energyof one shell to certain eigenvalues varying the probe location. Since L(z) commutes with Σ12, theconvenient basis is the eigenstates of Σ12, and we can express the eigenstates of L(z) in the form ofeq.(5.2.7). The eigen-equation of L(z) is written asL(z)|k,~a,±〉 = λk|k,~a,±〉, (5.3.3)85where|k,~a,+〉 = ∑min{k,2Jr}i=0 ai+1 |Jl, Jl − k + i〉 ⊗ |Jr, Jr − i〉 ,|k,~a,−〉 = ∑min{k,2Jr}i=0 ai+1 |Jl,−(Jl − k + i)〉 ⊗ |Jr,−(Jr − i)〉 . (5.3.4)The coefficients of |k,~a,±〉 are functions of z determined by the following recursive relations.a±0 = 0, a±1 = 1,D−Jl−k+n,Jr−na±n+1 =(C±(Jl−k+n−1,Jr−n+1) − λ±k)a±n −D+Jl−k+n−2,Jr−n+2a±n−1for n = 1, . . . ,min{k, 2Jr}+ 1,(5.3.5)whereD±ml,mr =√(Jl(Jl + 1)−ml(ml ± 1)) (Jr(Jr + 1)−mr(mr ∓ 1))Cml,mr = Jl(Jl + 1) + Jr(Jr + 1) + z2 − 2mlmr − 2z(ml −mr).(5.3.6)For the equations at n, we have a±n in the (n − 1)th-order polynomial of λk(z), and λk(z) issolved from the (min{k, 2Jr} + 1)th-order polynomial with respect to the boundary conditionD−Jl−k+n,Jr−n = 0.[Σ12, L(z)] = 0 represents the axial symmetry around the zˆ-axis. Along the zˆ-axis, the coherentstate of each shell has a fixed azimuthal angular momentum measured by Σ12. When the probeapproaches to the nearest shell, one of the eigenstates of L(z) shall converge to the correspondingcoherent state. It is reasonable to assume that eigenstate has the same angular momentum asthe nearby coherent states, and therefore we conclude the coherent state with angular momentumJl + Jr − k is the eigenstate |k,~a,±〉 at z = zk when λk(zk) is minimized.The next step is to analyze the extremum of all eigenvalues from (2Jl+1)(2Jr +1) eigenstates ofL(z) for k = 0 . . . Jl + Jr. We consider Jl 6= Jr and assume Jl > Jr without lost of generality. Withrespect to the number of roots solved in the polynomial of λk(z), eigenstates of L(z) are dividedinto the following three types.type I {|k,~ai,±〉|i = 1, . . . , k + 1} k = 0, . . . , 2Jr,type II {|k,~ai,±〉|i = 1, . . . , 2Jr + 1} k = 2Jr + 1, . . . , Jl + Jr − 1,type III {|k,~ai〉|i = 1, . . . , 2Jr + 1} k = Jl + Jr.(5.3.7)The corresponding eigenvalues are denoted by λ±k,i for the state |k,~ai,±〉. The number of the indexi reflects the specified root solved from the polynomial of λk with min{k, 2Jr}+ 1 roots. The typeIII eigenstates simply label the index reversely comparing different ± signs, so they are not specifiedby ± signs. At fixed k, we label the index i such thatλ+k,1(zmink,1 ) ≤ λ+k,2(zmink,2 ) ≤ · · · ≤ λ+k,min{k,2Jr}+1(zmink,min{k,2Jr}+1), (5.3.8)where zmink,i labels the stationary point for λ+k,i. There are 2∑2Jrk=0(k + 1) independent type Ieigenstates, 2∑Jl+Jr−1k=2Jr+1(2Jr + 1) independent type II eigenstates, and 2Jr + 1 independent typeIII eigenstates, and the total number of eigenstates are exactly (2Jl + 1)(2Jr + 1). Because86Cml,mr(z) = C−(ml,mr)(−z), we have~a+i (z) = ~a−i (−z) and λ+k,i(z) = λ−k,i(−z) in type I,II, (5.3.9)andλk,i(z) = λk,i(−z) in type III. (5.3.10)This means we need to consider |k,~ai,+〉 and |k,~ai,−〉 as the coherent states at the North pole andthe South pole respectively. We find only one extremum appears as the global minimum in typeI,II λ±k,i.In the shell k = 0, it is easy to write out the closed form solution for any given so(4) irrep (Jl, Jr).|k = 0, a = 1,±〉 = |Jl,±Jl〉 ⊗ |Jr,±Jr〉 (5.3.11)λ±0 (z) = (Jl − Jr ∓ z)2 + Jl + Jr (5.3.12)Comparing to the embedding operator, we have the same emergent sphere with radius Jl − Jr atthe innermost shell, and the eigenstate is parallel to the corresponding zero state. Similarly, weidentify another 2Jr outer shells from the i = 1 lowest eigenstates in the type I. These 2Jr type Ieigenstates are the only eigenstates with eigenvalues smaller than Jl + Jr,Jl + Jr = λ+0,1(zmin0,1 ) > λ+1,1(zmin1,1 ) > · · · > λ+2Jr,1(zmin2Jr,1), (5.3.13)and their minima are acquired at larger radii,Jl − Jr = zmin0,1 < zmin1,1 < · · · < zmin2Jr,1. (5.3.14)We identify thatzmink=1...2Jr,i=1 (5.3.15)are radii of emergent spherical shells from the Laplace operator, and these radii are in close agree-ment with the embedding operator, see fig 5.5. As the rest of eigenstates including type I with i 6= 1,type II, and type III, they all have eigenvalues significantly larger than Jl−Jr, the largest eigenvalueamong type I with i = 1. In conclusion, by restricting the total azimuthal angular momentum, typeI i = 1 eigenstates, |k,~a1,±〉, are identified as the coherent states corresponding to 2Jr + 1 coherentshells.Let us compare three states|ΛDk 〉 =(|k,~a〉|k − 1,~b〉), |Λappk 〉 =(|k, ~d〉0), |ΛLk 〉 = |k,~c〉 (5.3.16)as the eigenstate of Dirac operator, the approximate eigenstate of Dirac operator, and the coherent87150 160 170 180 190 200 210 220 230 240radius160180200220240260280300320340160 170 180 190 200 210 220 230radius00. 5.5: In so(4) irrep (Jl, Jr) = (200, 40), we present 6 samples at k = {0, 13, 27, 40, 54, 67}shells. For the upper plot, we draw black vertical lines for the Dirac radii, plus signs for the Laplaceradii, and dash lines for λk,1(z). For the lower plot, we present ζ =RLk−RDkRDk+1−RDk−1showing the differenceof Laplace and Dirac radius over the distance between two adjacent shells.88state of Laplace operator respectively. Based on the perturbation method in the thin layer limitJl  Jr, |k − 1,~b〉 is a small correction adding to |k,~a〉 for the ratio 〈k,~a|k,~a〉/〈k − 1,~b|k − 1,~b〉 ≈Jl/Jr, so we can simply take |k,~a〉 to be the coherent state for the Dirac operator, and |k, ~d〉 to bethe approximate coherent state. We expect all three coherent states converge to each other in thelarge N thin layers limit, so we have the consistent symbol map with respect to both point probes.In the following, we present the data of normalized |ΛDk 〉 and |ΛLk 〉 at (Jl, Jr) = (200, 40) for selectedshells k = 16, 32, 48, 64. The fig 5.6 plots the magnitude of coefficients versus their index at k = 16.We neglect the detail coefficients on other shells since they give consistent results comparing to thedata presented. It is clear that ~a and ~c and ~d are in close agreement with each other while ~b ismuch smaller by the factor Jr/Jl = 1/5. Taking a parallelogram constructed by two vectors ~a and~c in Rk+1, we can quantify the difference between these two unit vectors precisely by the area ofthe parallelogram. Fig 5.7 plots the parallelogram area on selected shells. The parallelogram areagrows from inner to outer shells, and it also indicates the lower half part of Dirac’s zerostate is lessnegligible at outer shells.0 200 400 600 800 1000 1200 1400index00. coefficientsFigure 5.6: Coefficients of coherent states at k = 16 shell in irrep (Jl, Jr) = (200, 40); coefficients atlarger indices are zeros and not presented; non-zero coefficients are found only at states respectingthe total azimuthal momentum= Jl + Jr − k. ~a denotes Dirac’s upper half zerostate, ~b denotesDirac’s lower half zerostate, ~c denotes Laplace’s coherent state, and ~d denotes the approximatezerostate with perturbation method.890 10 20 30 40 50 60 70k00.0020.0040.0060.0080.010.0120.014parallelogramFigure 5.7: Parallelogram area between ~a and ~c in irrep (Jl, Jr) = (200, 40); ~a and ~c agree eachother better at inner shells.5.4 Supergravity charge densityTo demonstrate the supergravity charge density formula [46,96], we first briefly review reference [97]for the derivation of familiar fuzzy sphere radius. Because of the D-brane democracy, the chargedensity in the large N limit is consistent either treating the fuzzy sphere as D0 or D2 branes. Thenwe apply the same computations on S2Jl,Jr . However, the computation concerning D2 branes doesnot agree with the one concerning D0 branes nor previous results from point probe methods. Thesolution is to isolate the block in the matrices of S2Jl,Jr with respect to constant total azimuthalangular momentum, and it is in agreement with our conjecture that each spherical shell has a fixedtotal azimuthal angular momentum.5.4.1 so(3) sphere S2JFollowing reference [97], we review the derivation of S2J ’s radius using supergravity charge densityformula concerning D0 or D2 branes.D0 charges Given a set of collective coordinates { ~X} for a group of D0 branes, the charge densityis described by the following Fourier transformation,D(~x) =1(2pi)3∫R3d3k e−ı~k·~xD˜(~k), where D˜(~k) = Tr(eı~k· ~X). (5.4.1)The conventional fuzzy sphere S2J is a good example to demonstrate how this formula works. First,we align the direction of ~k with x3 by a conjugation on S2J , so we haveTr(eı~k· ~X)= Tr(eıkCL3). (5.4.2)90Without lost of generality, we take Li from spin-J representation of su(2) for half-integer J, and wehaveTr(eıkCL3)=J∑m=−JeıkCm =J∑m=1/2eıkCm + e−ıkCm. (5.4.3)The point is to apply this lemma,∫R3d3xeı~k·~xF (|~x|, a) = eıak + e−ıak for F (r, a) = −12pia(1aδ(r − a) + ∂∂rδ(r − a)), (5.4.4)and therefore we findD(~x) =J∑m=1/21(2pi)3∫R3d3k eıkCm + e−ıkCm =J∑m=1/2F (|~x|, Cm). (5.4.5)The next step is to take the large N limit, and we see the D0 charges distribute evenly on a spherewith radius ρ = CJ as expected. Denote y = ρmJ and limJ→∞∑Jm=1/2ρJ =∫ ρ0 dy, we havelimJ→∞D(~x) = Jρ∫ ρ0 dyF (x, yCJρ )= −ρ22piC3J2∫ ρ0 dy1y2δ( ρCJ x− y)− 1y ∂∂y δ( ρCJ x− y)= ρ22piC3J21y δ(ρCJ x− y)|ρ0.