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Geometry from quantum mechanics : entanglement, energy conditions and the emergence of space Sabella-Garnier, Philippe-Alexandre 2016

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Geometry from Quantum MechanicsEntanglement, Energy Conditions and the Emergenceof SpacebyPhilippe-Alexandre Sabella-GarnierB.Sc. (Honours), McGill University, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2016c© Philippe-Alexandre Sabella-Garnier 2016AbstractThis thesis presents various examples of the application of quantum-mechanicalmethods to the understanding of the structure of space-time. It focuses onnoncommutative geometry and the gauge/gravity duality as intermediariesbetween quantum mechanics and classical geometry.First, we numerically calculate entanglement entropy and mutual infor-mation for a massive free scalar field on commutative and noncommutative(fuzzy) spheres. To define a subregion with a well-defined boundary in thenoncommutative geometry, we use the symbol map between elements of thenoncommutative algebra and functions on the sphere. We show that theUV-divergent part of the entanglement entropy on a fuzzy sphere does notfollow an area law. In agreement with holographic predictions, it is exten-sive for small (but fixed) regions. This is true even in the limit of smallnoncommutativity. Nonetheless, we find that mutual information (which isUV-finite) is the same in both theories. This suggests that nonlocality atshort distances does not affect quantum correlations over large distances ina free field theory.Previous work has shown that a surface embedded in flat R3 can be as-sociated to any three Hermitian matrices. By constructing coherent statescorresponding to points in the emergent geometry, we study this emergentsurface when the matrices are large. We find that the original matricesdetermine not only shape of the emergent surface, but also a unique Pois-son structure. We prove that commutators of matrix operators correspondto Poisson brackets. Through our construction, we can realize arbitrarynoncommutative membranes.Finally, we use the gauge/gravity correspondence to translate the pos-itivity of relative entropy on the boundary into constraints on allowablespace-time metrics in the bulk. Using the Einstein equations, we interpretthese constraints as energy conditions. For certain three-dimensional bulks,we obtain strict constraints coming from the positivity of relative entropywith a thermal reference state which turn out to be equivalent to a versionof the weak energy condition near the boundary. In higher dimensions, weuse the canonical energy formalism to obtain similar but weaker constraints.iiPrefaceChapter 2 is the merger of two published papers: Entanglement entropy onthe fuzzy sphere, by Joanna L. Karczmarek and Philippe Sabella-Garnier[1] and Mutual information on the fuzzy sphere by Philippe Sabella-Garnier[2]. The work in section 2.2 was carried out by Joanna Karczmarek. Thenumerical computations were carried out by the thesis author. The analysisof results was done jointly by Joanna Karczmarek and the thesis author,with the exception of section 2.4.2 which was done by the thesis author.A version of chapter 3 was published as Emergent geometry of mem-branes, by Mathias Hudoba de Badyn, Joanna L. Karczmarek, PhilippeSabella-Garnier and Ken Huai-Che Yeh [3]. The paper was drafted byJoanna Karczmarek, based on shared work by Joanna Karczmarek, KenYeh and the thesis author. Figures and numerical calculations were doneby the thesis author. Significant speed up in the numerical calculations waspossible thanks to a breakthrough by Mathias de Badyn.Chapter 4 is based in part on results published in Inviolable energy condi-tions from entanglement inequalities, by Nima Lashkari, Charles Rabideau,Philippe Sabella-Garnier and Mark Van Raamsdonk[4]: the results in sec-tion 4.2 were presented in that paper and are the joint work of Mark VanRaamsdonk, Charles Rabideau and the thesis author. The results in section4.3 have not been published and are the sole work of the thesis author.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation and outline . . . . . . . . . . . . . . . . . . . . . 11.2 Strings and branes . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.1 The holographic entropy bound . . . . . . . . . . . . 41.3.2 AdS/CFT: exact statement . . . . . . . . . . . . . . . 51.3.3 Extending AdS/CFT: the holographic dictionary . . . 71.4 Noncommutative geometry . . . . . . . . . . . . . . . . . . . 91.4.1 Definition and general properties . . . . . . . . . . . . 91.4.2 Noncommutative geometry and string theory . . . . . 101.5 Entanglement entropy . . . . . . . . . . . . . . . . . . . . . . 121.5.1 Definition and applications to field theory . . . . . . 121.5.2 Holographic entanglement entropy . . . . . . . . . . . 142 Entanglement entropy on the fuzzy sphere . . . . . . . . . . 162.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Noncommutative geometry . . . . . . . . . . . . . . . . . . . 202.2.1 Noncommutative plane . . . . . . . . . . . . . . . . . 202.2.2 Noncommutative sphere . . . . . . . . . . . . . . . . 212.2.3 The polar cap on the noncommutative sphere . . . . 252.3 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.1 Entanglement entropy for quadratic Hamiltonians . . 27ivTable of Contents2.3.2 The free scalar field on the commutative sphere . . . 282.3.3 The free scalar field on the fuzzy sphere . . . . . . . . 322.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4.1 Entanglement entropy . . . . . . . . . . . . . . . . . . 342.4.2 Mutual information . . . . . . . . . . . . . . . . . . . 382.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Emergent geometry of membranes . . . . . . . . . . . . . . . 493.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Basic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3 Coherent state and its properties . . . . . . . . . . . . . . . . 553.3.1 Example: noncommutative plane . . . . . . . . . . . 553.3.2 Example: noncommutative sphere . . . . . . . . . . . 563.3.3 Looking ahead: polynomial maps from the sphere . . 573.3.4 Example: noncommutative ellipsoid . . . . . . . . . . 583.3.5 Polynomial maps from the sphere . . . . . . . . . . . 683.3.6 Local noncommutativity . . . . . . . . . . . . . . . . 713.3.7 Coherent states overlaps, U(1) connection and Fµν ona D2-brane . . . . . . . . . . . . . . . . . . . . . . . . 733.3.8 Nonpolynomial surfaces . . . . . . . . . . . . . . . . . 753.4 Large N limit and the Poisson bracket . . . . . . . . . . . . 773.5 Area and minimal area surfaces . . . . . . . . . . . . . . . . 813.6 The torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.7 Open questions and future work . . . . . . . . . . . . . . . . 864 Energy conditions from entanglement inequalities . . . . . 884.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.1.1 Relative entropy and Fisher information . . . . . . . 884.1.2 The first law of entanglement entropy . . . . . . . . . 894.1.3 The modular Hamiltonian for CFTs and the Einsteinequation . . . . . . . . . . . . . . . . . . . . . . . . . 904.1.4 Canonical energy . . . . . . . . . . . . . . . . . . . . 914.1.5 Energy conditions . . . . . . . . . . . . . . . . . . . . 934.2 Asymptotic constraints from positivity of relative entropy . . 944.2.1 Warm-up: spatial interval with the vacuum as a ref-erence state . . . . . . . . . . . . . . . . . . . . . . . 944.2.2 Finding a stricter reference state . . . . . . . . . . . . 974.2.3 Tilted intervals with a thermal reference state . . . . 994.3 Canonical energy: higher-dimensional constraints . . . . . . 100vTable of Contents5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109viList of Figures1.1 Schematic illustration of the AdS/CFT duality . . . . . . . . 71.2 The Ryu-Takayanagi prescription for calculating entangle-ment entropy between boundary regions A and A¯ . . . . . . . 142.1 Degrees of freedom on the sphere and their matrix counterparts. 252.2 Fraction of total number of degrees of freedom in a ring ofconstant area centered at polar angle θ on a commutativeand fuzzy sphere . . . . . . . . . . . . . . . . . . . . . . . . . 272.3 Scaled entanglement entropy for µ = 1 at different N as afunction of the power p of the cutoff mmax = Np . . . . . . . 332.4 Entanglement entropy vs. area of boundary on a commuta-tive sphere with N = 75 for µ = 1.0. . . . . . . . . . . . . . . 352.5 Slope a of Scomm vs. A at different N, with µ = 1.0 on thecommutative sphere. . . . . . . . . . . . . . . . . . . . . . . . 362.6 Entanglement entropy S as a function of angular size θ ofpolar cap C on the noncommutative sphere. N = 200 andµ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.7 Entanglement entropy for half of the fuzzy sphere as a func-tion of N for µ2 = 1 and µ2 = 0.001. . . . . . . . . . . . . . . 382.8 Entanglement entropy S scaled by N as the fractional area aof polar cap for different values of N on the fuzzy sphere. . . 392.9 Entanglement entropy for half of the fuzzy sphere as a func-tion of inverse mass. . . . . . . . . . . . . . . . . . . . . . . . 402.10 Entanglement entropy S scaled by N as the fractional area aof polar cap for a large mass on the fuzzy sphere, µ2 = 1000(N=50). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.11 Regions A and B between which we calculate the mutual in-formation I(A,B). . . . . . . . . . . . . . . . . . . . . . . . . 422.12 Mutual information for two polar caps separated by an annu-lus centered on the equator. . . . . . . . . . . . . . . . . . . . 43viiList of Figures2.13 Mutual information for two polar caps separated by an annu-lus of angular width δ centered on the equator, µ = 1.0 . . . . 442.14 Mutual information on the fuzzy sphere for two polar capsseparated by an annulus of angular width δ centered on theequator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.15 Mutual information between a polar cap A of fixed size anda region B separated by A by an annulus of size δ. . . . . . . 462.16 Coefficients of cos(2nθ) terms in a Fourier expansion for en-tanglement entropy S shown in figure 2.6. . . . . . . . . . . . 473.1 Difference between x3 at finite N and c, as a function of N . . 653.2 Magnitude, ∆, of the difference between the approximateeigenvector and the exact eigenvector as obtained numeri-cally, for the ellipsoid in figure 3.1. . . . . . . . . . . . . . . . 663.3 The difference between the actual eigenvalue x3 and the clas-sical (large N) position c for 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). . . . . . . . . . . . . . . . 693.4 θ23/θ12 for the example in figure 3.3. This ratio appears todecrease like J−1. . . . . . . . . . . . . . . . . . . . . . . . . . 723.5 Magnitude of the overlap between the eigenstate correspond-ing to the point p at the north pole and the eigenstate corre-sponding to a point p’ a distance |d| away. . . . . . . . . . . . 743.6 Angle φ between the normal vector ~n computed using equa-tion (3.8) and the noncommutativity vector ijkθjk, for thesurface in equation (3.89) at a point given by x = 0.5, y = 0. 763.7 Relative error in the noncommutative area as given in equa-tion (3.103) compared to the classical area, for an ellipsoidwith major axes 6, 3 and 1. . . . . . . . . . . . . . . . . . . . 82viiiAcknowledgementsFirst of all, I would like to express my sincere gratitude to my supervisor,Joanna Karczmarek. This thesis would not have been possible without herinsight, vision and perseverance. I am also thankful for her support andguidance throughout the last five years.I am also grateful to Mark Van Raamsdonk for including me in theproject of which chapter 4 is part of, for shared work on some of the resultstherein and for feedback on that chapter. I am thankful to my other col-laborators, Nima Lashkari, Charles Rabideau and Ken Yeh, for our sharedproductive work and to Matt Beach for feedback on chapter 4. For feedbackon this whole manuscript, I thank members of my PhD committee: MarcelFranz, Janis McKenna and Gordon Semenoff.These last few years would not have been the same without being sur-rounded by the current and former string theorists of Hennings 418, whocreated an environment conductive to thinking, both technical and not. Inparticular, I want to thank Laurent and Charles for spirited discussions onlife, the universe and everything.Finally, I would like to thank my family, friends and loved ones formaking me a better person so that I could focus on being a better physicist.This work was funded in part by the Fonds de Recherche du Que´bec–Nature et Technologies and by the Walter C. Sumner foundation.ixChapter 1IntroductionIt is not unreasonable to imaginethat information sits at the coreof physics, just as it sits at thecore of a computer.John Archibald Wheeler, 19981.1 Motivation and outlineThe principal challenge facing theoretical physics has long been the inabilityto formulate a consistent quantum theory of gravity. Since Einstein’s generaltheory of relativity (GR), gravity has been recognized as a geometrical effect:the fundamental degrees of freedom of the theory are packaged in the metricthat measures distances in space-time. The metric can be regarded as afield, and the classical field equations obtained from the Einstein-Hilbertaction have been experimentally verified. However, that field theory is notrenormalizable. In other words, applying the usual quantization techniquesthat have yielded the Standard Model of particle