(5.4.6)We ignore the y = 0 term because it does not contribute to the total charge from integrating overthe space R3. Finally, we havelimJ→∞D(~x) =ρ2piC3J2δ(ρCJx− ρ) = N4piρ2δ(x− ρ), (5.4.7)where ρ = CJ and N = 2J as the total number of D0 branes.D2 charges Given a set of collective coordinates { ~X} for a group of D2 branes, the charge densityis described by the following Fourier transformation,Dij(~x) =1(2pi)3∫R3d3k e−ı~k·~xD˜ij(~k), where D˜ij(~k) = Tr(−ı [Xi, Xj ] eı~k· ~X). (5.4.8)Parallel to the discussion on D0 charges, we demonstrate the formula with S2J again because thoseD0 branes spreading around can be considered as one spherical D2 brane in the large N limit. First,we bring the direction of ~k to be along x3 by a conjugation, R−1SO(3)LiRSO(3) = RjiLj for the SO(3)91rotation matrix RSO(3), and deriveTr−ı [Xi, Xj ]︸ ︷︷ ︸C2ijlLleı~k· ~X = C2ijkRkl Tr(LleıkiRjiCLj)= C2k ijlklTr(L3eıkCL3)= C2k ijlkl∑Jm=1/2m(eıkCm − e−ıkCm) .(5.4.9)The point is to apply this lemma∫R3d3xeı~k·~xFij(|~x|, a) = ijl klk(eıak − e−ıak)for Fij(~x, a) =−ijl2piaxlx∂∂xδ(x− a), (5.4.10)and therefore we findDij(~x) =−C2piijlxlxJ∑m=1/2∂∂xδ(x− Cm). (5.4.11)The next step is to take the large J limit, and we see the D2 brane spread around a sphere withradius ρ = CJ as expected. Denote y = ρmJ and limJ→∞∑Jm=1/2ρJ =∫ ρ0 dy, we havelimJ→∞Dij(~x) =Jρ∫ ρ0 dy−C2pi ijlxlx∂∂xδ(x− yCJρ )= 12pi ijlxlx∫ CJ0 dy′ ∂∂y′ δ(x− y′)= 12pi ijlxlx δ(x− y′)|y′=CJy′=0 .(5.4.12)Once again, we ignore the y′ = 0 term because it does not contribute in the total charge fromintegrating Dij(~x) over R3. Finally, we found a spherical D2 brane with radius CJ aslimJ→∞Dij(~x) =12piijlxlxδ(x− CJ). (5.4.13)5.4.2 so(4) sphere S2Jl,JrD0 charges Repeat the same procedure on S2Jl,Jr , we show the D0 brane charges distributing overa thick spherical shell. In the so(4) irrep (Jl, Jr), we have the collective coordinates asXi = CΣi4 = C(J li ⊗ 1− 1⊗ Jri). (5.4.14)Similarly, we first align ~k in the x3 direction with the SO(3) subsymmetry in SO(4). so(4) generatorsare Lorentz bi-vectors WΣµνW−1 = RαµRβνΣαβ , so we can find conjugation WΣi4W−1 = RjiΣj4 forRjiki = δj3. Thus, we have the Fourier modes at ~k asD˜(~k) = Tr(eıkCΣ34)= Tr(eıkCΣ12). (5.4.15)92In the second equality above, we applied the SO(4) symmetry to replace Σ34 by Σ12. Because ofthe branching rule Σ12 = J l3 + Jr3 = ⊕Jl+Jrj=Jl−JrJ3 for Ji in the spin j = Jl − Jr, . . . , Jl + Jr irrep ofsu(2), we haveTr(eıkCΣ12)=Jl+Jr∑j=Jl−Jrj∑m=−jeıkCm. (5.4.16)For the simplicity again, we assume Jl− Jr to Jl + Jr are half-integers to avoid m = 0 terms. Afterthe inverse Fourier transformation with lemma (5.4.4), we findD(~x) =Jl+Jr∑j=Jl−Jrj∑m=1/21(2pi)3∫R3d3ke−ı~k·~r(eıkCm + e−ıkCm)=Jl+Jr∑j=Jl−Jrj∑m=1/2F (|~x|, Cm).(5.4.17)Rewrite Jl − Jr = J and Jl + Jr = ξJ for ξ > 1, we factorize F (|~x|, Cm) = 1C3J3F ( xCJ , mJ ). Inthe large J limit, by denoting z = mJ , y =jJ , and limJ→∞∑ξJj=J∑jm=1/21J2=∫ ξ1 dy∫ y0 dz, we havelimJ→∞D(~x) = limJ→∞ξJ∑j=Jj∑m=1/21C3J3F ( xCJ ,mJ )= −12piC3J∫ ξ1 dy∫ y0 dz1z2δ( xCJ − z)− 1z ∂∂z δ( xCJ − z)= 12piC3J∫ ξ1 dy1z δ(xCJ − z)|y0.(5.4.18)The z = 0 term is again ignored. We concludelimJ→∞D(~x) ={12piC21x for 1 ≤ xCJ ≤ ξ,0 otherwise.(5.4.19)S2Jl,Jr is like a thick spherical shell of D0 charges12piC21x distributing from x = C(Jl − Jr) tox = C(Jl + Jr).S2Jl,Jr has the same density distribution as stacked S2J {Xi = ⊕Jl+Jrj=Jl−JrJi}. In the finite J scenario,these stacked fuzzy spheres with radii Rj = Cj have constant gap ∆Rj = Rj+1−Rj = C, and eachshell has Nj = 2j + 1 D0 branes, so we see density D(Rj) = 12piC21Rjis consistent inNj = D(Rj)4piR2j∆Rj . (5.4.20)Even though the density profile is the same, S2Jl,Jr could have different number of D0 charges pershell and different gap ∆Rj increasing with j.To compare with the point probe methods, we can find the number ofD0 charges on each coherentshell by the rank of the collection of coherent states. Based on the North pole eigenstate of theLaplace operator, we have the other eigenstate around the kth shell as RSO(3)|k,~ai,±〉 by a rotationmatrix RSO(3)(nˆ, θ) = exp(ıθijknˆiΣjk)for a point rotated from the North pole with respect to the93axis nˆ by the angle θ. RSO(3)(nˆ, θ) is in the direct sum of spin j = Jl− Jr, . . . , Jl + Jr irrep of su(2)because of the branching rule so(4)→ so(3). We find i = 1 type I state on the kth shell consists ofstates in irreps of j = Jl +Jr, . . . , Jl +Jr−k, and therefore the collection of i = 1 type I eigenstatesover the kth shell hasrank(k) =Jl+Jr∑j=Jl+Jr−k(2j + 1) = (k + 1) (2(Jl + Jr) + 1− k) . (5.4.21)This observation also indicates the coherent state on the kth shell overlaps with other coherentstates on every outer shells. On the other hand, the Dirac operator hasJl+Jr+12∑j=Jl+Jr−k+ 12(2j + 1) = (k + 1) (2(Jl + Jr) + 2− k) (5.4.22)linear independent zero states. There are more linear independent zero states around the shellbecause of the Hilbert space doubled by the σ matrices. However, if we consider only half of theHilbert space, the Dirac operator sees the same (k+1) (2(Jl + Jr) + 1− k) linear independent statesaround the kth shell just like the Laplace operator. Hence the point probe methods identify thenumber of D0 branes but fail to detect the density profile. The summation of the ranks from allshells is higher than the total number (2Jl + 1)(2Jr + 1) of D0 branes because a point like D0 braneappears on separated shells. We found charges in spin j = Jl + Jr − k irrep of su(2) extending fromthe kth shell to the outermost shell. Because of the gap between shells, we don’t consider them asextended D1 branes, which shall have one commutative dimension.D2 charges Now we apply the charge density formula considering S2Jl,Jr as D2 branes. Similarly,we first align ~k in the x3 direction by the SO(3) subsymmetry in SO(4). S2Jl,Jr has the chargedensity in the momentum space at ~k asTr−ı [Xi, Xj ]︸ ︷︷ ︸C2Σijeı~k· ~X = C2Rmi Rnj Tr (ΣmneıkCΣ34)= C2k ijlkl Tr(Σ12eıkCΣ34),(5.4.23)where in the second equality, we applied Tr(Σ23eıkCΣ34)= Tr(Σ31eıkCΣ34)= 0 and R1iR2j−R2iR1j =ijlkl/k. It is unexpected that S2Jl,Jr does not appear the same as a stack of S2J , which have Fouriermodes as C2k ijlkl Tr(Σ12eıkCΣ12), even though they have the same D0 charge distribution.94Apply the lemma (5.4.10) for a = ml ±mr onTr(Σ12eıkCΣ34)=Jl∑ml=−JlJr∑mr=−Jr(ml +mr)eıkC(ml−mr)=Jl∑ml=1/2Jr∑mr=1/2(ml +mr)(eıkC(ml−mr) − e−ıkC(ml−mr))+ (ml −mr)(eıkC(ml+mr) − e−ıkC(ml+mr)) ,(5.4.24)we find that the inverse Fourier transformation givesDij(~x) =−C2pi ijlxlxml=Jlmr=Jr∑ml,mr=1/2ml±mr 6=0ml+mrml−mr∂∂xδ (x− C(ml −mr))+ ml−mrml+mr∂∂xδ (x− C(ml +mr)) .