physics is not possible:the resulting quantum theory would either lose all predictive power or haveunacceptable divergences.One could ask whether it is necessary to have a quantum theory ofgravity: would it not suffice to have quantum fields evolving on a classicalspace-time? The answer, unfortunately, is no. When the regimes of appli-cability of classical general relativity and quantum field theory overlap, theresult is a breakdown of at least one of the two. The most striking exampleof this breakdown is the black hole information paradox. If we consider fieldtheory in the vicinity of a classical black hole, we are inevitably lead to theconclusion that information must be lost. This is a violation of one of thefundamental principles of quantum mechanics: unitarity. If we are unwillingto accept the breakdown in quantum mechanics, it is therefore necessary to11.1. Motivation and outlinetreat the black hole, a gravitational system, as a quantum object.1Any connection between quantum mechanics and general relativity wouldthus be of considerable interest. The current leading candidate for a fullyquantum theory of gravity is superstring theory. One of the products ofsuperstring theory is the holographic duality: in its most conservative form,it is a duality between strongly-coupled supersymmetric Yang-Mills theorywith a large number of colours living in R3,1 and classical gravity in anAdS5 × S5 background. It is also conjectured to hold more generally, defin-ing a quantum theory of gravity on an asymptotically Anti de Sitter spacebackground as being dual to a conformal field theory defined on the spatialboundary of the geometry.Another connection between quantum mechanics and geometry is fea-tured in string theory. Often, the space in which the strings evolve is non-commutative. Noncommutative spaces can be seen as a quantization ofordinary spaces: position coordinates are promoted to operators in muchthe same sense as position and momentum in quantum mechanics. Matterfields are represented by other operators, and are on a similar footing asposition. By mixing quantum mechanical concepts such as states and oper-ators with geometrical concepts such as area and volume, we can hope tolearn more about an eventual quantization of gravity.One final concept can help us make deep connections between geome-try and quantum mechanics: quantum information. Recently, the idea thatspace-time fundamentally consists of information has been the focus of alot of research. That research is built on a decidedly quantum property:entanglement, and the entropy associated with it. Entanglement entropy isaffected in known ways by the structure of space, and can in turn informus of that structure. Furthermore, using the holographic duality entangle-ment entropy in field theory can be related to the area of certain surfacesin gravitational theories. This relationship between the quantum and thegeometrical has been exploited to study space-time from a quantum per-spective and even, in some contexts, to derive the Einstein field equationsfrom quantum-mechanical considerations [6, 7].This thesis is focused on advancing the understanding of geometry fromquantum mechanical properties, using the above connections between the1Recently, the information paradox has been sharpened into the firewall paradox[5].Some argue that there must be a violation of either semi-classical gravity or quantummechanics right behind the horizon of the black hole. The most conservative solutionadvocated by the authors is that there is a “firewall” of high energy behind the horizonof the black hole, even though classically the horizon can be made to have very littlecurvature.21.2. Strings and branestwo. The rest of this introduction is used to lay the groundwork for thefollowing chapters by describing in more detail holographic duality, noncom-mutative geometry and entanglement entropy. In Chapter 2, entanglemententropy and related quantities are used to study the geometrical distributionof the degrees of freedom of a free field theory defined on a noncommutativesphere. Chapter 3 describes the emergence of a wide class of noncommu-tative two-dimensional surfaces in string theory. These surfaces can be un-derstood through the spectrum of a particular operator. The states in thatspectrum correspond to points on the surfaces and are mathematically anal-ogous to the coherent states of a quantum harmonic oscillator. In Chapter 4,constraints on classical geometry obtained from constraints on entanglemententropy through the holographic duality are presented. Finally, Chapter 5 isa summary of these results and a presentation of possible future extensionsof this work.1.2 Strings and branesString theory was initially conceived as a theory of the strong nuclear force.In the late 1960s, the original idea was to model mesons as tiny strings toexplain their apparent classification along so-called Regge trajectories of theform l = α(0) + α′s, where l is the angular momentum and s the square ofthe centre-of-mass energy of the particle and α′ is a constant characterizingthe family [8]. After the development of QCD in the 1970s, this property wasunderstood as arising from the string-like properties of flux tubes connectingthe quark and anti-quark forming the meson (e.g. [9]). The success ofQCD and the need for a large number of dimensions to avoid anomaliesfrom quantization (either d = 26 for the bosonic string or d = 10 for whatbecame known as the superstring) led researchers to abandon string theoryas a model of hadrons [8].In 1974, Joe¨l Scherk and John Schwarz approached string theory as afundamental theory in its own right instead, interpreting the massless spin-2state of its spectrum as a graviton and the massless spin-1 states as gaugebosons [10]. They found that such a theory would have no UV divergencesand naturally unified general relativity and quantum field theory. In the1980s, five consistent supersymmetric string theories were classified [11, 12]:they all live in ten space-time dimensions, have no tachyon in their spec-trum (a problem with bosonic string theory) and have supergravity as alow-energy limit. In the late 1980s and early 1990s, higher-dimensional dy-namical objects on which open strings can end (now called D-branes) were31.3. Holographyshown to appear naturally in the spectrum of string theories [13] and sourceRamond-Ramond fields present in supergravity [14], strengthening the un-derstanding of dualities between the different string theories.At the 1995 Strings conference, Ed Witten took this further by suggestingthat the five different string theories were in fact just different limits of oneover-arching, eleven-dimensional theory: “M-theory” [15]. The story camefull-circle in 1997 when Juan Maldacena showed that type IIB superstringtheory on a background of AdS5 × S5 is itself dual to large-N N = 4 super-Yang-Mills theory on 3 + 1 dimensions (i.e. a supersymmetric version ofQCD) [16].Maldacena’s duality is the most famous example of what is now knownas holographic duality (or sometimes informally as AdS/CFT). This dual-ity associates to each physically independent state of a quantum theory ofgravity in the classical limit2 a physical state of a strongly-coupled quantumtheory of a large number of fields. The gravitational state should describea space-time that asymptotically approaches (d + 1)-dimensional Anti deSitter space and the quantum field theory should be thought of as living onthe causal boundary of that space-time. Although no formal proof of theholographic duality in general exists, specific implementations (includingMaldacena’s original one) have withstood numerous tests and consistencychecks (see, e.g. [17]).1.3 Holography1.3.1 The holographic entropy boundThe starting point for holography is the famous result of Bekenstein [18]and Hawking[19] that the entropy of a black hole is given bySBH =A4GN, (1.1)where A is the area of the black hole’s event horizon and GN is Newton’sgravitational constant. The proportionality of entropy to area can be estab-lished by classical arguments [18]. Very schematically, consider throwing aclassical object (say, a balloon full of hot gas) of energy E and entropy Sinto a black hole. We have that E ∼ S. The area of a black hole isA = 4pir2S = 16piM2 , (1.2)2By classical limit, we mean the limit in which both loop and higher curvature correc-tions are suppressed.41.3. Holographywhere rs is the Schwarzschild radius and M is the mass of the black hole.Thus, a change of mass ∆M will correspond to a change of area∆A ∼M∆M (1.3)(for ∆M M). If we identify ∆M ∼ E, we are led to∆S ∼ ∆A . (1.4)This argument can be made precise, fixing the proportionality constant, byconsidering quantum fields evolving on a fixed black hole space-time[19].One is then led to the conclusion that black holes not only have an entropybut also a definite temperature at which they radiate.Equation 1.1 has an important consequence: the entropy contained inany region is bounded by the area of the boundary of the region. To see this,suppose we had a system with an entropy S that is entirely contained in aregion of area A < 4GNS. This system must have less mass than the blackhole with the same area as the region, otherwise it would have collapsed intoa black hole. If we then add just enough mass and energy to the systemfor it to collapse into a black hole of area A, then, by the second law ofthermodynamics, the resulting black hole would violate 1.1. It must thenbe that for any region of space:S ≤ A4GN. (1.5)Relation 1.5 is known as the holographic entropy bound. If we regard en-tropy as a capacity to store information, then we see that the maximumamount of information stored in a gravitational system is proportional tothe area of the system. This is to be contrasted with a non-gravitationalsystem, for which the information stored is generically extensive (i.e. pro-portional to its volume). Hence the idea, still crude at this stage, that agravitational theory is a “hologram”: a higher-dimensional image of a lower-dimensional non-gravitational theory.1.3.2 AdS/CFT: exact statementThe general idea of holography can be made much sharper by appealing tostring theory, without assuming that string theory is the “correct” theoryof quantum gravity. The canonical example of holography is as follows.3. Consider a stack of N  1 coincident D3-branes in a ten-dimensional3This was initially introduced in [16] The canonical review is [17]. Here, we mostlyfollow the presentation of [20]51.3. Holographyflat space-time. There are two free parameters in string theory: the stringcoupling constant gs and the string tension 1/α′. When λ ≡ gsN is small,string perturbation theory is well defined. In that case, we can think ofour setup as containing three types of string interactions: open-open, open-closed and closed-closed. The open-open interactions happen on the stackof D-branes and are described by N = 4 super Yang-Mills with ’t Hooftcoupling λ. 4 The supersymmetry ensures that the theory has conformalinvariance that survives quantization. If we then take a low-energy limit, theopen-closed interactions vanish because the interaction is irrelevant in theWilsonian sense. When λ is large, the perturbative description can no longerbe used. However, supergravity is a valid approximation to string theoryin that limit, and the supergravity configuration with the same conservedcharges as the stack of D3-branes is a black brane, with a metric of the formds2 = H(r)−1/2ηµνdxµdxν +H(r)1/2dxmdxm , µ, ν = 0..3 ,m = 4..9 ,(1.6)H(r) = 1 +L4r4, L2 = 4pigsNα′2 , r2 = xmxm . (1.7)A black brane is essentially black hole with a planar horizon instead of aspherical one. In these coordinates, the horizon is at r = 0. As with blackholes, excitations living near the horizon appear red-shifted to far awayobservers. Therefore, taking r → 0 ensures that we have low-energy modes.The metric in that limit takes the formds2 =r2L2ηµνdxµdxν +L2r2dr2 + L2dΩ25 . (1.8)(We have re-written the metric in the m directions in spherical coordinates).This is the metric of AdS5×S5. There are other possible low-energy modes:they correspond to small excitations of massless fields at any r. The onlymassless fields are gravitons. In string theory, gravitons are the masslessexcitations of closed strings.We have described two different regimes of the same low-energy setup:the small λ regime corresponds to a particular supersymmetric gauge the-ory as well as massless excitations of closed strings and the large λ regimecorresponds to supergravity in an AdS5×S5 background and massless exci-tations of closed strings. Now, let’s make an additional, innocuous looking4That is to say, a large N SU(N) gauge theory with four distinct supersymmetries andcoupling constant g2YM = gs61.3. HolographyFigure 1.1: Schematic illustration of the original AdS/CFT duality. Nshould be taken to be large. The conjecture is that, starting from the top-left corner, we can follow the arrows in a different order and still end up inthe bottom-right corner.assumption: let’s assume that the large λ and low-energy limits commute.Factoring out the closed strings, we have thatN = 4 SU(N) Super Yang-Mills with N →∞, λ→∞=Type IIB Supergravity on AdS5 × S5 . (1.9)This is Maldacena’s original AdS/CFT duality. It is summarized in figure1.1. Many others can be obtained by starting with something different thana single stack of D3-branes. The general result is always a duality betweena strongly-coupled conformal field theory and