(5.4.25)We keep assuming Jl, Jr are half-integers for consistency. If we compute a stack of S2J instead, wefindDij(~x) =−C2piijlxlxJl+Jr∑J=Jl−JrJ∑m=1/2∂∂xδ(x− Cm), (5.4.26)and emergent shells with radii CJ for J = Jl − Jr, . . . , Jl + Jr in the large N limit.In the spirit of the Dbranes democracy, the D2 charge distribution shall agree with the D0charge distribution if the given matrices represent surfaces condensed by D0 charges. However, wedo not see such agreement here. The continuum limit of the supergravity charge formula is notvery rigorous, and we get different results if we take the large N limit before the inverse Fouriertransformation. Despite the existence of the limitation, we already see differences in Fourier modesof D2 and D0 charges for S2Jl,Jr . The Fourier modes of D0 charges is the same for S2Jl,Jrand a stackof S2J , D˜(k) = Tr[eıkCΣ12]. The Fourier modes of D2 charges for S2Jl,Jr is D˜ij(k) ∝ Tr[Σ12eıkCΣ34],but a stack of S2J have D2 charges D˜ij(k) ∝ Tr[Σ12eıkCΣ12]. There is no agreement between thetwo.5.4.3 D2 charge density with fixed angular momentumIn the last section, we find that the D2 charges distribution computed naively from the supergravityformula disagree with any other approaches. Hence, we propose to compute the blocks of S2Jl,Jr sep-arately in the momentum space, and we find that the combination of blocks in the coordinate spacegives the consistent result of spherical shells. Each block consisted of states restrictively with thesame azimuthal angular momentum represents each individual shell. It is a similar idea like probingS2Jl,Jr with the Laplace operator, the overlapped stacking layers of S2Jl,Jrare inseparable unless eachblock of constant azimuthal angular momentum is probed separably. The critical difference is thatwe compute the inverse Fourier transformation of each shell rather than all shells together. This95Fourier transformation is an improper integral giving Cauchy principle values, and summing prin-cipal values from improper integrals is different from the principal value of the improper integral ofthe sum.First, by choosing the convenient representation basis, we compute D2 charges in the momentumspace by Tr[Σ34eıkcΣ12]rather than Tr[Σ12eıkcΣ34]in (5.4.23). To compute the M -th shell withthe azimuthal angular momentum ml +mr = Jl +Jr−M for M = 0, . . . , 2Jr, we restrictively tracethe block with states satisfying ml +mr = Jl + Jr −M asRTr[Σ34eıkcΣ12]=∑ml,mrml+mr=Jl+Jr−M(ml −mr) eıkc(ml+mr)= (Jl − Jr) (M + 1) eıkc(Jl+Jr−M).(5.4.27)The M -th shell’s D2 charges in the momentum space becomesD˜ij(~k,M) = c2 (Jl − Jr) (M + 1) ijl klkeıkc(Jl+Jr−M). (5.4.28)The inverse Fourier transformation5 follows by1(2pi)3∫R3d3ke−ı~k·~x klk eıkρ= ı(2pi)3ddxl∫R3d3ke−ı~k·~x eıkρk= 1(2pi)3ddxl(4piır2−ρ2 − 2pi2ρ δ(r − ρ)),where ρ = c(Jl + Jr −M).(5.4.29)The M-th shell’s D2 charge distribution is the real part,Dij(~x,M) = −c2 (Jl − Jr) (M + 1)4piijl1ρddxlδ(r − ρ). (5.4.30)It is clear that the imaginary part of the charge distribution in the position space shall vanish oncewe consider all shells together since the previous result concerning all shells in (5.4.25) is purelyreal, so we can simply extract the real part to be the result.The next step is to add up all shells in the large N limit. By denoting z = MJ , Jr = J , andJl = ξJ for ξ > 1, we replace the summation by the integration limJ→∞∑2JrM=01J =∫ 20 dz and5The detail computation of the inverse Fourier transformation is provided in the appendix.96foundlimJ→∞2Jr∑M=0Dij(~x,M)= limJ→∞− cijlxl4pir2Jr∑M=0(Jl−Jr)(M+1)(Jl+Jr−M)ddrδ(r − ρ)= limJ→∞− cJijlxl4pir2∫0dz (ξ−1)(zJ+1)ξ+1−zddrδ(r − ρ)= − ijlxl4pir limJ→∞2∫0dz (ξ−1)(zJ+1)ξ+1−zddz δ(r − cJ(ξ + 1) + cJz)=ijlxl4pir limJ→∞(cJ2 (ξ2−1)r2HcJ(ξ+1)cJ(ξ−1) − 2Jδ (r − cJ(ξ − 1))).(5.4.31)This is a result consistent with point probe methods for D2 branes distributing from the innermostshell Jl − Jr to the outermost shell Jl + Jr with charge density cJ2l −J2rr2, which decays faster thanthe D0 charges in the same range, see (5.4.18).If we carefully examine the computation in (5.4.31), the integration by parts substitution isequivalent tolimJ→∞2J∑M=0f(M) ddM δ(r − ρ)= limJ→∞f(M)δ(r − ρ)|M=2JM=0 −2J∑M=0df(M)dM δ(r − ρ).(5.4.32)Therefore, we can rewrite the D2 charge distribution in the large N limit aslimJ→∞2Jr∑M=0Dij(~x,M)=cijlxl4pir(2Jr∑M=0(Jl−Jr)(Jl+Jr+1)(Jl−Jr+M)2 δ (r − (Jl − Jr +M))− (Jl−Jr)(2Jr+1)Jl+Jr δ (r − (Jl + Jr)) + δ (r − (Jl − Jr))),(5.4.33)and it reveals layers of D2 branes at radii Jl − Jr +M in agreement with the result concerning D0branes.5.5 Appendix5.5.1 Fourier transform of the Green’s function of the Helmholtz equationThe supergravity D2 charge density of one shell of S2Jl,Jr has the Fourier modes like the termklk eıak in the momentum space. In the following, we examined the details of computing Fouriertransformation back and forth.Referring to the well-known Fourier transformation of the Green’s function of the Helmholtz97equation, in references [98,99], the inverse Fourier transformation follows by1(2pi)3∫R3d3ke−ı~k·~x klk eıkρ= ı(2pi)3ddxl∫R3d3ke−ı~k·~x eıkρk= ı(2pi)3ddxl(4pir2−ρ2)PV= 1(2pi)3ddxl(4piır2−ρ2 − 2pi2ρ δ(r − ρ)), where ρ = c(Jl + Jr −M).(5.5.1)The derivative and the integral are exchangeable because the singularity has no xl dependence. PVindicates the Cauchy principle value in the bracket.If we switch the position and momentum spaces, the computation of∫R3d3ke−ı~k·~x eıkak is parallelto the Fourier transformation of the Green’s function of the Helmholtz equation, which is the spatialpart of the well-known spherical wave.(∇2 + a2)ΦS(~x) = −δ(~x), where ΦS(~x) = eıar4pir. (5.5.2)Rewrite ΦS(~x) and δ(~x) in Fourier series,(∇2 + a2) eıar4pir=1(2pi)3∫R3d3keı~k·~x (−k2 + a2) Φ˜S(~k), (5.5.3)δ(~x) =1(2pi)3∫R3d3keı~k·~x, (5.5.4)the Helmholtz equation suggests the Fourier mode of the spherical wave to beΦ˜S(~k) =1k2 − a2 . (5.5.5)To recover the spherical wave rigorously, 1(2pi)3∫R3d3keı~k·~xΦ˜S(~k) is an improper integral computeddirectly with the residual theorem, see reference [98]. The common trick dealing the singularity atk = a is to set a→ a± ı by an infinitesimal  and take the Cauchy principle value to be the integralresult.1(2pi)3∫R3d3k eı~k·~xk2−a2 =1(2pi)2ır∞∫0dk(eıkr − e−ıkr) kk2−a2= 1(2pi)2ır∞∫−∞dkeıkr kk2−a2= 1(2pi)2ırlim→0+∮γ dkeıkr kk2−(a+ı)2= 1(2pi)2ırlim→0+2piı Resk→a+ı(keıkrk2−(a+ı)2)= eıar4pir ,(5.5.6)where the closed integral path γ is defined as the real axis joining the infinite upper half circle, sothe only enclosed residue is at k = a + ı. On the other hand, sending a → a − ı results in e−ıar4pir .Finally, we learned the computation of∫R3d3ke−ı~k·~x eıkak from the inverse Fourier transformation of98the spherical wave ∫R3d3xe−ı~k·~xeıar4pir= lim→0+1k2 − (a+ ı)2 . (5.5.7)If one is not happy about the ı appearing in the Fourier modes, another expression in terms ofdistributions is available in reference [99].∫R3d3xe−ı~k·~xeıar4pir=1k2 − a2 +piı2aδ(k − a) (5.5.8)Without the ı term, 1k2−a2 is just a particular solution of the Helmholtz equation in the momentumspace, and the general solution shall include piı2aδ(k − a), which is the solution of the homogeneouspart of the Helmholtz equation.(−k2 + a2) piı2aδ(k − a) = 0 (5.5.9)Following reference [99], we demonstrate how the odd and even parts of eıar4pir are transformed frompiı2aδ(k − a) and 1k2−a2 respectively.1(2pi)3∫R3d3keı~k·~x piı2aδ(k − a) = 1(2pi)3∞∫0dk4pi kr sin krpiı2aδ(k − a)= ı sin ar4pir(5.5.10)1(2pi)3∫R3d3k eı~k·~xk2−a2 =1(2pi)2ır∞∫−∞dk keıkrk2−a2PV= 1(2pi)2ır[∮γ′ dkkeıkrk2−a2 + piıResk→a(keıkrk2−a2)+ piı Resk→−a(keıkrk2−a2)]= cos ka4pir(5.5.11)After the second equal sign, the improper integral took the Cauchy principle value for the integrationalong the real axis except two infinitesimal gaps over the sigularties k = ±a, and it is equivalent tothe contour integral over γ′ together with two half-residues. The closed contour γ′ is taken to be theinfinite upper half circle joining the line along the real axis with two infinitesimal half circles across±a from above, where the infinite half circle has no contribution in the integral because of Jordan’slemma. No singularity is enclosed in γ′, so the contour integration vanishes. Two half-residuescontributed at k = ±a correspond to the two counterclockwise infinitesimal half circles extractedfrom γ′. Finally, we recover the spherical wave eıar4pir without extending a from a real number to thecomplex plane.99Chapter 6Fuzzy three sphere in R46.1 IntroductionThe fuzzy three-sphere proposed several years ago in reference [93] is an interesting matrix configu-ration that does not converge to the expected three-sphere in the commutative limit. It has a radiusanomaly in that the radius converges to different values in different points of view. With the newprobing methods developed in previous chapters, we have an improved realization of a matrix con-figuration’s non-commutative structure with coherent states, and we found the fuzzy three-sphereis actually a thick three-sphere, such that the radii found in different methods are within the thickshell. An emergent point on the fuzzy three-sphere expands