semi-classical gravity in anasymptotically-AdS space-time with one more noncompact dimension (andsome number of compact dimensions).1.3.3 Extending AdS/CFT: the holographic dictionaryThere is a general dictionary between quantities in the field theory andquantities in the gravitational theory. First, the field theory can be thoughtof as living on the boundary of the gravitational theory. That statement is abit of an abuse of language: strictly speaking, the two theories are defined ondifferent space-times and do not “talk” to each other, since they are different71.3. Holographyformulations of the same theory. But the image of the field theory livingon the boundary of AdS is a useful one, because some notions of localityremain. Take z to be the extra holographic dimension (with the boundarylocated at z = 0) and x to be the boundary directions. As z → 0, the metriccan be written asds2 =L2z2(dz2 + ηµνdxµdxν), µ, ν = 0 · · · d− 1 . (1.10)An on-shell bulk scalar field φ(x, z) with mass m goes asφ(x, z) ∼ z∆φ(x) as z → 0 , (1.11)with m2 = ∆(∆− d). In the field theory, such a field is dual to an operatorO(x) = COφ(x) (1.12)with conformal weight ∆ and CO some constant. This can be generalized toinclude spin. Notice that this procedure assigns to each bulk field a specificboundary operator, but the reverse operation is not uniquely defined. Thisreflects the added coordinate reparametrization freedom in the bulk. If thebulk metric is written in Fefferman-Graham coordinates [21], coordinates ofthe formds2 =L2z2(dz2 + ηµνdxµdxν + hµν(x, z)dxµdxν), (1.13)with hµν(x, z) a power series in z, then the lowest order term in z sets theexpectation value of the CFT stress tensor. Higher-order terms in hµν arefixed by imposing the Einstein equation order-by-order. The z coordinateplays the role of an energy scale in the boundary field theory. Notice thatthe metric diverges as z → 0, so to calculate most quantities in the bulk itis necessary to regulate it by imposing an IR cutoff: z ≥  for some small .This cutoff corresponds to a UV cutoff in the boundary field theory. To seethis, it is easier to do a change of coordinates u = L2z . In these coordinates,the AdS metric takes the formds2 =L2u2du2 +u2L2ηµνdxµdxν (1.14)and the boundary is located at u = ∞. A conformal field theory must beinvariant under a dilatation transformation: xµ → λxµ, which scales theenergy scale as E → E/λ. The corresponding symmetry in the bulk isxµ → λxµ, u → u/λ: it is easy to see that it is an isometry that leaves the81.4. Noncommutative geometrymetric invariant. Hence the identification of z with L2/E: the holographiccoordinate is related to the energy scale, with regions near the boundaryassociated with the UV of the field theory and regions far from the boundarywith the IR. Correlators and other similar quantities can be matched byequating the partition functions of the two theories:ZCFT = ZAdS (1.15)and taking appropriate functional derivatives. Of course, the full quantumgravity partition function is not known. Depending on the desired appli-cation, it can either be taken to be defined by 1.15, or we can do a saddlepoint approximation, takingZAdS ≈ e−IAdS,classical . (1.16)It is conjectured that 1.15, from which the whole dictionary can be derived,should hold for a large class of strongly-coupled CFTs and correspondingclassical gravity theories. One of the prime reasons for this is that theisometry group of AdSd+1, SO(2, d), is isomorphic to the conformal groupin d dimensions (i.e. the group of symmetries of a d-dimensional CFT).1.4 Noncommutative geometry1.4.1 Definition and general propertiesThe basic idea behind noncommutative geometry is simple: coordinates arereplaced by operators that do not commute with each other. The mostimmediate consequence of this change is that, following the logic of theHeisenberg uncertainty principle, it is impossible to determine exactly aposition along more than one coordinate axis. In other words, there is aminimal area in position space that can be resolved (analogous to a cell ofsize ~ in phase space for ordinary quantum mechanics). To understand thisbetter, we can look at a very simple example: the noncommutative plane.DefineX =`2(a† + a) , Y = i`2(a† − a) , (1.17)where a, a† are the usual harmonic oscillator raising and lowering operatorswith [a, a†] = 1. The commutation relation between the coordinate operatorsis then[X,Y ] = i`22(1.18)91.4. Noncommutative geometryand the uncertainty principle then gives us that for any state〈∆X∆Y 〉 ≥ `24. (1.19)But we must of course define what we mean by “state” in this context. Wewant to be able to associate states to points on a two-dimensional plane.We could, for example, define X eigenstates. But these states would thenhave no sensible position along the Y axis (and vice-versa). The sensiblething to do is then to define states |α〉 that saturate the uncertainty bound1.19 in a symmetric way:〈α|∆X|α〉 = 〈α|∆Y |α〉 = `2. (1.20)We recognize these states as equivalent to the coherent states of a harmonicoscillator:|α〉 = e−|α|2/2eαa† |0〉 . (1.21)These states have the property thata|α〉 = α|α〉 , (1.22)and so〈X〉 = ` Re (α) , 〈Y 〉 = ` Im (α) . (1.23)Thus, we cover the whole complex plane with these states. Notice that themap is not one-to-one since the states are overcomplete:|〈β|α〉|2 = e−|α−β|2 . (1.24)We can go on to define functions of noncommutative coordinates and builda noncommutative analogue of quantum field theory on a noncommutativemanifold (see, e.g., [22]).1.4.2 Noncommutative geometry and string theoryIn the context of string theory, noncommutative geometry is interesting be-cause it is the natural description of the position of D-branes[22, 23]. Inperturbative string theory, a Dp-brane is a p-dimensional surface on whichopen strings can end [24]. But Dp-branes also have dynamics of their own,as we have seen when discussing AdS/CFT. For N coincident (stable) Dp-branes, the low-energy limit of the action in the absence of background fieldsis 10-dimensional U(N) super-Yang-Mills theory dimensionally reduced to101.4. Noncommutative geometry(p+1)-dimensions [25]. The 9− p scalars coming from the gauge field com-ponents in the dimensionally reduced directions correspond to the positionsof the Dp-branes in the directions normal to the branes. These scalars (inthe gauge theory sense) are N ×N matrices. In the ground state, they allcommute and the eigenvalues are just the positions of the branes. However,in general they can be non-commuting and we must conclude that the braneslive in a non-commutative geometry. We can consider the example of D0branes to see this more clearly.10-dimensional SYM reduced to 0+1 dimensions has an action of theform [26] 5S =12g∫dttr(X˙iX˙i − 12[Xi, Xj ]2 + 2ΘT Θ˙− 2ΘTΓi[Θ, Xi])(1.25)Where i = 1 . . . 9, Xi are N × N matrices of scalars, Θi are matrices ofspinors, Γi are the usual gamma matrices. The bosonic part of the action issimply quantum mechanics for matrices of bosons 6. Setting all the fermionsto zero, we can write:H =∫dttr (ΠiΠi + V (Xi)) (1.26)with [Πi, Xj ] = iδij1N×N . In the absence of background fields, the energy isminimized by having the matrices commute. But it is possible to introducecoupling between the D0-branes and a background R-R field to modify thepotential. For example, a constant F4 flux with F0ijk = fijk along threedirections and zero in the other directions leads to a potential of the form[25, 27]V = −14[Xi, Xj ][Xi, Xj ]− i3fijkXiXjXk (1.27)which has [Xi, Xj ] = ifijkXk as a solution. This corresponds to a fuzzysphere: it has a solution of the form Xi = fLi, where Li are the usual spin-JSU(2) generators (N = 2J + 1). If we define R such that f = R√J(J+1)thenwe have that3∑i=1XiXi =R2J(J + 1)3∑i=1LiLi = R21N×N (1.28)5As an aside, it is remarkable that this action is conjectured to be the exact action of11-dimensional M-theory in a frame with infinite boost [26] and that we therefore expectno higher-order corrections6Note that the classical degrees of freedom are matrices, even before quantization111.5. Entanglement entropywhich is clearly the matrix analogue of∑3i=1 xi = r2, the equation describinga sphere [22, 28] . It should be noted that the geometry is recognized hereby the structure of the algebra of the non-commutative coordinates. Thisis also the case when dealing with the noncommutative plane, as outlinedearlier (where the algebra was the infinite-dimensional Heisenberg algebra).1.5 Entanglement entropy1.5.1 Definition and applications to field theoryOne of the latest and most promising entries in the holographic dictionaryconcerns entanglement entropy. Recall that a quantum state of two or moredegrees of freedom is said to be entangled when it cannot be written as atensor product. For example, consider a system of two spins (or “qubits”)that can be either up or down. The state|φ〉 =|↑↑〉 (1.29)is not entangled, whereas the state|ψ〉 = 1√2(|↑↑〉+ |↓↓〉) (1.30)is entangled. The entanglement entropy of a subsystem A is defined as thevon Neumann entropy of the density matrix obtained by tracing over thecomplement A¯ (see, e.g. [29]):ρA = TrA¯ (|ψ〉〈ψ|) , (1.31)SA = −TrA (ρA log ρA) . (1.32)We define functions of matrices by their Taylor series, as usual. For thestate 1.29, if we take A to be the first qubit we have thatρA = 〈↑|↑↑〉〈↑↑|↑〉+ 〈↓|↑↑〉〈↑↑|↓〉=|↑〉〈↑| ,and so 7SA = −(1 log 1 + 0 log 0) = 0 . (1.33)7Strictly speaking, we should consider SA = − lim→0((1− ) log(1− ) +  log )121.5. Entanglement entropyIn contrast, for the state 1.30 we have:ρA =12(|↑〉〈↑| + |↓〉〈↓|) ,SA = −(12log12+12log12)= log 2 .This is the result we would expect since there are two possible states A¯ couldbe in if we only know ρA.The construction can be trivially extended to a system consisting of anarbitrary finite number of degrees of freedom. If we impose ultraviolet andinfrared regulators to a quantum field theory, we can think of the resultingsystem as having a finite number of degrees of freedom. If we take A tobe a subregion of the space on which the field theory is defined, we canexamine geometrical properties associated to entanglement entropy. Thefirst striking property is that in a local field theory regulated by a small-distance regulator , the entanglement entropy for a region A in the groundstate of the theory follows an area law[30]:SA ∼ |∂A|d−1+ · · · , (1.34)where |∂A| is the area of the boundary, d is the number of space dimensionsand · · · stand for less divergent terms. This property is dependent on thelocality of the field theory. Since the interactions are local, in the regulatedtheory only adjacent degrees of freedom are entangled. A single degreeof freedom occupies a volume of order d, and the volume of degrees offreedom entangled between A and A¯ is |∂A|. This means that there aren|∂A|d−1 microstates consistent with the density matrix for A, where n is someconstant factor fixed by the details of the theory, hence the area law.The area law divergence of entanglement entropy masks a lot of structurevisible in the IR. To get rid of the divergence, we can consider mutualinformation. The mutual information between two disjoint regions A and Bis defined asI(A,B) = SA + SB − SA∪B . (1.35)Since A and B are disjoint, the area of A ∪ B is the sum of the areas of Aand B, and so the leading-order divergences cancel out 8.8Notice that the divergence would also cancel out if we had a volume law instead of anarea law, for the same reason.131.5. Entanglement entropyFigure 1.2: The Ryu-Takayanagi prescription for calculating entanglemententropy between boundary regions A and A¯1.5.2 Holographic entanglement entropyFor strongly-coupled systems with a gravity dual, Ryu and Takayanagi[31]have given a prescription for calculating entanglement entropy. If A and A¯are regions living on the boundary of AdS, we have thatSA =|γA|4GN, (1.36)where γA is a bulk codimension-2 spacelike surface anchored on the boundarybetween A and A¯ (as seen in figure 1.2) with minimal area.Note that γ is non-trivial because of the structure of the metric of Antide Sitter space, which diverges near the boundary. It is this divergence thatallows a surface to reduce its area by probing deeper into the bulk. Forexample, consider the Poincare´ patch of AdS2+1. The spatial part of themetric isds2 =L2z2(dz2 + dx2), (1.37)where z is the holographic direction and z = 0 is the boundary. If we takeA = x ∈ [−R,R] and parametrize our minimal curve as x(z), the length of141.5. Entanglement entropythe curve is given byL =∫ds = 2L∫ z0dzz√1 + x˙2 , (1.38)where  regulates the boundary divergence and z0 is the point of deepestpenetration in the bulk. Using the Euler-Lagrange equation, we have thatx˙z√1 + x˙2= K , (1.39)with K an integration constant. By symmetry, ˙x(z0) diverges, and so wesee that K = 1z0 . The expression can be integrated, and using the fact thatx(z = 0)2 = R2 we get that the minimal surface is given byx2 + z2 = R2 . (1.40)Finally, evaluating 1.38, we haveS =14GN2L log(2R)(1.41)In the holographic dictionary, AdS3 is dual to the vacuum state of a stronglycoupled CFT with central charge c = 3L2GN9, so that we get that the entan-glement entropy for an interval of length l = 2R would beS =c3log(l), (1.42)which is exactly the result calculated through more elaborate field theorymethods [34]. Note that to obtain the field theory result, one must impose aUV cutoff corresponding to a minimal lengthscale . This is consistent withthe statement made earlier that the holographic direction corresponds to anenergy scale.The Ryu-Takayanagi prescription is valid for static bulks. The Hubeny-Rangamany-Takayanagi prescription [35] extends it to non-static cases, butis not necessary for the purposes of this thesis. A proof of the Ryu-Takayanagiprescription [36] as well as a prescription to include quantum corrections toit [37] are also known, but beyond the present scope.The striking feature of 1.36 is the direct connection it makes betweenquantum mechanical entanglement and classical geometry. Interpreted in aliteral sense, it seems to imply that geometry is a feature that emerges fromthe entanglement of field theory degrees of freedom [38, 39]. We will returnto this idea in Chapter 4.9This statement actually predates AdS/CFT and can be obtained by examining theasymptotic symmetries of AdS3, as in [32, 33]15Chapter 2Entanglement entropy on thefuzzy sphere2.1 IntroductionIn this Chapter, we study the geometric structure of degrees of freedom onthe noncommutative sphere using entanglement entropy.Field theories on noncommutative spaces are interesting for many rea-sons, one of them being that these theories are inherently nonlocal and assuch might serve as toy models for certain phenomena in quantum gravity.For example, it has been conjectured that black hole horizons scramble in-formation so fast that in the membrane paradigm the horizon of a black holecannot be modeled by a local theory [40]. It has been suggested that non-commutative theories might in fact be examples of fast scramblers, since (atleast at strong coupling) they exhibit enhanced thermalization rates com-pared to local theories [41].As discussed in Chapter 1, noncommutative gauge theories arise nat-urally in string theory as the effective theories for low energy degrees offreedom on a D-brane with a worldvolume magnetic field [42]. Using thisfact, it is possible [43, 44] to find holographic duals to such theories. Intrigu-ingly, the dual geometry is an ordinary manifold , with noncommutativityencoded in the shape of the holographic dual. Since the dual has ordinarygeometry, it can be used to study intrinsically geometric observables such asthe entanglement entropy associated with some region of space and obtainedby tracing out all degrees of freedom residing outside of this region. If weare able to provide an interpretation for such holographically defined geo-metric observables in the noncommutative field theory, we can study hownoncommutative space emerges both from the gravitational dual and thenoncommutative algebra on the field theory side. Entanglement entropy isa particularly interesting geometric observable as it provides information onhow degrees of freedom at different points are coupled, probing nonlocal-ity of the theory and perhaps teaching us about its scrambling behaviour162.1. Introduction(for example, it was shown in [45] that possessing extensive entanglemententropy is a necessary condition for scrambling).Previous work [46, 47] has studied holographic entanglement entropy andmutual information, uncovering some interesting properties of these observ-ables in strongly coupled noncommutative gauge theories.10 In particular,[47] argued that nonlocality leads to extensive (volume-law) holographic en-tanglement entropy, instead of the more usual area-law behaviour. Further-more it was also shown in [47] that mutual information for strongly-coupledtheories behaves differently in commutative and non-commutative theorydue to UV/IR mixing.While the type of UV divergence (area or volume law) of entanglemententropy is usually attributable to locality in the UV, mutual information iscalculated from the finite parts of entanglement entropy. It provides a boundon the range of correlations [49] and can therefore be seen as a measure ofnon-locality in the IR.To interpret the findings of [46, 47] within field theory, we must answerthe following questions: Can one divide the Hilbert space of a field theoryon some noncommutative geometry into two components associated withthe inside and the outside of some geometric region? If not, what preciselyis the meaning of holographic entanglement entropy in field theory, and ifyes, is the volume-law behaviour observed through a holographic descrip-tion a property associated with strong coupling or would it be also seen atweak coupling? This last question is further motivated by the fact that, forexample, the enhancement in thermalization timescale mentioned above isnot seen in perturbation theory [41].In this Chapter, we shed some light on these issues by considering oneof the simplest nontrivial noncommutative field theories: the theory for afree scalar on a noncommutative (or fuzzy) two-sphere.11,12 The main ad-vantage of working with the noncommutative sphere is that the field theoryis UV-finite and expressible as a finite size matrix model. Further, the ma-trix model for a free scalar is purely quadratic and therefore entanglemententropy is straightforward to compute.We compare both the entanglement entropy and the mutual informationto that of the free scalar field on the commutative sphere, which we regularizeby discretizing the polar angle θ.10See also an earlier work [48].11The theory of a free scalar on a noncommutative plane is equivalent to the free scalaron a commutative plane and therefore not interesting.12Previous work entanglement entropy on a fuzzy sphere includes [50, 51], where theentanglement entropy for half the sphere was computed.172.1. IntroductionTo describe the noncommutative two-sphere, we embed the sphere inthree dimensions and represent the three Cartesian coordinates by SU(2)generators in the N = 2J+1 dimensional irreducible representation [22, 28],as described in Chapter 1.We obtain entanglement entropy associated with a polar cap region Cwhose size is controlled by θ (see figure 2.1) for both theories. Using a mapfrom operators (matrices) to functions on the sphere—called a symbol [22]—we determine which entries of the matrix Φ correspond to degrees of freedominside this polar cap and which correspond to the outside. Thus, we writethe Hilbert space H of our matrix model as a product of two smaller spaces:one corresponding to the inside of the polar cap, HC and one correspondingto the outside of the polar cap, HC¯ , with H = HC ⊗ HC¯ . We computeentanglement entropy using the usual definition:S = −TrC (ρC ln ρC) , (2.1)whereρc = TrC¯ |ψ〉〈ψ| (2.2)is the density matrix associated with Hc when the entire quantum systemis in a state |ψ〉 (which we will take to be the vacuum).Of course, it is not possible to draw a sharp boundary for a region on anoncommutative sphere. For our procedure, the boundary of the region Ccan be thought of as having a thickness of√θ where θ = R2/N is the mag-nitude of the coordinates. We can compare the length-scale√θ = R/√Nto the UV cutoff of the theory, . The UV cutoff is most easily obtainedby dividing the area of the sphere 4piR2 by the total number of degrees offreedom in our noncommutative model, N2. Since a small region of area 2should contain exactly one degree of freedom,  is approximately R/N . Wesee that√θ is parametrically larger than .Since, in the large N limit, the noncommutative sphere is supposed toreduce to the commutative one, we might expect that the entanglemententropy on a noncommutative sphere would agree with that on a commuta-tive sphere for regions whose diameter is larger than√θ. If that were thecase, we would not be able to discover any deviation in the noncommutativecase, as regions whose boundary has thickness√θ cannot be smaller than√θ. Fortunately, as has been observed in [46, 47], strong deviations fromcommutative behaviour should be seen in entanglement entropy for regionswhose size is of order θ/ = R, which corresponds to the entire sphere (or,equivalently, the IR cutoff of our theory). The reason why noncommutativeentanglement entropy for regions larger than√θ (but smaller than θ/) does182.1. Introductionnot agree with its commutative counterpart lies in UV/IR mixing: becauseof this mixing, a noncommutative theory with a UV cutoff  is expected tohave nonlocal behaviour up to a length-scale θ/ [52]13. Thus, we expectdeviation from commutative behaviour at least for regions whose area is asmall (but finite in the commutative limit) fraction of the total sphere area.Any such deviation we see can be interpreted as a result of UV/IR mixingon the noncommutative sphere.In fact, this is precisely what we discover: for small regions, entangle-ment entropy on the noncommutative sphere grows linearly with the areaof the region (and not with the length of its boundary), and hence followsthe volume-law.14 For regions whose area is comparable to the total areaof the sphere, the entanglement entropy receives higher power corrections.However, while in [47] it was shown that the entanglement entropy for a fieldtheory with some effective noncommutativity scale aθ at strong coupling un-dergoes a phase transition between volume-law at length-scales below a2θ/and area-law at length-scales above that, on the noncommutative spherethere is no such phase transition. This is due to the compactness of themanifold and the resulting IR cutoff which was absent in the holographiccalculation. The phase transition is replaced with crossover behaviour nearthe IR cutoff (for a region whose size is half that of the whole sphere) andthe higher power corrections mentioned above lead to a smooth behaviour.We also find that mutual information between two regions separated byan annulus of variable width is the same for the commutative and noncom-mutative theories, evaluating that quantity numerically. For the particularcase of a free conformal scalar field, we validate our results by an analyticalcalculation.13 In [52], the UV/IR connection was studied on the noncommutative plane. TheUV/IR mixing on the fuzzy sphere has been studied using the one-loop effective actionin several interacting theories (see for example [53] and [54]). Here we simply use theresults from flat noncommutative geometry as a guide to interpreting our results. Noticethat entanglement entropy could potentially be sensitive to UV/IR mixing not detectedby, for example, divergences in the two point functions. In this case our results could beinterpreted as evidence of previously undiscovered UV/IR mixing.14 Even though we are working in two spacial dimensions, we will continue to usehigher-dimensional terminology and refer to entropy growing with the area of the regionas volume-law behaviour and entanglement entropy proportional to the length of theboundary of the region as area-law behaviour.192.2. Noncommutative geometry2.2 Noncommutative geometryIn this section, we review the concepts of a symbol and a corresponding starproduct as a way to encode the noncommutative structure of geometry. Webegin by reviewing the better-known example of a noncommutative plane,and then show how the same tools can be applied to treat the noncommu-tative sphere. Our general approach is similar to that in [55], though thedetails are somewhat different. In section 2.2.3 we use our symbol map onthe fuzzy sphere to obtain our desired mapping between the polar cap andmatrix elements.2.2.1 Noncommutative planeAs discussed in Chapter 1, the noncommutative plane has as its structurealgebra the Heisenberg algebra. This is the algebra generated by two oper-ators xˆ and yˆ with the commutation relation[xˆ, yˆ] = iθ . (2.3)√θ has the units of length and is the fundamental length-scale of noncom-mutativity.A common treatment of noncommutative geometry uses a map s whichtakes elements Aˆ of the structure algebra to functions on the correspondingcommutative manifold (in this case, the ordinary two-dimensional plane),s(Aˆ) = fA(x, y). The function fA is called the symbol of the algebra elementAˆ. The symbol map is not unique: there are many different definitions of s,corresponding to different ways to order the algebra element s−1(fA). Forevery symbol map s there exists a so-called star product ∗ with the propertythats(AˆBˆ) = s(Aˆ) ∗ s(Bˆ) . (2.4)As AˆBˆ 6= BˆAˆ, ∗ cannot be a commutative product, but it is associative.We follow the approach due to Berezin [56] and define the symbol as anexpectation value in a coherent state:s(Aˆ) = fA(α, α¯) = 〈α|Aˆ|α〉 . (2.5)To derive a star product compatible with the Berezin symbol, we need touse the fact that |α〉 is, up to a normalization factor, a holomorphic functionof the complex variable α:|α〉 = e−|α|2/4θ∞∑n=0(α/√2θ)n√n!|n〉 . (2.6)202.2. Noncommutative geometryThis implies that 〈β|A|α〉/〈β|α〉 is holomorphic in α and antiholomorphicin β. Thus, since 〈β|β〉 = 1, we have〈β|A|α〉〈β|α〉 = e−β ∂∂α〈β|A|β + α〉〈β|β + α〉 = e−β ∂∂α eα ∂∂β〈β|A|β〉〈β|β〉 = e−β ∂∂α eα ∂∂β fA(β, β¯) ,(2.7)and, similarly,〈α|A|β〉〈α|β〉 = e−β¯ ∂∂α¯ eα¯ ∂∂β¯ fA(β, β¯) . (2.8)We can now compute an explicit expression for the star product, usingthe completeness relation for coherent states:(fA ∗ fB)(β, β¯) = 〈β|AB|β〉 = 12piθ∫d2α〈β|A|α〉〈α|B|β〉=12piθ∫d2α |〈β|α〉|2[e−β∂∂α eα ∂∂β fA(β, β¯)] [e−β¯∂∂α¯ eα¯ ∂∂β¯ fB(β, β¯)]=12piθ∫d2α(eβ∂∂α+β¯ ∂∂α¯ |〈β|α〉|2) [eα ∂∂β fA(β, β¯)] [eα¯ ∂∂β¯ fB(β, β¯)]=12piθ∫d2α |〈β|α+ β〉|2[eα ∂∂β fA(β, β¯)] [eα¯ ∂∂β¯ fB(β, β¯)]=12piθ∫d2α e−|α|2/2θ[eα ∂∂β fA(β, β¯)] [eα¯ ∂∂β¯ fB(β, β¯)]= e2θ ∂∂ζ∂∂η¯ fA(β + ζ, β¯)fB(β, β¯ + η¯)|ζ=η¯=0 (2.9)This is known as the Vorol product.2.2.2 Noncommutative sphereA rather similar approach allows us to study the noncommutative sphere.The structure algebra is simply the algebra ofN×N hermitian matrices, Mn.In this algebra, as we have already discussed in the Introduction, we singleout three matrices Li satisfying the SU(2) commutation relations. Since theLi form an irreducible representation of SU(2), these three matrices generateall of Mn.