into a non-commutative cell with a cer-tain shape determined by the dispersion of the coordinate expectation, and the fuzzy three-sphereis, in fact, the collection of non-commutative cells.Before bringing in the fuzzy three-sphere proposed in reference [93], we shall introduce the well-known fuzzy four-sphere S4cl(5) proposed in reference [89] since the fuzzy three-sphere is taken asan equatorial slice from S4cl(5). Reference [89] studied the fuzzy four-sphere as a candidate for thelongitudinal 5-brane in M-theory wrapped around the longitudinal direction in which the L5-braneis constructed in terms of matrix variables according to the Matrix theory conjecture [50]. Thefuzzy four-sphere S4cl(5) has matrix coordinatesXˆµ =R4√Nf(Nf + 4)(γµ ⊗ 1⊗ · · · ⊗ 1 + · · ·+ 1⊗ · · · ⊗ γµ)SYM⊗Nf for µ = 1, ..., 5, (6.1.1)where γµ are cl(5) generators, and SYM⊗Nf means the symmetrized Nf -fold tensor product. Thesymmetric tensor product representation is in the representation spaceHNf =(C4 ⊗ · · · ⊗ C4)SYM⊗Nfin which C4 denotes the representation spaces of cl(5). For the basis {e1, e2, e3, e4} in C4, the basisof HNf can be written as|e(i1 , ei2 , . . . , e iNf )〉for symmetrizing index i1 . . . iNf . (6.1.2)100Here, HNf has dimensions CNf+33 for choosing Nf indices from {e1, e2, e3, e4}.The commutators Σˆµν =Nf(Nf+4)4ıR24[Xˆµ, Xˆν]are the generators of SO(5) in irrepso(5)[0,Nf2 ], theirreducible representation with the highest weight [0, Nf2 ]. S4cl(5) is a sphere because of the rotationalsymmetryXˆi = RjiVˆ−1Xˆj Vˆ , where Vˆ = exp( ı2Φ · Σˆ)(6.1.3)for a constant rotation matrix Rji = exp(Φji). The radius of S4cl(5) comes from the sum of squarematrix coordinates5∑µ=1Xˆ2µ = R241. (6.1.4)Instead of the symmetric tensor product representation, we choose to express S4cl(5) in terms offour bosonic oscillators aα obeying[aβ, a†α]= δαβ for numerical convenience. The fuzzy four-sphereS4cl(5) can be written asXˆµ =R4√Nf(Nf + 4)a†α (γµ)αβ aβ. (6.1.5)Correspondingly, HNf is realized as the Nf -particle Fock spaceHNf = a†i1 . . . a†iNf|0〉, {i1, . . . , iNf} = {1, 2, 3, 4}. (6.1.6)The orthonormal Fock state is denoted for short as|~n = (n1, n2, n3, n4)〉 = (a†1)n1(a†2)n2(a†3)n3(a†4)n4√n1!n2!n3!n4!|0〉, (6.1.7)which corresponds to the basis of Hn in a symmetric tensor product representation as|n1, n2, n3, n4〉 = | e(1, . . . , e1︸ ︷︷ ︸×n1, e2, . . . , e2︸ ︷︷ ︸×n2, e3, . . . , e3︸ ︷︷ ︸×n3, e4, . . . , e4)︸ ︷︷ ︸×n4〉. (6.1.8)The matrix element can be found as〈~n|a†iaj |~m〉=√ni(nj + 1)δ~n−iˆ+jˆ~m . (6.1.9)As proposed in reference [93], four matrix coordinates of S4cl(5) are projected in the sub-representationspace to represent the meridian of the perpendicular axis along the fifth coordinate, and this merid-ian is the fuzzy three-sphere embedded in R4. By taking gamma matrices in eq.(6.1.5) as~γ =(σ(2, 1), σ(2, 2), σ(2, 3), σ(1, 0), σ(3, 0)), (6.1.10)101the fifth matrix is diagonalized asXˆ5 =R4√Nf(Nf + 4)(N1 +N2 −N3 −N4) ,where Ni = a†iai. (6.1.11)Given an integer ND ≡ n1 + n2 − n3 − n4, the Nf -particle Fock space, i.e., HNf , is divided intosubspaces asHresND ={|n1, n2, n3, n4〉|∑ini = Nf , n1 + n2 − n3 − n4 = ND, ni ≥ 0}. (6.1.12)Each subspace HresND has dimensions Dim(HresND) =(Nf+ND2 +1)(Nf−ND2 +1)by counting all possibledistributions with Nf+ND2 = n1 + n2 andNf−ND2 = n3 + n4. Within HresND , we have the constant〈Xˆ5〉 = R4ND√Nf(Nf+4). The idea in reference [93] is that the three-sphere slice of S4cl(5) at altitudex5 = 〈Xˆ5〉 is described by Xˆ1,2,3,4 projected in HresND .The operator PND =∑~nd∈HresND|~nd〉〈~nd| projects states in HNf to HresND , and the four matrixcoordinates of the fuzzy three-sphere areXI=1,2,3,4 = PRXˆIPR, where PR = PND=1 + PND=−1. (6.1.13)Taking ND = ±1 restricts Nf to be an odd integer. There are two reasons for the combinedprojection (6.1.13) at the equator. First, the projection in a single slice PNDXˆIPND = 0 leavesonly a zero matrix, so one needs a bigger projection. Second, only the two adjacent restrictedrepresentations with ND = 1 and ND = −1 correspond to slices of equal radius and preserve thesum of squared coordinates of the fuzzy three-sphere as a scalar matrix:4∑I=1X2I =R242(Nf + 1)(Nf + 3)Nf(Nf + 4)≡ (R3)2 . (6.1.14)The slicing division along the x5 axis preserves the SO(4) symmetry for each three-sphere sliceof a fixed ND as follows. The branching rule of SO(5) decomposed into SO(4) isirrepso(5)[0,Nf2]→ ⊕ND irrepso(4)[Nf +ND4,Nf −ND4] (6.1.15)for ND = Nf , Nf −2, . . . ,−Nf , where irrepso(4)[Nf+ND4 , Nf−ND4 ] denotes the irrep of SU(2)⊗SU(2) ∼=SO(4) with the highest weights Nf+ND4 andNf−ND4 for each SU(2). In particular, our specific repre-sentation with eq.(6.1.10) reproduces SO(4) generators in ⊕ND irrepso(4)[Nf+ND4 , Nf−ND4 ] by[XˆI , XˆJ]102with Xˆ{I=1,2,3,4} of S4cl(5). We can list generators of SO(4) asfor i, j, k = 1, 2, 3,[Xˆi, Xˆj]= 2ıijka†σ(0, k)a,[Xˆ4, Xˆk]= 2ıa†σ(3, k)a,(6.1.16)It is clear that these are block diagonalized by examining 〈~nND1 |[XˆI , XˆJ]|~nND2〉 = 0 among discrete|~nND1〉 ∈ HresND1 and |~nND2〉 ∈ HresND2 since these commutators conserve ND = n1 + n2 − n3 − n4.Referring to the rotational symmetry of S4cl(5) in eq.(6.1.3), we confirm the SO(4) symmetry of eachND slice withVˆ −1S3PNDXˆIPND VˆS3 = RJI PNDXˆJPND , (6.1.17)where VˆS3 = exp(ı2ΦIJ ΣˆIJ)for I, J = 1, . . . , 4.We emphasize that the radius of the fuzzy three-sphere is found as a different value in manyapproaches. The original radius in reference [93] follows from∑4I=1 PRXˆ2IPR = R24 − R24Nf(Nf+4),and the radius converges to R4 in the large N limit as the meridian of S4cl(5). However, the ideais to have the sum of squared coordinates as a scalar matrix proportional to the squared radiusas in eq.(6.1.14), the author then corrected the radius to be R3 in reference [91, 100], even thoughthe three-sphere meridian at latitude 〈X5〉 = ±R4√Nf(Nf+4)≈ 0 should have radius R4 rather thanR3 ' R4√2 in the large N limit. Furthermore, our result in section 6.3 using the method of theLaplace probe suggests another different three-sphere radius. The explanation for the fuzzy three-sphere’s different radii is the main point of this chapter. In the following sections, we will explainthat the undetermined radius of the fuzzy three-sphere is due to its non-convergent geometry inthe large N limit. We will do this by studying the local non-commutative structure with coherentstates.6.2 Embedding operatorAs explained in Chapter 1 and Chapter 4, the embedding operator cannot be applied in the even-dimensional background because of the broken symmetry. Here, we explicitly demonstrate theanomaly in applying the embedding operator on the three-sphere in which the coherent state canonly be found at the origin regardless of its radius.By crossing out a dimension of the 2k + 1-dimensional embedding operator E2k+1, the surfaceequation of 2k matrices can be decomposed asdet (E2k+1) = −det(E+2k)det(E−2k), (6.2.1)whereE±2k = γi ⊗ (Xi − xi)± ı1⊗ (Xd−1 − xd−1) for i = 1, . . . , 2k − 1. (6.2.2)103Either E+2k or E−2k can be the dimensional reduced embedding operator with a zero eigenvalue on theemergent surface, and they result in the same emergent surface equation, det(E+2k)= det(E−2k)= 0,for Hermitian coordinates.The embedding operator applied on the three-sphere is written asE+4 =((X3 − x3) + ı(X4 − x4) (X1 − x1)− ı(X2 − x2)(X1 − x1) + ı(X2 − x2) −(X3 − x3) + ı(X4 − x4))= 2ıRNf (a†2a4 + a†3a1)− x34+ 2ıRNf (a†4a1 − a†2a3)− x12−2ıRNf(a†3a2 − a†1a4)− x12+ 2ıRNf(a†4a2 + a†1a3)+ x34− , (6.2.3)where we denote for short by x12± = x1 ± ıx2, x34± = x3 ± ıx4, and RNf = R4√Nf(Nf+4) .In the restricted Fock space HresND=1 ⊕ HresND=−1, we find the term a†iaj in the three-sphere isnilpotent, i.e. (a†iaj)2=(a†jai)2= 0 for i ∈ {1, 2} and j ∈ {3, 4}. (6.2.4)Consequently, X12± = X1 ± ıX2 and X34± = X3 ± ıX4 are nilpotent as well.(X12±)Nf+1 = (X34± )Nf+1 = 0. (6.2.5)A nilpotent matrix is equivalently a matrix with only the zero eigenvalues.However, the nilpotent X12/34± is in contradiction to finding a coherent state |λc〉 outside theorigin. With proper rotation, one expects the corresponding eigenstate of E±4 as|Λ〉 =(|λc〉0)for Ered4 |Λ〉 = 0, (6.2.6)and it leads to the conflicting results,X34+ |λc〉 = x34+ |λc〉 and X12+ |λc〉 = x12+ |λc〉, (6.2.7)unless all xi collapses to zero. One could further assume the non-parallel zero state, but it is notthe case as we saw in the exact computation at lower N, which shows |~x| > 0→ det(E±4 ) 6= 0.6.3 Laplace probeApplying the Laplace probe L = ∑i (Xˆi − xi)2 defined in reference [24] on the three-sphere resultsinLS3(x1,2,3 = 0, x4) =(R23 + x24)1− 2x4RNf (a†1a3 + a†4a2 + a†3a1 + a†2a4) , (6.3.1)1040 0.5 1 1.5 2x400.511.522.53Figure 6.1: Given (Nf = 5, 15;R4 = 1), we plotted the minimum eigenvalue of LS3(x4) at vari-ous probe location x4. Enum is the lowest eigenvalue computed numerically, and Eλ is computedanalytically with eq.(6.3.3). We found Eλ = Enum for any given {Nf , x4} and confirmed Eλ is thelowest eigenvalue.where the probe is restricted to be along x4 axis with the spherical symmetry. One can easily spotan eigenstate in HresND=+1 ⊕HresND=−1|λS3〉 =1√2(|N, 0, N + 1, 0〉+ |N + 1, 0, N, 0〉) , where N = Nf − 12, (6.3.2)and the corresponding eigenvalueEλ(x4) = x24 +R242(Nf+1)(Nf+3)Nf(Nf+4)− x4R4 Nf+1√Nf(Nf+4)=(x4 − R42 Nf+1√Nf(Nf+4))2+R244(Nf+1)(Nf+5)Nf(Nf+4).(6.3.3)Fortunately, we recognize Eλ(x4) as the ground state energy by numerical data, see fig. 6.1, andtherefore |λS3〉 is the generalized coherent state for the Laplace operator LS3(x4) as described inreference [24]. The three-sphere radius as seen by the Laplace probe is R42 instead of R3 = R4√2 foundin the quadratic sum of coordinates.Based on the location of the probe with the lowest possible ground state energy Eλ, which isreferred to as the displacement energy in reference [24], the emergent surface is the three-sphere,S3E =~x|x2 =(R42Nf + 1√Nf (Nf + 4))2 . (6.3.4)105Another way in reference [24] of measuring the emergent surface is through the symbol map,S3|λ〉 = {~x|xi = 〈λ|Xi|λ〉} , (6.3.5)and the deviation between two ways of measuring the surface is bounded by the displacementenergy Eλ(x4). On the fuzzy three-sphere, we actually found two surfaces agree each other perfectlyS3E = S3|λ〉 since〈λ|X4|λ〉 = R42Nf + 1√Nf (Nf + 4)and 〈λ|X1,2,3|λ〉 = 0. (6.3.6)A criterion of emergent surface discussed in reference [24] is that the displacement energy variesmuch significantly in the normal direction than the tangent direction, and this is examined viaHessian matrix with componentsHij =∂2Eλ(~x)∂xi∂xj. This criterion on the fuzzy three-sphere is confirmedbelow. With the spherical symmetry, we have the displacement energy at arbitrary ~x asEλ(~x) =(|~x| − R42Nf + 1√Nf (Nf + 4))2+R244(Nf + 1) (Nf + 5)Nf (Nf + 4). (6.3.7)The Hessian is computed directly asHij(~x) =(2− R4|~x|Nf + 1√Nf (Nf + 4))δij +xixjR4|~x|3Nf + 1√Nf (Nf + 4), (6.3.8)and the matrix at the x4 North pole isH(x4 =R42Nf + 1√Nf (Nf + 4)) = diag(0, 0, 0, 2). (6.3.9)The hierarchy is clear that Hij = 0 except H44 = 2 indicating the significant variation in the x4direction normal to the three-sphere.There are more degenerate coherent states other than eq.(6.3.2). Considering exchanging indicesof bosonic operators as (1, 3)↔ (2, 4),1√2(|0, N, 0, N + 1〉+ |0, N + 1, 0, N〉)is another eigenstate with the same eigenvalue Eλ(x4). The set of degenerate coherent states is therepresentation space of SO(3), which is the local symmetry at a fixed point on the three-sphere. Inthe following, we first construct the orbit on the base of two known ground states, then we identify itas the spin Nf2 representation of SO(3), and finally we confirm the SO(3) orbit exhausts the moduliby numerical data.1060 10 20 30 40 50 60 70 80 90index0. 6.2: Given Nf = 11, the black dots are the complete eigenvalues ofLS3(x4 =R42Nf+1√Nf(Nf+4)), and the red line segment indicates the lowest eigenvalue Eλ(x4). Thereare 12 = Nf + 1 ground states supporting our argument of (Nf + 1)-fold degeneracy in the groundstate.The SO(3) orbit is constructed as|λ(α, β)〉 =[(αa†1 + βa†2)N (αa†3 + βa†4)N+1+(αa†1 + βa†2)N+1 (αa†3 + βa†4)N] |0〉, (6.3.10)where α, β ∈ C and satisfy the normalization condition 〈λ|λ〉 = 1 derived in eq.(6.3.15). It isstraightforward to findLS3(x4)|λ〉 = Eλ(x4)|λ〉 (6.3.11)with the same ground state energy Eλ(x4) using the lemmaai(αa†i + βa†j)n=(αa†i + βa†j)nai + αn(αa†i + βa†j)n−1. (6.3.12)This SO(3) rotation of indices (1,2) and (3,4) spans out 2(N +1) = Nf +1 dimensions in HresND=+1⊕HresND=−1, so the SO(3) orbit is in the spin Nf2 representation. The numerical data in fig. 6.2 showthat the degeneracy of the Laplace operator is consistent with Nf + 1, and therefore we claim theSO(3) orbit gives the complete ground states.We can expand the orbit as|λ(α, β)〉 =N∑i=0N+1∑j=0αi+jβNf−i−jCNi CN+1j√i!j!(N − i)!(N + 1− j)!(| (i,N − i, j,N + 1− j)〉+ | (j,N + 1− j, i,N − i)〉)(6.3.13)by formulas of binomials, and we find the overlap between two coherent states as〈λl(αl, βl)|λr(αr, βr)〉 = 2N !(N + 1)!(α∗l αr + β∗l βr)2N+1, (6.3.14)107which leads to the normalizing condition,1 = 2N !(N + 1)!(|α|2 + |β|2)2N+1. (6.3.15)The (Nf + 1)-fold degeneracy suggest that the three-sphere slice is a stack of (Nf + 1) overlappedmembranes, and it is consistent with the three-sphere’s source, fuzzy four-sphere, which has Nfoverlapping layers according to the 4-brane charge computation at leading order in Nf [89]. Eventhough the fuzzy three-sphere can not be interpreted as a D-brane conclusively [93], we can stillconstruct a U(Nf +1) connection living on the worldvolume of the three-sphere with the degeneratecoherent states as shown below.Following reference [22], we computed the Berry connection,2viAi = −2ıvi〈λ(~x)|∂i|λ(~x)〉, (6.3.16)with vi tangent to the surface and confirmed that Ai corresponds to the U(Nf + 1) gauge field. Theorthonormal basis for the set of coherent states at the pole of (x1 = x2 = x3 = 0, x4) is denoted by|λp〉 for p = 1 . . . Nf + 1, and we can write the element of the expectation matrix of an operator Oat ~x on the fuzzy three-sphere as〈O(~x)〉pq = 〈λp|R†so(4)(~x)ORso(4)(~x)|λq〉, where Rso(4)(~x) = exp( ı2Φµν(~x)Σµν). (6.3.17)Recall that we found 〈Σij〉pq are SO(3) generators in the spin Nf/2 representation and 〈Σi4〉pq = 0at the x4 pole, it is straightforward to compute the Berry connection in the xj direction as−2ı 〈λp(~x)|∂j |λq(~x)〉 = 〈λp|R†so(4)(~x)Σj4Rso(4)(~x)|λq〉= RajRb4 〈λp|Σab|λq〉(6.3.18)in which 〈λp|Σab|λq〉 is an element of the Nf + 1 dimensional SO(3) generator, and Aabj ∝ RajRb4 isthe component of the U(Nf + 1) gauge field.It may seem that the fuzzy three-sphere is a legitimate three-sphere with Nf + 1 overlappinglayers according to the Laplace probe, except for the radius. In the commutative limit, we expectedthe fuzzy three-sphere to have the radius as R4 because it is the meridian cut from S4cl(5), and thesum of squared coordinates indicates that the radius to be R4√2, and the Laplace probe finds theradius at R42 .To resolve this conflict, we computed the dispersion of surface expectation at the x4 axis asΘt =3∑i=1〈λ| (Xi − xi)2 |λ〉 = R244(Nf+1)(Nf+5)Nf(Nf+4),Θn = 〈λ| (X4 − x4)2 |λ〉 = 0,(6.3.19)where Θn and Θt are normal and tangent to the surface respectively. Θn = 0 corresponds to the108-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.800. 6.3: The dashed circle depicts the three-sphere SL with radius RL found by the Laplaceprobe. The dashdotted circle depicts the three-sphere SQ with radius RQ due to the squared rootsum of coordinates. The red dotted line represents the non-commutative cell at the x4 North pole,which is a ball with diameter 2√Θt in x1,2,3. The fuzzy three-sphere is the region bounded by SQand SL as a result of the dispersion Θt.local flatness property of S4cl(5), see reference [89], such that the non-commutative cell encircled bythe coherent states is tangential to the emergent surface. The non-vanishing Θt → R244 in the largeN limit suggest that the coherent state is highly delocalized, and the non-commutative cell has theshape of a two ball with radius√Θt → R42 because of the SO(3) symmetric degenerate coherentstates. In conclusion, we found that the fuzzy three-sphere is a thick three-sphere owing to thesignificant dispersion. The inner radius found by the Laplace probe is R42 , and the outer radiusfound be the quadratic sum is R4√2, and the thickness is due to Θt as shown in fig. 6.3.The fuzzy three-sphere shrinks after being removed from the fuzzy four-sphere, so we suspect thatthe radius of the three-slice on the four-sphere was extended longer by the stack of other slices alongthe x5 axis. Our future work is to keep