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. (2.10)212.2. Noncommutative geometryTo define an analog of the coherent state,15 let nˆ be a unit 3-vector (or apoint on a unit sphere). Then, define Lnˆ = nˆiLi. A coherent state at pointRnˆ on the sphere of radius R is then |nˆ〉, whereLnˆ|nˆ〉 = J |nˆ〉 , 〈nˆ|nˆ〉 = 1 . (2.11)The coherent state at the north pole is the state with the largest angularmomentum in the 3-direction; a coherent state at any other point can beobtained from the one at the north pole by a SU(2) rotation. Recall theWigner formula,16|〈m|nˆ〉| =√(2J)!(J +m)!)(J −m)!(cosθ2)J+m (sinθ2)J−m, (2.12)where θ is the polar angle at point nˆ on the unit sphere: the angle betweenthe positive 3-axis and nˆ.One consequence is that, if the angle between two unit vectors nˆ1 andnˆ2 is χ = arccos(nˆ1 · nˆ2), then|〈nˆ1|nˆ2〉| =(cosχ2)2J=(nˆ1 · nˆ2 + 12)J. (2.13)For large J , the overlap between the states |nˆ1〉 and |nˆ2〉 decreases sharplyas the angle between them is increased. If we let χ = 2/√J , we have(cosχ2)2J ≈ (1− 12J)2J, (2.14)which approaches 1/e for large J . Thus, the effective width of the coherentstates on a sphere of radius R is proportional to RN−1/2. A single coherentstate covers an area proportional to R2/N , which is natural given that thenoncommutative sphere should contain N unit noncommutative ‘cells’.17It is easy to convince oneself that for nˆ in the 1-3 plane, 〈m|nˆ〉 can be realwhen we take L2 to be purely imaginary (and therefore antisymmetric). Torestore the phase of 〈m|nˆ〉 for all directions nˆ, write nˆ in polar coordinates:nˆ = (sin θ cosφ, sin θ sinφ, cos θ). As we just discussed, for the azimuthal15For another approach to coherent states on the fuzzy sphere, see [57]16 See for example [58] equation (3.8.33).17In string theory, we would say that the spherical D2-brane has N units of flux piercingit, corresponding to N D0-branes dissolved in its worldvolume. The effective theorydescribing N D0-branes is written in terms of N × N hermitian matrices, lending theD2-brane the noncommutative structure we are studying.222.2. Noncommutative geometryangle φ = 0, 〈m|nˆ〉 is real. For all other angles, we rotate around the 3-axisto obtain〈m|nˆ〉 =√(2J)!(J +m)!)(J −m)!(12sin θ)J (tanθ2)−me−imφ . (2.15)Now, consider a complex variable α = R tan (θ/2) eiφ. This is simply thecomplex coordinate arising from a stereoscopic projection. This coordinatedoes not cover the entire sphere (it is singular at the point θ = pi), but acomplementary complex coordinate, α˜ = R tan ((pi − θ)/2) e−iφ, does. Sinceα˜(α) = R2/α is a holomorphic function, together these two complex coor-dinates define a complete complex structure.We will now change notation and denote the coherent states with |α〉instead of |nˆ〉. Just as it was with the coherent state on the plane, up to anormalization factor our coherent states on the sphere are holomorphic inthe complex variable α.|α〉 =(12sin θ)J J∑m=−J√(2J)!(J +m)!)(J −m)!(αR)−m |m〉 . (2.16)These coherent states are overcompleteN4piR2∫4d2α(1 + |α/R|2)2 |α〉〈α| = 1 , (2.17)with respect to the SU(2) invariant measure on the sphere, 4d2α(1+|α/R|2)2 .For any matrix operator A in Mn, consider its Berezin symbol fA(α) =〈α|A|α〉. The Berezin symbol is a function on the sphere which correspondsto the matrix in Mn. Since equation (2.7) is valid on the sphere (as it reliesonly on the coherent states being holomorphic), we have:(fA ∗ fB)(β, β¯) = 〈β|AB|β〉 = NpiR2∫d2α(1 + |α/R|2)2 〈β|A|α〉〈α|B|β〉(2.18)=NpiR2∫d2α(1 + |α/R|2)2 |〈β|α〉|2[e−β∂∂α eα ∂∂β fA(β, β¯)] [e−β¯∂∂α¯ eα¯ ∂∂β¯ fB(β, β¯)].To simplify our computation, we will compute only the star product at232.2. Noncommutative geometrythe north pole, β = 0. We have(fA ∗ fB)(0, 0) = NpiR2∫d2α(1 + |α/R|2)2 |〈0|α〉|2[eα ∂∂β fA(β, β¯)]β=0[eα¯ ∂∂β¯ fB(β, β¯)]β=0(2.19)=NpiR2∫d2α(1 + |α/R|2)2(11 + |α/R|2)2J [eα ∂∂β fA(β, β¯)]β=0[eα¯ ∂∂β¯ fB(β, β¯)]β=0.Now, on the surface of it, this integral does not appear convergent; however,we have:∂p∂βpf(β, β¯)|β=0 = 0 for p > 2J . (2.20)To see that this is the case, just write the Berezin symbol for any operatorA asfA(β, β¯) =j∑n,m=−j〈β|n〉〈n|A|m〉〈m|β〉=J∑n,m=−J(2J)! 〈n|A|m〉√(J +m)!(J + n)!(J − n)!(J −m)!βJ−mβ¯J−n(1 + ββ¯)2J(2.21)∂p∂βp acting on a (n,m) term in the above sum is nonzero only if p = J −mand n = J . Thus, p is at most 2J .Returning to our expression for the star product, we now have that(fA ∗ fB)(0, 0) = NpiR22J∑p,q=01p!q!∫d2α αpα¯q(1 + |α/R|2)2J+2[∂p∂βpfA(β, β¯)]β=0[∂q∂β¯qfB(β, β¯)]β=0= N2J∑p=0R2p(2J − p)!p!(2J + 1)![∂p∂βpfA(β, β¯)]β=0[∂p∂β¯pfB(β, β¯)]β=0=2J∑p=0(2J − p)!p!(2J)![R2p∂p∂αp∂p∂β¯pfA(α, α¯)fB(β, β¯)]α=β=0(2.22)This is the star product derived and used, for example, in [59].If the functions fA and fB are very smooth, only the first few terms willcontribute. We can then write(fA ∗ fB)(0, 0) ≈∑p=0(2J)−pp![R2p∂p∂αp∂p∂β¯pfA(α, α¯)fB(β, β¯)]α=β=0= eR22J∂∂α∂∂β¯ fA(α, α¯)fB(β, β¯)|α=β=0 , (2.23)242.2. Noncommutative geometryθ1 N(1-cos(θ)) x(N-1/2)NFigure 2.1: Degrees of freedom on the sphere and their matrix counterparts.which reduces to the Vorov product with the noncommutativity parameterR2/(4J).2.2.3 The polar cap on the noncommutative sphereNow, consider an operator |m1〉〈m2| and its Berezin symbol fm1,m2(α) =〈α|m1〉〈m2|α〉. We havefm1,m2 = (phase)(2J)!√(J +m1)!)(J −m1)!(J +m2)!)(J −m2)!×(cosθ2)2J+(m1+m2) (sinθ2)2J−(m1+m2)(2.24)For large J , the function(cosθ2)2J(1+x) (sinθ2)2J(1−x)(2.25)has a sharp peak at θ such that cos θ = x. Therefore the Berezin symbolfm1,m2(α) is largest when the vector nˆ makes an angle θ0 = arccos(m1+m22J)with the vertical axis. For θ close to θ0, we can write(cosθ2)2J(1+x) (sinθ2)2J(1−x)≈[14(1 + x)1+x (1− x)1−x]Je−J(θ−θ0)2.(2.26)Therefore, the Berezin symbol of the operator |m1〉〈m2| is appreciable onlywhen the polar angle θ is within 1/√J of θ0.252.2. Noncommutative geometryThis implies that the degrees of freedom corresponding to a polar capC of angular radius θ (i.e., all points on the sphere whose polar angle isless than θ) can be identified, in the large J limit, with the set of matrixelements {〈m1|Φ|m2〉 | m1 +m2 > 2J cos(θ)}. In particular, to compute theentanglement entropy for half the sphere, we should include the degrees offreedom in ‘half’ the matrix. This was conjectured, but not proven, in [50].Note that it does not matter whether the (anti)diagonal degrees of freedomare included or not, as the answer will be the same in any pure state.Since our coherent states have a width proportional to R/√N , theboundary of our polar cap region C can be thought as having a thicknessof the same size, R/√N . In other words, if we consider the subspace ofMn spanned just by the matrix elements indicated in figure 2.1, the corre-sponding functions on the sphere would have support on the polar cap Cand Gaussian drop-off ‘tails’ controlled by R/√N outside of the polar capC.Taking into account the inherent ambiguity on whether the matrix de-grees of freedom directly on the line delimiting the boundary are included inthe polar cap or not and splitting the difference, we find that the degrees offreedom above the kth “anti-diagonal” correspond to a range of polar angles[0, θ] withcos θ = 1− kN − 12. (2.27)The apparent distribution of UV degrees of freedom on the fuzzy spheredoes not converge to that of the commutative sphere. Using equation (2.27),we can calculate that distribution on the fuzzy sphere along the polar direc-tion by simply counting the number of elements in the corresponding partof the matrix. The fraction of matrix degrees of freedom corresponding tofield degrees of freedom in a polar cap of area A grows roughly as A2, incontrast to the constant density of degrees of freedom on a commutativesphere. We can examine this in more detail by looking at rings centered atvarying polar angles θ. Consider two polar caps described by matrix degreesof freedom corresponding to triangles ending at k and k+1. These caps havean area of 2pik/N and 2pi(k+ 1)/N respectively, making the area of the ringformed by removing the smaller cap from the larger cap 2pi/N (independentof the position of the ring). But this ring is represented in the matrix bythe k degrees of freedom in the kth anti-diagonal line. Therefore, the frac-tion of degrees of matrix degrees of freedom describing a ring centered onθ is kN(N+1) =1−cos θN+1 . However, on a regularized commutative theory wewould expect a fixed density of degrees of freedom per unit area. These two262.3. Setup0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6θ0.0000.0020.0040.0060.0080.010Fraction of degrees of freedomFigure 2.2: Fraction of total number of degrees of freedom in a ring ofconstant area centered at polar angle θ. The solid red line is for the fuzzysphere at N = 100, the dashed blue line corresponds to a fixed density.distributions are illustrated in figure 2.2. It is interesting that, despite thisapparent difference between the two theories, the UV-finite mutual informa-tion is the same, as we will show.2.3 Setup2.3.1 Entanglement entropy for quadratic HamiltoniansSince we are dealing with free scalar field theories, the Hamiltonian for boththe commutative and noncommutative theories is quadratic. The sphere isa compact surface, therefore we do not need to worry about IR divergences.The noncommutative sphere comes with a built-in UV regulator, and we willregulate the theory on the commutative sphere. Therefore, the Hamiltonianfor each of these theories will reduce to that for a finite number N of coupledharmonic oscillators:H =12N∑i,j=1(δijpipj + xiKijxj) , (2.28)with Kij a positive-definite matrix and [xi, pj ] = iδij . It is straightforward,if tedious, to diagonalize the Hamiltonian and write down an explicit solutionfor the ground state. Knowing the ground state, we can find the associated272.3. Setupdensity matrix and then perform the partial trace to obtain the reduceddensity matrix for the “region” i ≤ I, and finally calculate the entanglemententropy from that reduced density matrix (which will ultimately dependonly on the matrix K and the “boundary” I). This is the approach thatwas originally taken in [60, 61]. A more efficient method to perform thesame calculation is outlined in [62]: define[XI ]ij =12[K−1/2]ij , [PI ]ij =12[K1/2]ij i, j ≤ I ,CI =√XI · PI , (2.29)and the entanglement entropy for i ≤ I isSI = Tr[(CI +121)log(CI +121)−(CI − 121)log(CI − 121)].(2.30)The mutual information between two disjoint regions A and B,I(A,B) = SA + SB − SA∪B , (2.31)can also be calculated using (2.30) as long as we perform the proper unitarytransformations on the Hamiltonian to map the region of interest to the firstI oscillators.2.3.2 The free scalar field on the commutative sphereWe wish to calculate the entanglement entropy between a polar cap and itscomplement for the ground state of a real scalar field on a sphere of radius1. This is a divergent quantity, therefore we must start by regularizing thefield theory. The most natural regularization scheme on a spherical geome-try is to expand functions in spherical harmonics and cut off the expansionat some highest mode N . However, this is not the most useful procedure inthis situation for two reasons: the Hamiltonian expressed in terms of spher-ical harmonics modes is diagonal (which is usually a desirable feature butleads to zero entanglement entropy) and there is no simple way to associatecontiguous regions to ranges of modes.Given that the regions that interest us are polar caps, the next mostnatural regularization scheme (in the spirit of the one presented in [61]) issimply to cut up the continuous polar θ variable into an evenly-spaced meshand expand the azimuthal coordinate φ in Fourier modes. The field theoryHamiltonian isH =12∫dΩ(Π2 + |∇Φ|2 + µ2Φ2) , (2.32)282.3. Setupwith the usual canonical commutation relation [Φ(Ω),Π(Ω′)] = iδ(Ω − Ω′).We can write H as:H =12∫dΩ(Π2 +(∂Φ∂θ)2+1sin2 θ(∂Φ∂φ)2+ µ2Φ2). (2.33)Define:18Φ(θ, φ) =1√pi sin θ[b0√2+∞∑m=1(am sinmφ+ bm cosmφ)],Π(θ, φ) =1√pi sin θ[d0√2+∞∑m=1(cm sinmφ+ dm cosmφ)]. (2.34)One can check that the Fourier coefficients are (m > 0):am =√sin θpi∫ 2pi0 dφΦ(θ, φ) sinmφ , bm =√sin θpi∫ 2pi0 dφΦ(θ, φ) cosmφ ,cm =√sin θpi∫ 2pi0 dφΠ(θ, φ) sinmφ , dm =√sin θpi∫ 2pi0 dφΠ(θ, φ) cosmφ ,b0 =√sin θ2pi∫ 2pi0 dφΦ(θ, φ) , d0 =√sin θ2pi∫ 2pi0 dφΠ(θ, φ) , (2.35)and that the non-vanishing commutation relations are:[am(θ), cm′(θ′)] = [bm(θ), dm′(θ′)] = iδ(θ − θ′)δmm′ . (2.36)The first term of the Hamiltonian is (taking a0 = c0 = 0 for simplicity ofnotation):12∫dθ∞∑m=0(c2m + d2m) . (2.37)The second term is:12∫dθ sin θ∞∑m=0[(∂∂θam√sin θ)2+(∂∂θbm√sin θ)2], (2.38)and the third and fourth are:12∞∑m=0∫dθ(m2sin2 θ+ µ2)(a2m + b2m)(2.39)18A similar scheme was presented in [63].292.3. SetupWe relabel our terms so thatΦm = bm, Πm = dm (m ≥ 0)Φm = a−m, Πm = c−m (m < 0) . (2.40)Now, we discretize the polar coordinate:θ → θn = n piNn = 1 . . . N − 1 . (2.41)Since our continuous coordinates are now approximated by a mesh, we mustreplace the integrals above with Riemann sums. For the first and thirdterms, we use the trapezoidal rule: the integral is approximated as theaverage of the left and right Riemann sums, with an error of O(1/N2) (asopposed to O(1/N) for just a left or a right sum). For the second term, wepick a middle Riemann sum, evaluating the summands at the half-point ofeach interval. This also has an error of O(1/N2). The above terms become:12∞∑m=−∞piN(12Πm(θ1)2 +N−2∑n=2Πm(θn)2 +12Πm(θN−1)2 +O(N−2)),12∞∑m=−∞N−2∑n=1piNsin θn+ 12[(∂∂θΦm(θ)√sin θ)2]θ=θn+ 12+O(N−2) ,12∞∑m=−∞piN(12(m2sin2 θ1+ µ2)Φm(θ1)2 +N−2∑n=2(m2sin2 θn+ µ2)Φm(θn)2+12(m2sin2 θN−1+ µ2)Φm(θN−1)2 +O(N−2)).