on analyzing other three-sphere slices at higher altitude withcoherent states of the Laplace probe. Because of the fuzziness, we no longer require the quadraticsum of coordinates to be a scalar matrix, which is the reason for restricting the equatorial slice,and we can examine that if every longitudinal slice of the fuzzy four-sphere has the topology of athree-sphere.109Chapter 7Critical phenomena of the massiveYang-Mills matrix model7.1 IntroductionOne root of the matrix model is the generating functional of the two-dimensional (2D) Euclideanquantum gravity coupled with conformal matter, which can be interpreted as string theory in theanalytic continuation of Euclidean gravity to Minkowski spacetime. In string theory, the stringpartition function sums over worldsheets of any geometries and topologies, and the integral overthe worldsheet metric can be approached by summing random discretizations of the surface into Nequilateral polygons. Then, based on the idea in reference [101] in quantum chromodynamics thatthe discretization with n-sided polygon is dual to the Feynman diagram of the one-matrix modelwith a n-valent vertex, such that the partition function of the random polygonization is the sameas the free energy of the matrix model, references [102–105] computed the generating functional ofrandom polygonizations with matrix integral including every genus, and the polygonization revealsthe 2D quantum gravity non-perturbatively.On the other hand, the 2D quantum gravity coupled to the conformal matter field with centralcharge c ≤ 1 is exactly solvable by using the Liouville theory, and the resolved free energy coincideswith the matrix model description in the double scaling limit, see references [106–108], where thedouble scaling limit means takingN →∞ and g → g∗ while fixing N(g∗ − g)γ12 = const.. (7.1.1)In the double scaling limit, the matrix model reproduces the Liouville theory by the singular partof the partition function near the critical coupling g∗,Z = (g∗ − g)2−γ0f((g∗ − g)N2γ1), (7.1.2)110for some analytic function f and critical exponents,γ1 = 2− γ0 = 112(25− c+√(1− c)(25− c)). (7.1.3)The double scaling limit corresponds to the continuum limit of polygonization with infinite numbersof infinitesimal tiles, while the generating functional takes account of surfaces with every genus non-perturbatively. The specific scaling of g with N near the critical point gc implies the double scalinglimit is a fixed point of the renormalization group flow of N .The renormalization group approach in one-matrix model in reference [109] was proposed as anapproximate method for a arbitrary central charge, and the renormalization group flow equationreproduces the string partition function and the critical exponents qualitatively comparing to thesoluble cases c ≤ 1. By coarse graining the matrix dimension from N+1 to N , reference [109] foundthat the matrix partition function ZN (g) =∫dφN exp(−Ntr ∑k≤11kgkφkN)satisfiesZN+1(g) =(1 +1Nr(g) +O(N−2))N2ZN(g +1Nβ(g) +O(N−2)), (7.1.4)and therefore the free energy F (N, g) = − 1N2logZN (g) satisfies the Callan-Symanzik like renormal-ization group equation1 (N∂∂N+ 2)F (N, g) = r(g) + β(g)∂F∂g(7.1.5)at the linear order by approximating(N ∂∂N + 2)F (N, g) ≈ − 1N log ZN+1(g)ZN (g) . At the fixed pointg∗, which is the root of β(g∗) = 0, the coupling constant g stops running with the dimensionalrenormalization, and the scaling law eq.(7.1.2) for Z = F (N, g) is recovered at the critical exponentsγ1 =2β′(g∗)and γ0 = 2− 2β′(g∗). (7.1.6)In this chapter, we apply the method of dimensional renormalization from [109] to the massiveYang-Mills (YM) type matrix model eq.(2.2.20), which comes from the bosonic part of the IKKTmatrix model regularized by Lagrange constraints λ2 tr(XiXi − L2). These constraints come froman approximation of saturated IR cutoff observed in the Monte Carlo simulation. Here, we drop theintegration∫dλ of the Lagrange multiplier λ, and we think of λ as the running mass with respect tothe dimensional renormalization group flow. Dropping∫dλ restores the IR cutoff to an inequality,and the λ mass term generates finite dimensional solutions in the massive YM model. This way,we find the free energy of the massive YM model also satisfies the Callan-Symanzik equation neara fixed point λ∗(L), which varies with the cutoff L.IKKT model has remarkable phenomena of emergent geometry and emergent gravity, but it is1There is a summation∑k βk(g)∂F∂gkin the case of multiple couplings.111unfortunately not solvable like the Gaussian matrix models. Especially, emergent geometry mightnot reflect the exact curvature due to gravity without considering the non-commutative structureon the emergent surface. In references [26,67], the induced metric derived from the emergent surfaceequation is found different than the effective metric in the view of a scalar field on the emergentsurface. Studying the critical phenomena of IKKT model with renormalization group approachmight pave a path towards realizing the emergent gravity beyond the Liouville theory.As explained in Chapter 1, the IKKT model is only convergent with the temporal and spatialcutoffs1Ntr (X0)2 ≤ κL2 and 1Ntr (Xi)2 ≤ L2, (7.1.7)and one is able to find interesting solutions of Friedmann-Robertson-Walker type expanding universewith SO(3) symmetry in the Monte Carlo simulations [21, 62, 63].2 Since the quadratic cutoffs aresaturated in the simulations, references [62, 66] then studied the IKKT model as the massive YMmodel by turning cutoffs into rigid constraints, which are mass terms (Lagrange multipliers) addedin the action. The mass term is also crucial for YM matrix model to have finite-dimensional non-commutative solutions as pointed out in reference [26]. Then, many classical solutions representingemergent spacetime can be found analytically in the massive YM model.In particular, the solution known as the generalized fuzzy sphere constructed by m + n + 1generators in the the Lorentz algebra SO(n,m+2) is worth extensive study, see references [25–27,66, 67, 94, 95]. Without the IR cutoff, the generalized fuzzy sphere is unstable due to its negativemass. By including the fermion term and quantum effects, [77] showed the generalized fuzzy spherecan be stabilized via the soft supersymmetry breaking mechanism with a small positive mass. Inthis chapter, we consider a straightforward setup for the purely bosonic generalized fuzzy sphere byintroducing the IR cutoff, which can be sent to infinity. The generalized fuzzy sphere, depending onthe signature selected, can be a (m + n)-dimensional sphere, one-sheet or two-sheets hyperboloidsembedded in Rm+n+1 or R1,m+n. In the Minkowski background, references [26, 67] pointed outthat the emergent metric of the generalized fuzzy sphere changes from Euclidean to Lorentzianaccompanied by a rapid spatial expansion, and it can be interpreted as the big bang of the emergentuniverse, which begins before the physical time appears.Our subject is the generalized fuzzy sphere S2so(4) embedded in R3, which emerges as a thickspherical shell. While this lowest dimensional case is simple enough for analytic and numericalcomputations, it still captures most of the physical behaviors of the generalized fuzzy sphere.3Following the analysis of the non-commutative structure of S2so(4) in Chapter 5, we study its criticalphenomena with dimensional renormalization group flow introduced in this chapter. Rather than2Other regulators of general exponents, 1NtrXp ≤ Lp, have been numerically examined in [110]. Qualitativelythe same emergent universes are found for p within a finite range. While at p larger than 4, the cutoff loses therestriction, and the result has no sensible large N limit. We simply choose the quadratic regulator for the universaland convergent result since we are mainly interested in the classical solutions.3The obvious difference with the higher generalized fuzzy sphere is that the coherent structure becomes highlydegenerate because of the high rank of local rotational symmetry.112forcing the IR cutoffs at eq.(7.1.7) to be saturated as in [62,66], we relax the model by consideringthe mass term without integrating the Lagrange multiplier λ and restore the inequalities, and weare interested in the particular S2so(4) with λ near the fixed point of the massive Yang-Mills matrixmodel.Unlike the conventional fuzzy sphere S2so(3) constructed in the spin J representation of su(2)with one spin number J , S2so(4) is constructed in the two spin number (Jl, Jr) representation ofsu(2)Jl ⊗ su(2)Jr , which is the double cover of so(4). Given N constituent quantum particlesdistributed in the space of size L3, while S2so(3) is fixed by J(N,L), S2so(4) is undetermined in theparameter space {(Jl, Jr)|N = (2Jl + 1)(2Jr + 1)} because the ad hoc mass λ (Lagrange multiplier)is only resolved from the equation of motion [Xi, [Xi, Xj ]] = λXj and the saturated cutoff 1N trX2 =L2.7.2 Renormalize group flow of NThe framework is the D-dimensional massive Yang-Mills matrix model with the Euclidean signature,which is the regulated bosonic IKKT model after Wick rotation. It is described by the actionSE (XN , λ) of N by N Hermitian matrix XN as follows.SE (XN , λ) = tr(−14 [Xµ, Xν ][Xµ, Xν ]− λ2 (XµXµ − L2))for ZN (λ) =∫ LdXNe−NSE(XN ,λ).(7.2.1)The index subtraction is carried out by δµν . By scaling X = Rx with respect to the squared effectiveradius 1N tr(X2) = R2, we impose an IR cutoff for R ≤ L in the integration of ∫ dX as∫ LdX =L∫0dRRD−1∫dxδ(1− trx2) . (7.2.2)The coarse-graining process decomposes the partition function of N + 1-dimensional matricesXN+1 =(XN vv† α), (7.2.3)to the partition function of N -dimensional matrices XN with transformed action. In the sense ofrenormalization in the coordinate space, we assume the matrices redistribute themselves to fill upthe same emergent space from N + 1 to N . It is straightforward to computetr(X2N+1 −X2N)= 2v†v + α2 (7.2.4)113andCYM (α) = tr([XµN+1, XνN+1]2 − [XµN , XνN]2)= 2v†(µvν)v†(µvν) − 4v†µ(Xν − αν)(Xν − αν)vµ+4v†µ((Xν − αν)(Xµ − αµ)− [Xµ, Xν ] )vν .(7.2.5)For the time being, it is safe to ignore v†(µvν)v†(µvν) in CYM to perform the Gaussian integrationfor our qualitative analysis. Also, we consider the large N limit and set αµ = 0 since the termsincluding α are relatively of the order 1/N . The setting, αµ = 0, can be improved by the saddlepoint method as in reference [109]. Hence, we approximateSE (XN+1) ≈ SE (XN ) + v†µWµνvν +λ2L2, (7.2.6)whereWµν = δµν(XξXξ − λ)+XµXν − 2XνXµ. (7.2.7)In the following, by restrictingXξXξ = R21N + ∆ (7.2.8)for the limited spectral radius4 ρ(∆)  R2, we impose the spherical topology on the matrix con-figuration {X} with thickness limited in the subleading order. We also assume W is positive-semidefinite5, which is necessary for the Gaussian integral. With these two assumptions, we canintegrate out v and v† in the partition function asZN+1(m)≈ ∫ dXN e−(N+1)(SE(XN ,λ)+λL22 ) ∫ dvdv† exp [−(N + 1)v†µWµνvν]=∫dXN e−(N+1)SE(XN ,λ)(piN+1)NDe−tr logW−(N+1)λL22 ,(7.2.9)and we approximately truncate the following expansion up to the one-loop level O(X4) astr logW = ND logR2 + tr log (1 + ω)≈ ND logR2 +2∑p=1(−1)p−1p tr ωp,(7.2.10)whereω = WR2− 1ND= R−2∆− λ−X21 X1X2 − 2X2X1 · · ·X2X1 − 2X1X2 ∆− λ−X22 · · ·....... . . . (7.2.11)4Given a square matrix M, we denote its spectral radius as ρ(M), which is the largest absolute value of M’seigenvalues.5If W is singular, the Gaussian integral is computed by replacingW with lim→0+W + i in the analytic continuation.114The attempt of higher order perturbation is not worthy because higher and lower order terms aremixed up together in the trace. In the one-matrix model with polynomial potentials, references [111,112] demonstrated that one can absorb the additional interactions induced by higher order terms intothe lower order action and acquire the non-linear RG equation with the help of reparametrizationinvariance and Schwinger-Dyson equations. Nevertheless, the critical exponents are not improvedsignificantly comparing to the exact computation with central charge c ≤ 1, so we choose to limitthe computation up to the leading order for massive YM matrix model.Based on our assumption of dominatingR2 inW , we further expand the action−(N+1)SE (XN+1, λ)to leading order as−(N + 1)SE (XN , λ)− tr logW − (N + 1)λL22≈ (N+14 − 1R4 ) tr [Xµ, Xν ]2 −ND logR2 + (N+1)Nλ2 R2− (N+1)22 λL2 + 3N2 + (D + 1)NλR2 +O( λ2R4, ∆2R4, ∆λR4).(7.2.12)The action has leading order at the largest effective radiusR, which is restricted byR2 = 1N trXµXµ ≤L2, so we expand the action in eq.(7.2.12) near R2 ≈ L2 as6(N+14 − 1L4)tr [Xµ, Xν ]2 +((N+1)Nλ2 − NDL2)R2− (N+1)22 λL2 + (3+2D)N2 −ND logL2 + (D + 1)NλL2 +O( λ2R4, ∆2R4, ∆λR4).(7.2.13)The comparison with the action of XN ,−NSE = N4tr [Xµ, Xν ]2 +N2λ2R2 − N22λL2, (7.2.14)suggests the rescaling of the matrix field X → ρY , such that(N + 14− 1L4)ρ4 =N4and((N + 1)λ2− DL2)ρ2 =Nλ′2, (7.2.15)so we can renormalize the partition function with a new coupling constant λ′, i.e., −(N+1)SE (XN+1, λ)→−NSE (XN , λ′). With respect to eq.(7.1.4), the new coupling constant,λ′ ≈ λ+ 1N(λ2− 2DL2+2λL4)+O(N−2), (7.2.16)resolved at the order of 1/N leads to the β function,β (λ) =λ2− 2DL2+2λL4, (7.2.17)6Considering the rescaling X → ρY , we should expand the action near the scaled cutoff R2 ≈ L2ρ2. However, thenew coupling constant λ′ at the leading order is the same as eq.(7.2.16).115with the non-trivial fixed point,λ∗ ≈ 4DL2L4 + 4for β(λ∗) = 0. (7.2.18)In particular, the small Lagrange multiplier λ∗ at L  1 confirms the numerical observation inMonte Carlo simulations [21,63] that the IR cutoffs are automatically saturated at large N , and themass terms can be removed by sending L→∞.7.3 The thickness of the generalized fuzzy sphereAs introduced in Chapter 5, the equation of motion,δij [Xi, [Xj , Xk]]− λXk = 0, (7.3.1)from the action SE in eq.(7.2.1) has a solution, known as the generalized fuzzy sphere S2so(4), whichis described by the collective coordinates,Xi =√λ2(J li ⊗ 1− 1⊗ Jri)for i, j, k = 1, 2, 3, (7.3.2)from generators of so(4) in terms of its double cover su(2)Jl ⊗ su(2)Jr . Embedded in R3, S2so(4) is athick spherical shell with radius extended from√λ2 |Jl − Jr| to approximately√λ2 |Jl + Jr|. Besidesthe thickness, S2so(4) with different parameters (Jl, Jr) has different non-commutative structure de-termined by the symbol map with respect to its coherent states |Λ(~x)〉, see Chapter 5 for details.For example, S2so(4) with fixed dimensions N = (2Jl + 1)(2Jr + 1) is much more fuzzy if the layer isthicker when Jl ≈ Jr by considering the deviation 〈(Xi−〈Xi〉)2〉 from the surface expectation 〈Xi〉.In this section, we look for reparametrization, S2so(4)(Jl, Jr) = S2so(4)(N,L), with mass λ at the fixedpoint λ∗ = 12L2 +O(L−4) in the renormalization group flow of N .Naively, the generalized fuzzy sphere with fixedN constituents, S2so(4) (Jl, Jr|N = (2Jl + 1)(2Jr + 1)),prefers thinner shells at larger radius when Jl  Jr if we only seek for the lower value of action SE .Applying S2so(4) as the ground state in the action SE at eq.(7.2.1),SE(X = S2so(4), λ) = −3λ28N (Jl(Jl + 1) + Jr(Jr + 1)) = −3λ4trXiXi, (7.3.3)we find the sphere saturating the constraint tr(XiXi − L2) = 0 is most stable, and therefore themass of the stabilized sphere is found to beλ =2L2Jl(Jl + 1) + Jr(Jr + 1). (7.3.4)With the cutoff 1N trXiXi ≤ L2, we found the target S2so(4) with larger N needs to be thicker116to have its mass at the fixed point. The target S2so(4)(N,L) can be comprehended instinctivelyregarding the surface density σ ≡ NL2as follows. In the large N limit, the mass of the thickestsphere with Jl ≈ Jr and the mass of the thinnest sphere with Jl >> Jr are respectivelyλthick =4L2N+O(N−2) and λthin =8L2N2+O(N−4) (7.3.5)according to eq.(7.3.4). Only when the surface density is between the range√23< σ =NL2<L23, (7.3.6)we can find the target S2so(4)(N,L) with fixed point mass λthin < λ∗ =12L2< λthick. Within thisdensity range, we find larger surface density thickens S2so(4)(N,L).Nevertheless, the naive perturbative calculation of the surface density of S2so(4)(N,L) at eq.(7.3.6)is inaccurate near the thick limit Jl ≈ Jr because the thin spherical shell assumptions in eq.(7.2.8)and positive-semidefinite W are not satisfied. The quadratic sum of matrix coordinates of S2so(4)can be written asXiXi =λ2(ζ1N −Jl+Jr⊕J=Jl−JrJ(J + 1)12J+1), where ζ ≡ 2 (Jl(Jl + 1) + Jr(Jr + 1)) . (7.3.7)Comparing to eq.(7.2.8), we have in generalR2 =λζ4and ∆ =λ2(ζ21N −Jl+Jr⊕J=Jl−JrJ(J + 1)12J+1), (7.3.8)and the thin restriction is not satisfied as R2 = ρ(∆) = λJl(Jl + 1) when Jl = Jr.In conclusion, imposing the spherical topology at eq.