(2.42)We evaluate the derivative by taking the symmetric difference around thepoint it is evaluated at. This has an error of O(1/N2), so the error on thatpart of the Hamiltonian does not change orders of magnitude. We defineΦm,n =√piNΦm(θn) and Πm,n =√piNΠm(θn). The commutation relationsare now [Φmn,Πm′n′ ] = iδmm′δnn′ and the above terms take the form:12∞∑m=−∞(12Π2m,1 +N−2∑n=2Π2mn +12Π2m,N−1),12∞∑m=−∞(N−2∑n=1N2pi2sin θn+ 12[Φm,n+1sin θn+1− Φm,nsin θn]2),302.3. Setup12∞∑m=−∞(12(m2sin2 θ1+ µ2)Φ2m,1 +N−2∑n=2(m2sin2 θn+ µ2)Φ2m,n+12(m2sin2 θN−1+ µ2)Φ2m,N−1).(2.43)We make a final set of re-definitions to ensure we have both canonicalcommutation relations and properly scaled momenta in the Hamiltonian:Π˜m,n =1√2Πm,n and Φ˜m,n =√2Φm,n for n = 1 and n = N − 1. We omitthe tilde for simplicity. The Hamiltonian is now:19H =12∞∑m=−∞[N−1∑n=1Π2mn +N2pi2(sin θ3/22 sin θ1Φ2m,1 +sin θN−3/22 sin θN−1Φ2m,N−1+N−2∑n=22 cos( pi2N)Φ2mn −√2 sin θ3/2√sin θ1 sin θ2Φm,1Φm,2−√2 sin θN−3/2√sin θN−1 sin θN−2Φm,N−1Φm,N−2 − 2N−3∑n=2sin θn+1/2√sin θn sin θn+1Φm,nΦm,n+1)+14(m2sin2 θ1+ µ2)Φ2m,1 +N−2∑n=2(m2sin2 θn+ µ2)Φ2m,n +14(m2sin2 θN−1+ µ2)Φ2m,N−1].(2.44)H decouples into Hm’s that do not depend on the sign of m, so we canwrite:H = H0 + 2∞∑m=1Hm . (2.45)We can evaluate the contributions to entanglement entropy and mutualinformation coming from each of these Hm’s using the method outlinedpreviously and obtain the total entanglement entropy withS = S0 + 2∞∑n=1Sm . (2.46)When |m| > N , the diagonal terms in Hm are generically larger than theoff-diagonal ones. In other words, the Φm,n decouple at large m, which19Using the fact thatsin((n−1/2)piN)+sin((n+1/2)piN)sin(npiN )= 2 cos pi2N.312.3. Setupsuggests that the sum (2.46) converges and therefore we can approximateit numerically by cutting it off at some mmax = Np for some power p > 1.To see this, we compute the entanglement entropy for a polar cap of smallsize (θ ≈ 35◦ and large size (θ ≈ 90◦) for different maximal m of the formmmax = Np (at different N). The results are shown in figure 2.3. At m ∼N4/3, the result differs from the asymptotic value by less than 0.05%.2.3.3 The free scalar field on the fuzzy sphereRecall that the noncommutative sphere is obtained by replacing Cartesiancoordinates xi, i = 1, 2, 3 withXi = RLi√J(J + 1), (2.47)where Li are the generators of the N = 2J + 1-dimensional irreduciblerepresentation of SU(2), i.e. [Li, Lj ] = iijkLk. A real scalar field on thefuzzy sphere corresponds to anN×N Hermitian matrix Φ, and the Laplacianacting on the field is− 1R2[Li, [Li,Φ]] , (2.48)since the Li generate rotations. Integration on the fuzzy sphere is a trace4piR2NTr(·) , (2.49)with the prefactor chosen so that the identity function maps to the unitmatrix. The Hamiltonian for a free scalar field on the fuzzy sphere is thenH =4piR2N12Tr[Φ˙2 −R−2[Li,Φ]2 + µ2Φ2]. (2.50)This is at most quadratic in every matrix element [Φ]ij , we can thereforecalculate the entanglement entropy between any subset of those and the restusing equation (2.30). For example,20 by labeling the entries of Φ asΦ =Φ1Φ2+iΦ3√2Φ4+iΦ5√2Φ7+iΦ8√2. . .Φ2−iΦ3√2Φ6Φ9+iΦ10√2. . . . . .Φ4−iΦ5√2Φ9−iΦ10√2. . . . . . . . .Φ7−iΦ8√2. . . . . . . . . . . .. . . . . . . . . . . . . . . (2.51)20This form is intuitively clear but not very efficient numerically. [50] gives an equivalentbut faster prescription.322.3. Setup1.0 1.1 1.2 1.3 1.4 1.5p0.08400.08450.08500.08550.0860S/NN=50N=75N=100(a) Small polar cap: θ ≈ arccos(0.8)1.0 1.1 1.2 1.3 1.4 1.5p0.1300.1320.1340.1360.1380.1400.1420.144S/NN=50N=75N=100(b) Large polar cap: θ ≈ pi2Figure 2.3: Scaled entanglement entropy for µ = 1 at different N as afunction of the power p of the cutoff mmax = Np332.4. Resultswe can write the Hamiltonian in the form of (2.28) with[K]ij ∼ −12∂2Tr([Lk,Φ]2)∂Φi∂Φj+ µ2δij . (2.52)Note that if µ = 0, K has a zero eigenvalue associated with the matrix Φ be-ing proportional to the identity. This flat direction leads to infinite entangle-ment entropy. To study the massless case, one could impose a tracelessnesscondition on Φ, thus eliminating the massless mode.2.4 Results2.4.1 Entanglement entropyCommutative sphereOn the commutative sphere, the entanglement entropy for polar caps be-haves exactly as we would expect it to, which is a good check on our dis-cretization scheme and numerics. We can clearly see in figure 2.4 that therelation between the length of the boundary and the entropy is linear, witha very small y-intercept that is an artifact of discretization:Scomm ≈ aA . (2.53)We can study the parameter a as a function of N , as seen in figure 2.5.We can see that it has a term linear in N, as expected, and a constant term:a = a1N + a2. Therefore, we can writeScomm = αA+ βA+ · · · , (2.54)where ‘· · · ’ stands for terms that go to zero as  → 0. Since  = piN , our fittells us that α = 0.074 and β = −0.068.Fuzzy sphereFigure 2.6 shows entanglement entropy on the fuzzy sphere for a polar capregion as a function of the polar angle θ for µ = 1 and N = 200. Anglesbeyond θ = pi/2 are not shown as entanglement entropy necessarily hasS(θ) = S(pi − θ) for a pure state such as the vacuum. The most interestingfeature is the small angle behaviour: S ∼ θ2. Notice also that S(θ) issmooth as a function of θ, including at θ = pi/2, indicating that there is nophase transition in the entanglement entropy. Such a phase transition was342.4. Results0 1 2 3 4 5 6 7A024681012SFigure 2.4: Entanglement entropy vs. area of boundary on a commutativesphere with N = 75 for µ = 1.0.observed for N = 4 Yang-Mills in [46, 47], but here it is absent, probablydue to the theory being defined on a compact manifold.In figure 2.7 we show the dependence of the entropy S for a half-sphere(θ = pi/2) as a function of N . The behaviour is clearly linear (though withan offset dependent on the mass), leading us to conclude thatSN= F (θ) + O (N−1) corrections (2.55)where F (θ) is proportional to θ2 for small θ, is smooth for θ ∈ [0, pi] and hasthe property that F (pi − θ) = F (θ).The claim in equation (2.55) is further supported by figure 2.8, whichshows S/N as a function of the fractional area of the polar cap,a :=area(C)4piR2= sin2(θ/2) , (2.56)for several different values of N . The convergence to a fixed curve at largeN is evident.Result (2.55) is quite reasonable from the point of view of the matrixmodel. We can think of the square matrix in figure 2.1 as literally a squareblock of coupled harmonic oscillators. In this way of thinking, the couplingsarising from Hamiltonian (2.50) are only among nearest-neighbour oscilla-tors. Therefore, we would expect the entanglement entropy to follow an area352.4. Results0 10 20 30 40 50 60 70 80N0. 2.5: Slope a of Scomm vs. A at different N, with µ = 1.0 on thecommutative sphere. The red fit line takes the form: a = 0.0236N − 0.0682law. The length of the boundary is the number of matrix elements lying onthe diagonal in figure 2.1, which is 2J(1 − cos θ) = 2Na, thus expect theentanglement entropy in this square array of oscillators to be proportionalto Na, which is what we see for small a in figure 2.8.Finally, we studied dependence of the entanglement entropy on the di-mensionless mass µ of the field. Figure 2.9 shows the entanglement entropyfor half the sphere as a function of µ−1. Over a wide range of masses, forµ < N , the entanglement entropy appears approximately independent ofthe mass. This is the region in which our result (2.55) is applicable. In fact,in this region, S must slowly rise with µ−1, since at µ = 0 the entropy isinfinite due to the appearance of a flat direction in the model. For largemasses µ > N , the entropy decreases to zero, as the kinetic term (whichcouples degrees of freedom at different points and is the source of entan-glement entropy) is overwhelmed by the mass term (which does not coupledegrees of freedom at different points). To understand this behaviour, let’sconsider a toy model of two coupled oscillators x1 and x2 (of equal masses)coupled by a potential(1 + µ2)x21 + (1 + µ2)x22 + 2x1x2 . (2.57)Entanglement entropy for one of these oscillators, in the vacuum of the sys-tem and as a function of µ−1 is shown in the inset to figure 2.9. This toyexample models the full Hamiltonian (2.50): the kinetic term in (2.50) con-tributes diagonal and off-diagonal terms (of order 1 in the toy Hamiltonian362.4. Results0 p8p43 p8p2010203040qSFigure 2.6: Entanglement entropy S as a function of angular size θ of polarcap C on the noncommutative sphere. N = 200 and µ = 1.372.4. Results                                   ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ μ2=10-3◆ μ2=10 20 40 60 80051015NSFigure 2.7: Entanglement entropy for half of the fuzzy sphere as a functionof N for µ2 = 1 and µ2 = 0.001. Both lines have a slope of 0.2.(2.57)), while the mass term contributes only diagonal terms. We see thatthe behaviour of the entanglement entropy in this toy model has the samequalitative features as the entanglement entropy on the sphere.When we study entanglement entropy for a field with a large mass, wefind that it grows with N faster than linear, and that its behaviour as afunction of θ at small θ and at fixed N is θa, with a power a > 2 (figure2.10). Notice that in the limit of infinite mass, the harmonic oscillators thatmake up our noncommutative sphere appear effectively uncoupled, so it isnot surprising that entanglement entropy is smaller at a large mass. Weleave an exploration of the details of this behaviour to future work.2.4.2 Mutual informationThe easiest UV-finite quantity to calculate from entanglement entropy isthe mutual information between two polar caps separated by an annuluscentered on the equator with width δ (see figure 2.11a). Figure 2.12 showsthe result of this calculation for a fixed angular separation of about 0.2pi:2121Because of the differences in regularization, polar caps on the commutative and fuzzyspheres do not actually have their boundaries at the same θ. We have picked here polar382.4. Resultsõõõõõõõõõõõõõõõõõõõõõõ õõóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó óó óó óó óó ó óó ó 󨨨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨.......................................................................................................................................................................................................õ N=25ó N=50¨ N=75.N=2000.0 0.1 0.2 0.3 0.4NFigure 2.8: Entanglement entropy S scaled by N as the fractional area a ofpolar cap for different values of N on the fuzzy sphere. Notice that S/Nconverges to a good large N limit. For small θ, S/N appears proportionalto a, consistent with an extensive entanglement entropy. µ = 1.392.4. Results▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽△△△△△△△△△△△△△△△△△△△△♢♢♢♢♢♢♢♢♢♢♢♢♢♢♢♢♢♢♢♢▽ N=25△ N=50♢ N=7510-2 0.1 1 10 10010-710-610-510-410-310-210-11/μS/N10-3 10-2 0.1 1 10 100 100010-1010-610-21001/μSFigure 2.9: Entanglement entropy for half of the fuzzy sphere as a functionof inverse mass. Vertical lines indicate mass at which the diagonal and off-diagonal elements of the Hamiltonian (2.50) matrix are of the same order(m ∼ N). The inset shows entanglement entropy for two coupled harmonicoscillators whose potential energy is given by equation (2.57).402.4. Results0.0 0.1 0.2 0.3 0.4 2.10: Entanglement entropy S scaled by N as the fractional area aof polar cap for a large mass on the fuzzy sphere, µ2 = 1000 (N=50).412.4. ResultsABδ/2δ/2(a) Regions on the sphererArBABε(b) Regions conformally mapped tothe planeFigure 2.11: Regions A and B between which we calculate the mutual infor-mation I(A,B).it is easy to see that it asymptotes to a finite value as N is increased forboth the commutative and fuzzy spheres and that the value for both casesis similar. They appear to be consistent. We can repeat the calculation forvarious widths of the central annulus. This is shown in figure 2.13, wherewe can also see the convergence to a finite value as N increases.The δ ∼ 0 and δ ∼ pi regions can be studied analytically for a conformallycoupled theory, i.e. for µ2 = 18R, where R = 2R2 = 2.0 is the curvature ofthe sphere. Following the argument in appendix A of [63] and using resultsfrom [64] , we note that R× S2 is conformally related to R1,2 byt± r = tan(τ ± θ2), φ = φ , (2.58)where the left-hand side coordinates are those on the plane and the right-hand side ones are those on the sphere. An entangling surface τ = 0, θ = θ0corresponds to a circle centered at zero with radius tan θ0/2. Polar capswith θ = pi±δ2 are then mapped to concentric disks on the plane centered atzero with radii r1,2 = tan(pi±δ4). The only conformally invariant quantitythat can be constructed from geometrical data on these two disks is thecross-ratiox =4r1r2(r1 − r2)2 = cot2(δ/2) . (2.59)Since mutual information is invariant under conformal transformations, itcaps that are separated by 0.2pi ≈ 0.628 for the commutative sphere and the closestpossible value on the non-commutative sphere: θ ≈ 0.609422.4. Results0 20 40 60 80 100 120 140 160N0.,B)CommutativeFuzzyFigure 2.12: Mutual information for two polar caps separated by an annuluscentered on the equator. On the commutative sphere, the annulus has awidth of 0.628 rad and I(A,B) goes to 0.12 faster than 1N . On the fuzzysphere, the annulus has a width off 0.609 rad and I(A,B) it goes to 