(7.2.8) and the IR cutoff 1N trXiXi ≤ L2introduce a fixed point as the negative mass λ∗ for the target S2so(4)(N,L). For a steady S2so(4)(N,L)to have a critical mass λ∗, the surface density of N constituents needs to be in a certain range.At the threshold of low surface density at σ =√23 , S2so(4)(N,L) collapses into a conventionalfuzzy sphere, which is constructed by SO(3) generators at fixed radius R = L + O(ρ−1L−1), andthe collapsed S2so(4)(N,L) has zero thickness. The collapse of thickness has a problem reported inreference [94], which finds the incorrect long-range gravity (not proportional to 1r2) in the emergentuniverse, see reference [94] for the gravity propagator.Above the lower bound, S2so(4)(N,L) becomes thicker because more quantum particles join inthe emergent surface until the surface reaches another upper bound at NL2= L23 , and the surfacebecomes a solid ball. Near this thick limit, the beta function at eq.(7.2.17) is incorrect since theGaussian integral can not be computed in perturbation method.Finally, when the Euclidean background is Wick rotated back to the Minkowskian signature, thegeneralized fuzzy sphere corresponds to an emergent hyperboloid like a rapidly expanding universe.The asymmetry temporal and spatial Lagrange multipliers are crucial in finding the big bang from117the signature exchange [67], and we could investigate the generalized fuzzy ellipsoid or hyperboloidwith similar renormalization group approach in the future works.118Chapter 8ConclusionString theory inherits the century-old idea of quantized spacetime intrinsically at a scale smallerthan the string length. Matrix models construct the dynamical and probabilistic spacetime in thelanguage of non-commutative geometry and extend the description of the Universe before physi-cal time emerges at the point when its signature changes from Euclidean to Minkowskian. Thisthesis seeks to examine emergent spacetime in matrix models by effectively reconstructing severalmacroscopic characteristics of the Universe.We built up the matrix coordinates for arbitrary two-dimensional surfaces with spherical topol-ogy, and we demonstrated that the methodology applies on surfaces with aother genus by examiningthe non-commutative torus. With the coherent states that we constructed on the emergent surface,we found the conditions on the matrices to converge to the given surface in the commutative limit; wederived the unique Poisson structure determined by the matrices, and we proved the correspondencefrom Poisson brackets to commutators. Thus, we established the codimension-one non-commutativemembranes embedded in R3, in which the physical properties are computed via the expectationswith respect to coherent states. The conditions on emergent surfaces in the large N (smooth) limitthat we defined are generally applicable in any dimensions, and the conditions help in establishinga physically meaningful non-commutative spacetime in future studies.We generalized our work to higher dimensions and proposed the embedding operator for matrixconfigurations embedded in an odd-dimensional background based on the Dirac operator, whichmeasures the interaction between the D0 brane probe and the membrane. As a by-product, wefound a new non-commutative flat space restoring rotation symmetry, which was broken in theearlier works. We applied a similar method using the Laplace operator instead of the Dirac operatoron matrix configurations embedded in the background with even dimensions. Our coherent staterealization resolved the issue of the inconsistent radius of the fuzzy three-sphere.We studied the generalized fuzzy sphere as a toy model for the emergent universe arising in theIKKT matrix model. We examined the emergent geometry with two point-probe methods, includingthe Dirac type probe and the Laplace type probe, and the supergravity charge density formula. Wefound that the generalized fuzzy sphere is a thick spherical shell of overlapping coherent layers, not119a codimension-one surface like other cases studied in point-prob methods. With Wick rotations, wecan extend our result to a fuzzy hyperboloid, which is to be considered as an emergent Universe.Only after decomposing matrices with respect to geometric symmetries for probing the emergentsurface locally, we obtained the consistent results in three different methods. It would be interestingto see how our trick of decomposing matrices can be applied to the non-compact emergent surfaces,which are constructed from non-compact Lie groups in infinite-dimensional representations. Thetrick of isolating a patch on an emergent surface in finite dimensional matrices would be useful innumerical studies of emergent Universe.The tangential and radial perturbations on a conventional fuzzy sphere S2so(3) correspond to aU(1) gauge field and a scalar field. This can be shown explicitly by rewriting the matrix modelaction, expanded near the fuzzy sphere matrices, into the non-commutative action of a U(1) gaugefield and a scalar field, see reference [113]. Because we found that the generalized fuzzy sphere S2so(4)is a thick spherical shell, one should be able to correspond the radial perturbation on S2so(4) to aU(1) gauge field instead of a scalar field in future works.We studied critical phenomena of the massive Yang-Mills (YM) matrix model, which is thebosonic part of the regularized IKKT model. By imposing spherical symmetry on the matrices,we found a fixed point of the dimensional renormalization group flow, where the free energy of themassive YM model obeys the Callan-Symanzik like equation. The fixed point we found confirmsthe numerical behaviour observed in Monte Carlo simulations [21, 63] and gives a particular largeN invariant value of the running mass for relating the partition function of a dynamical generalizedfuzzy sphere to a dual quantum gravity system, which is not yet clear. Along the lines of how theGaussian matrix model at the fixed point reproduces quantum gravity of two-dimensional conformalmatter, in principle, the gravity picture dual to the massive YM model could be found at a fixedpoint in future studies.Certainly, we are still far from validating the old hypothesis of non-commutative spacetime, butalong the way, we have refined the description of emergent geometry and learned new insights inphysics. We have bridged matrices and geometry, generalized the methodology to higher dimensions,and tackled objects with entangled internal structure by decomposition with respect to geometricalsymmetry. We have shown the capability of generating complex emergent geometry to effectivelydescribe the empirical world and found supports in non-commutative spacetime defined in matrixmodels.The important ingredients omitted in this thesis are Fermion matrices. Although supersymmetry(SUSY) has not yet been observed, it is essential to string theory. The derivations and evidencepresented in this thesis are in the context of the solely bosonic part. Adding Fermions alters thecomputations significantly, but the scheme of coherent states remains valid. In principle, one couldaccommodate the more complicated fermionic part using the perturbation method based on thegeometry of the fuzzy sphere reported in this thesis. One interesting future work is to study thegeneralized fuzzy sphere when SUSY is put back in the model.120Along the lines of matrix models as a non-perturbative realization of the holographic principle,one interesting future work would be to construct fuzzy hyperboloid for a matrix example of anti-deSitter/conformal field theory (AdS/CFT) correspondence. The generalized fuzzy sphere generatesan emergent hyperboloid, which could be thick, from the Lorentz algebra so(n,m + 2), where onecarefully takes the finite-dimensional doubleton representations for the non-compact Lie algebra[26, 114–117]. With gravity resolved in the effective metric [70, 94], the AdS side has a completematrix description. The dynamical matrix configuration of the fluctuating fuzzy AdS in the massiveYM matrix model could be described by CFT in the neighbourhood of the fixed point. The insightsof AdS/CFT correspondence could be demonstrated explicitly in such a system. 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