0.13faster than 1N .must have an expansion in powers of x. It was shown in [65] that as x→ 0(i.e. in the region where δ ∼ pi) the mutual information takes the form ofI(δ) =112x+O(x2) ≈ 112cot2δ2. (2.60)The δ ∼ 0 behaviour can be obtained by looking at the limit when |r1−r2| →0 and matching to the area law [63]. Using the result in [62], we know thatI(δ) ≈ 0.0397Aε, (2.61)where A is the length of the boundary between the two disks in flat spaceand ε is the distance between them (see figure 2.11b). We take A = 2pi√r1r2(the geometric mean of the boundary lengths) and ε = |r1 − r2| to obtain[63]I(δ) ≈ 0.0397 · 2pi√r1r2|r1 − r2| ≈ 0.125 cotδ2. (2.62)Figure 2.14 shows the mutual information for a conformal scalar on thefuzzy sphere at a high value of N, as well as curves for both the small andlarge δ behaviour of a conformal scalar on the commutative sphere. We cansee that these agree when we expect them to.432.5. Discussion0.0 0.5 1.0 1.5 2.0 2.5 3.0δ10-510-410-310-210-1100101I(A,B)Commutative, N=25Commutative, N=50Commutative, N=75Fuzzy, N=25Fuzzy, N=50Fuzzy, N=75Figure 2.13: Mutual information for two polar caps separated by an annulusof angular width δ centered on the equator, µ = 1.0To ensure that the symmetry of the previous setup does not lead tounusual cancellations, we can consider more generic regions A and B. Themost convenient configuration is to fix the size of A and vary the width δof the annulus. The results for a particular size of A are shown in figure2.15: we can again see a striking agreement between the commutative andnon-commutative theories.2.5 DiscussionAs discussed earlier, the number of degrees of freedom associated with a po-lar cap on the fuzzy sphere does not grow proportionately to its fractionalarea a, but rather to the square of a. This is not possible in the local the-ory (where degrees of freedom are more-or-less associated with points in theunderlying geometry). If degrees of freedom in a subregion of a noncommu-tative sphere reside predominantly near the boundary of the region, we canaccount for our entanglement entropy results as follows: just like in commu-tative theories, quantum correlations between a region and its complementdevelop only across the boundary, but since these degrees of freedom areconcentrated near the boundary, entanglement entropy grows more rapidlythan it would in a commutative theory. Such a ‘reshuffling’ of degrees offreedom when a finite region is considered would be an interesting way inwhich a noncommutative field theory can reproduce the entanglement en-442.5. Discussion0.0 0.5 1.0 1.5 2.0 2.5 3.0δ10-310-210-1100101I(A,B)Figure 2.14: Mutual information on the fuzzy sphere for two polar capsseparated by an annulus of angular width δ centered on the equator. Cal-culation done at conformal coupling (µ = 0.5) with N=300. The solid anddashed lines correspond to the analytical predictions (2.62) and (2.60) for acommutative sphere at small and large δ respectively.tropy predicted by holography.An alternative interpretation of our results can be obtained when werealize that our result for the entanglement entropy can be written in somesuggestive ways. In the linear regime, we have thatS = κNarea(C)R2= κarea(C)θ. (2.63)This should be compared with the general form of the result from [47], whichwasS ∼ area(C)2. (2.64)It is as if the UV cutoff  was replaced by√θ. Such a replacement wouldbe fairly natural in the case of area-law entropy, as√θ is the thickness ofthe boundary; however, it is hard to see how the fuzziness of a boundarydefinition can affect entropy in the volume-law regime. It would be veryinteresting to see what happens to the entanglement entropy if one candefine a region with a boundary whose thickness is less than√θ, ideally assmall as .A further intriguing fact about the entanglement entropy we have foundis that F (θ) in equation (2.55) is very close to a multiple of sin2(θ). This452.5. Discussion0.0 0.2 0.4 0.6 0.8 1.0 1.2δ10-1100I(A,B)CommutativeFuzzyFigure 2.15: Mutual information between a polar cap A of fixed size anda region B separated by A by an annulus of size δ. On the commutativesphere , A terminates at θ ≈ 0.44 rad and on the fuzzy sphere it terminatesat θ ≈ 0.45 rad. In both cases,N = 100 and µ = presented in figure 2.16; the first three coefficients are 18.6, -9.68 and0.443. The ratio of the first two coefficients is -1.92 ≈ -2, which leads toF (θ) ∼ sin2(θ) = a(1− a), where a = sin2(θ/2) is the fractional area of thepolar cap. The expressionS ≈ Nκa(1− a) , κ = 0.8 (2.65)is most suggestive. If degrees of freedom are pictured as uniformly dis-tributed on the sphere, the dependence of S on the area a is most eas-ily explained by assuming that every (localized) degree of freedom on thesphere has its entanglement uniformly spread over the whole sphere. This isin sharp contrast with local theories, where short-ranged interactions leadto quantum correlations between a region and its complement being estab-lished across the boundary surface, leading to the area-law for entanglemententropy [30]. In this interpretation, however, it is puzzling why the entan-glement entropy does not grow like the total number of degrees of freedom,N2. Perhaps the answer is that the localized degrees of freedom are onlyweakly entangled with each other. Or perhaps the N -dependence is causedby the thickness of the boundary (notice that replacing N with N2 in equa-tion (2.65) is equivalent to changing θ to 2 in equation (2.63), which turnsit into equation (2.64)).462.5. Discussion´´´ ´ ´ ´ ´ ´ ´ ´0 2 4 6 8-10-505101520Fourier mode nFouriercoefficientc nFigure 2.16: Coefficients of cos(2nθ) terms in a Fourier expansion for entan-glement entropy S shown in figure 2.6. Since entanglement entropy is sym-metric about θ = pi2 , the Fourier coefficients cn satisfy cn = c400−n for n =1 . . . 199, which implies S(θ) = c0 + 2∑199n=1 cn cos(2nθ) + c200 cos(2 · 200 · θ).472.5. DiscussionTo distinguish between these different interpretations of our results, fur-ther study is needed. In particular, it would be instructive to find a proce-dure for building regions with thinner boundaries.In contrast with the UV degrees of freedom, the behaviour of low-energymodes is unaffected, at least for free field theories. That much is clear sincethe mutual information for the commutative and fuzzy theories are identical.It would be interesting to understand this further. In particular, it wouldbe interesting to understand how the IR degrees of freedom arise from thematrix model.In light of our result, the differing behaviour of mutual information seenin [47] cannot be solely attributed to noncommutativity. Instead, it is likelycaused by a combination of noncommutativity, strong coupling and largeNc.48Chapter 3Emergent geometry ofmembranes3.1 IntroductionIn this Chapter, we go one step further and study the structure of noncom-mutative surfaces themselves (rather than fields on the surfaces, as in theprevious Chapter) through quantum mechanical properties: (approximate)eigenstates of operators.String theory contains many hints that spacetime might be a more com-plicated object—possibly even an emergent one—than a manifold. Mostof our understanding about non-perturbative string theory comes from thestudy of D-branes, extended objects that strings are allowed to end on.When N identical D-branes are considered, their coordinate positions aredescribed by N ×N hermitian matrices. If these matrix coordinates are si-multaneously diagonalizable, their eigenvalues are easily interpreted as thepositions 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-branes do not ‘view’ spacetime in the same way that ordinary pointparticles do. The standard string theoretic interpretation of such ‘fuzzy’configurations is through the so-called dielectric effect [27], where lower di-mensional D-branes ‘blow up’ to form a higher-dimensional D-brane. Lackof locality is related to the lower dimensional D-branes being ‘smeared’ overthe worldvolume of a higher dimensional emergent object.In most previous work, explicit geometric interpretation of the matrixcoordinates as a higher dimensional object has been limited to simple andhighly symmetric geometries, such as planes, tori and spheres.22 In theirpaper, [67], take this one step further: using the BFSS model they found ageometric interpretation of three matrix coordinates as a co-dimension one22One example of an attempt in a more general setup is [66], where a matrix configura-tion corresponding to a given surface was constructed using string boundary states if zeroenergy states of a certain Hamiltonian arising from the boundary action can be found.493.1. Introductionsurface embedded in three dimension. The argument was to consider a stackof D0-branes at an orbifold point, and then introduce an extra probe braneinto the system. By considering a fermionic string stretching between thestack and the probe brane, the emergent surface was defined as the locusof possible positions for the probe brane where the stretched string has amassless mode (indicating that the string has zero length). This lead to thefollowing effective Hamiltonian:Heff(xi) =∑i=1,2,3σi ⊗ (Xi − xi) , (3.1)where Xi for i = 1, 2, 3 are Hermitian, N × N , matrices corresponding tothe positions of the stack of D0-branes in a three dimensional flat transversespace, and xi are the positions of the probe brane. The fermionic mode ismassless when Heff has a zero eigenvalue. Thus, the surface corresponding tothe three matrices Xi is given by the polynomial equation det(Heff(xi)) = 0.This defines a co-dimension one surface in flat R3 space parametrized by(x1, x2, x3).We use equation (3.1) as the starting point for a concrete and explicitstudy of geometry of the emergent surface, identifying zero eigenvectors ofHeff with coherent states underlying noncommutative geometry of the emer-gent surface. In this way, we proceed in reverse of the usual quantizationprocedure (as described by Berezin in [56]), constructing first the coherentstates and from them an emergent manifold. Our approach provides a setof coherent states corresponding to a suitably nondegenerate set of threeHermitian matrices, and leading to an emergent geometry at large N . Con-versely, we give a procedure to find the matrices (and therefore the coherentstates) corresponding approximately to smooth genus-zero surfaces embed-ded in three dimensions and equiped with a Poisson structure. It would beinteresting to compare the resulting coherent states with those constructedby other methods, such as coherent states for general Lie groups [68, 69]or those arising from geometric quantization [70]. See [71] for a review ofcoherent states in dynamical systems and [72] for a review of coherent statesin noncommutative geometry.Our approach to noncommutative geometry most similar to that es-poused in [73] (see also [74] and references therein), but with an explicitconstruction for the coherent states associated with points on the surface.The results can also be thought of as a concrete realization of the abstractidea in the classic work by Kontsevich, [75]. Related recent work includes[76, 77], though our construction appears more general as it allows us tovary the local noncommutativity independent of the shape of the surface.503.1. IntroductionWe focus on configurations where a smooth and well-defined surfacearises from matrices with a large size N . Rather than assume it a priori,we prove a correspondence principle between matrix commutators and aunique Poisson bracket on the emergent surface arising from the matrixconfiguration (X1, X2, X3). This explicit correspondence makes the usualprocedure of going from matrix models to surfaces much less ad hoc, whichmight be of use when quantizing membrane actions by replacing them with amatrix model. We demostrate how easy it is to construct and study surfaceswith desired properties using our approach on several nontrivial examples,including the torus.For most of the paper, we focus on the following question: under whatconditions would a sequence of noncommutative geometries, each arisingfrom a matrix configuration (X1, X2, X3) and labeled by an increasing ma-trix size N , converge to a smooth limit? which quantities characterize thesurface in this limit?Since the polynomial equation det(Heff(xi)) = 0 has degree 2N , generi-cally, the locus of its solutions does not need to be smooth in the large Nlimit. When some generic matrices Xi are scaled so that the range of theireigenvalue distributions remains finite at large N , the resulting surface isgenerically quite complicated and does not have a large N limit. As a sim-ple (but not generic) example, let Xi = diag(σi + a1i , . . . , σi + aNi ), where σiare the Pauli matrices and aki are real numbers. The resulting surface is aunion 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.In the degenerate case where all aki are zero, the surface is a single sphere ofradius one centered at the origin. However, it still does not correspond toa smooth geometry, rather, it is a very fuzzy sphere with SU(N) symmetry.To obtain a smooth geometry, we can instead consider Xi = Li/J , withLi forming the irreducible representation of SU(2) with spin J (this is thestandard construction of the noncommutative sphere, see section 3.3.2 fordetails). This sphere has radius 1 independent of J . As N = 2J + 1→∞,the noncommutative sphere reproduces the ordinary one.When the the large N limit exists and is smooth, the emergent surfacewill be characterized by its geometry (the embedding into flat R3 space)and by a Poisson structure defining (together with N) a noncommutativegeometry in the large N limit. In section 3.2, we will make some defini-tions and introduce our approach. In section 3.3, we will analyze, analyt-ically and numerically, a series of examples from which a general picturewill emerge. In section 3.4 we will prove the correspondence principle anddiscuss smoothness conditions which determine how large N has to be for513.2. Basic setupa given noncommutative surface to be well described by the correspondingmatrices. In section 3.5, we will discuss the issue of area and derive thematrix equation for minimal area surfaces. In section 3.6, we construct asmooth torus embedded in R3. Finally, in section 3.7 we discuss topics forfuture work.3.2 Basic setupSince our emergent surface is given by the locus of points where the effectiveHamiltonian Heff in equation (3.1) has a zero eigenvalue, for each point pon the surface Heff has (properly normalized) zero eigenvector |Λp〉:Heff |Λp〉 = 0 . (3.2)The above equation defines (in non-degenerate cases) a two dimensionalsurface embedded in three dimensional space. We will take the three dimen-sional space to be flat; the metric on the emergent two dimensional surfacewill 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〉 . (3.3)This equation can be thought of as an analogue of an eigenvalue equa-tion: while the three matrices Xi cannot be simultaneously diagonalized,the above equation says that if we double the dimensionality of the spaceunder consideration, there are special vectors |Λp〉 on which the action of Xiis described by only three parameters. In analogy with the Berezin approachto noncommutative geometry [56], we would like to think of these states ascoherent states corresponding to points on the noncommutative surface.23In the Berezin approach, every point p is associated with a coherent state|αp〉. One then defines a map from any Aˆ to a function on the noncom-mutative surface via s(Aˆ) = 〈αp|Aˆ|αp〉. This function is usually called thesymbol map. From it one can find the corresponding star-product and therest of the usual machinery of noncommutative geometry.The first difficulty we see with |Λp〉 being the coherent state is that ouroperators Xi (and their functions) cannot be seen as acting on |Λp〉 dueto dimension mismatch. We can simply ‘double’ these operators by using23A somewhat similar approach but with a different effective Hamiltonian, and appli-cable only in the infinite N limit, was recently made in [78].523.2. Basic setup12 ⊗Xi instead (1k will denote the k × k identity matrix). However, whileit is true that〈Λp| 12 ⊗Xi |Λp〉 = xi(σ) , (3.4)this approach is somewhat artificial. We will see that there is a more nat-ural solution: for large N , when the emergent noncommutative surface issmooth in the sense discussed in the Introduction of this Chapter, the eigen-vector |Λp〉 is approximately a product, |Λp〉 = |a〉 ⊗ |αp〉, where |αp〉 is N -dimensional and |a〉 is 2-dimensional. In the next section, we will examineexamples in which the zero eigenvectors of Heff do factorize in this mannerwhen N is large. A way to measure the extent of the factorization is to writeany (2N)-dimensional vector as|Λp〉 =[ |α1〉|α2〉], (3.5)with ||α1||2 + ||α2||2 = 1, and to defineAp =√||α1||2||α2||2 − |〈α1|α2〉|2 , (3.6)which can be thought of as the area of the parallelogram defined by the twovectors |α1〉 and |α2〉. We will be arguing that, in the large N limit, Ap is oforder N−1/2, implying that |α1〉 and |α2〉 are indeed approximately paralleland we can write|Λp〉 =[a|αp〉b|αp〉]+O(1/√N) . (3.7)(By O(1/N−1/2) we mean that the norm of the correction vector decreaseswith increasing N like 1/N−1/2.) It will then be the N -dimensional vector|αp〉 that will play the role of a coherent state corresponding to point p.The complex coefficients (a, b) of the 2-vector |a〉 determine the directionof the normal vector n at point p given by (x1, x2, x3). To see this, considermoving p slightly to (x1 +dx1, x2 +dx2, x2 +dx3), where (dx1, dx2, dx3) is aninfinitesimal tangent to the surface. First order perturbation theory impliesthat 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〉 (3.8)is a vector normal to the surface at a point p. This is an exact statementand does not rely on our factorization assumption. Incidentally, we have533.2. Basic setupthe formula |n|2 = 1 − 4A2p, so the normal vector is close to being a unitnormal when the factorization condition holds. When we use equation (3.7),we obtain that the normal vector is (a¯b+ ab¯, i(ab¯− a¯b), a¯a− b¯b). Thus, thecoefficients (a, b) fix the direction of the normal vector. Conversely, thenormal vector fixes the coefficients (a, b) up to an overall irrelevant phase.Next, we will try to define local noncommutativity on the surface. Thelocal noncommutativity can be thought of in two different ways: the sizeof ‘fuzziness’ (or uncertainty) of the operators Xi in the state |Λp〉, or thesize of the commutators of the Xis when acting on |Λp〉. In a coherentstate, these two notions should be equal, and they turn out to be equalhere, strengthening our case that |Λp〉 can be thought of as a coherent state.Using σiσj = iijkσk = −σjσi for i 6= j and σ2i = 1, we have a nice littleidentity(Heff)2 = 12 ⊗∑i(Xi − xi)2 + 12iijkσi ⊗ [Xj , Xk] . (3.9)Then, since 〈Λp|(Heff)2|Λp〉 = 0, we have〈Λp|12 ⊗∑i(Xi − xi)2|Λp〉 = −12iijk 〈Λp|σi ⊗ [Xj , Xk]|Λp〉 . (3.10)When the vector |Λ〉 is indeed a product, we can use equation (3.7) to makethe following definition: the local noncommutativity on the noncommutativesurface isθ = 〈αp|∑i(Xi − xi)2|αp〉 = 12ijk θij nk , (3.11)whereθij := 〈αp| − i[Xi, Xj ]|αp〉 . (3.12)The LHS of expression (3.11) is a sum of squares of uncertainties inthe operators Xi, while the RHS depends on the commutators. The par-ticular combination of commutators is of interest: with our factorizationassumption, the commutator term picks up only the contributions that aretransverse to the normal, for example, if the normal vector n is pointing inthe x3 direction, only [X1, X2] contribute to θ. In fact, it will turn out that,in the large N limit, ijkθij is nearly parallel to nk. Thus, we can also writeθ asθ = 〈αp|√∑i 6=j−[Xi, Xj ]2 |αp〉 . (3.13)543.3. Coherent state and its propertiesAs for the first expression in equation (3.11), it will turn out that if wetake the normal vector to 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 complexexamples. In the process, we will construct the approximate eigenvector|αp〉 and study corrections to the large N limit described above.3.3 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]. (3.14)In this convention, we can write Heff in a natural way in terms of N × NblocksHeff =[X3 − x3 (X1 − iX2)− (x1 − ix2)(X1 + iX2)− (x1 + ix2) −(X3 − x3)], (3.15)We will now examine a series of examples of increasing complexity, al-ways focusing on a point where the normal vector to the surface is pointingstraight up (in the x3 direction).Our final conclusion will be that at such a point, the zero-eigenvector ofHeff has the form given in equation (3.7):|Λ〉 =[ |α〉0]+ O(N−1/2). (3.16)|α〉 with 〈α|α〉 = 1 will be the coherent state associated with this particularpoint on the surface, −i〈α|[X1, X2]|α〉 will correspond to the local value ofnoncommutativity at this point. This result is easily generalizable to anyorientation of the surface using an SU(2) rotation of the Pauli matrices.3.3.1 Example: noncommutative planeConsider the example of a noncommutative plane: let X3 = 0, and let[X1, X2] = iθ. Out of necessity, X1 and X2 are infinite dimensional opera-tors. This will not be the case when we are considering compact noncom-mutative surfaces. We haveHeff =[ −x3 A† − α¯A− α x3], (3.17)553.3. Coherent state and its propertieswhere A = X1 + iX2, A and A† are the lowering and raising operators ofa harmonic oscillator with [A,A†] = 2θ, and α = x1 + ix2. The loweringoperator A has eigenstates |α〉, called the coherent states, corresponding toevery complex number α: A|α〉 = α|α〉. We thus have a zero eigenvector forHeff with x3 = 0:|Λ(α)〉 =[ |α〉0]. (3.18)The noncommutative plane is flat and has constant noncommutativity.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 noncommutative surface should look like the noncommutative plane.This is the observation that will allow us to write our definition of a largeN (smooth) limit.3.3.2 Example: noncommutative sphereHere we have Xi = Li/J where Li form the N -dimensional irrep of SU(2):[Li, Lj ] = iijkLk and where J = (N − 1)/2 is the spin. It is useful toconsider the usual raising and lowering operators, L± = L1 ± iL2. Withoutloss of generality, consider that point on the noncommutative surface whichlies on the x3 axis. With x1 = x2 = 0, Heff isHeff =[L3/J − x3 L−/JL+/J −(L3/J − x3)]. (3.19)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. (3.20)It is easy to see that|Λ〉 =[ |J〉0](3.21)is a zero eigenvector of Heff if x3 = 1. Thus, the noncommutative spherehas radius 1.2424 This is a different definition of the radius of the noncommutative sphere than theusual one, which is based on the quadratic Casimir of the SU(2) irrep, and which givesthe radius to be√N2 − 1/J =√N+1N−1 .563.3. Coherent state and its properties3.3.3 Looking ahead: polynomial maps from the sphereA large class of surfaces that can be studied using our tools are surfacesthat are generated from polynomials of the normalized SU(2) generatorsconsidered above:Xi = polynomial(L1/J, L2/J, L3/J) , (3.22)where the polynomials in three variables have degrees and coefficients thatare independent of N . In this case, we expect that at large N the noncom-mutative surface will approach an algebraic variety given by the image ofthe unit sphere under the polynomial maps used to construct Xi.Concretely, consider a surface S in R3 constructed as follows: let p1, p2and p3 be three polynomials discussed, in three variables w1, w2 and w3.Then, consider the image in R3 under these three polynomial maps of thesurface∑i(wi)2 = 1, ieS ={(x1, x2, x3) | xi = pi(w1, w2, w3) and∑i(wi)2 = 1}. (3.23)We will restrict our considerations to surfaces which are non-self-intersecting,meaning that the polynomial map is one-to-one. The corresponding non-commutative surface is specified by three N ×N matrices Xi which can bewritten as corresponding polynomial expressions in Li:Xi = sym (pi(L1/J, L2/J, L3/J)) , (3.24)where, to avoid ambiguity, the ‘sym’ map completely symmetrizes any prod-ucts of the three non-commuting matrices Li. This symmetrization will turnout to play little role in what follows: re-ordering the terms of order k leadsto small—suppressed by a power of J—corrections in the coefficients of thepolynomials of order less than k.Now, consider an arbitrary point p = (y1, y2, y3) on the surface S. Actingwith SO(3) on the space (x1, x2, x3), arrange for the normal vector to S atthe point p to point along the positive x3-direction, and acting with SO(3)on the space (w1, w2, w3), arrange for the pre-image of the point p to be thenorth 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) , (3.25)x3 = y3 + c(w3 − 1) + p(2)3 (w1, w2, w3 − 1) ,573.3. Coherent state and its propertieswhere ci, a, b and c are real numbers and where p(2)i (·) are polynomialsof degree at least 2. To avoid a coordinate singularity, we should havec1c4 − c2c3 6= 0. Then, using a rotation of w1 and w2 (in other words,rotating the unit sphere around the north pole), we can set c3 zero andc4 > 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 interms of the metric on the sphere. If the metric on the sphere is gS2 , thenthe induced metric on the surface isgab :=(CT gS2C)ab, where C =[c1 c2c3 c4]. (3.26)This implies that√det g/√det gS2 = detC, which is a useful fact to keepin mind.Without loss of generality, we are interested in the eigenvector of Heff ata point such that the normal to the surface is pointing along the 3-direction.We now want to show that the corresponding zero-eigenvector of Heff hasthe form shown in equation (3.16).Before we plunge into analyzing this rather general setup, we will narrowthe example down to a simpler one which nonetheless contains most of thesalient features of our general approach.3.3.4 Example: noncommutative ellipsoidHere, we will consider a stretched noncommutative sphere. The most genericclosed quadratic surface in three dimensions is an ellipsoid, with three or-thogonal major axes positioned at some arbitrary position in the three di-mensional space under consideration. In other words, we will allow Xi tobe arbitrary linear combinations of L1/J , L2/J and L3/J . Under the gen-eral framework described above, this amounts to setting the higher degreepolynomials p(2)i to zero:Xi = AijLj/J, where A = c1 c2 a0 c4 b0 0 c . (3.27)The classical, or infiniteN , surface is given by xi = Aijwj with∑i(wi)2 =1. It is easy to check that at a point x = (a, b, c), this surface has a normalvector which is pointing along the positive x3-direction. We will thereforeconsider finding the exact location of the surface at a point with x = (a, b, x3)583.3. Coherent state and its propertieswhere 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) ] ,(3.28)whereA =(c1 + c4)− ic22JL+ +(c1 − c4) + ic22JL− . (3.29)What we need to do is find a good approximation to the zero eigenvectorof Heff(x3), together with an estimate for the (hopefully small) differencex3 − c. We conjecture that such a vector is in some way similar to that inequation (3.21): the ‘top part’ is large compared with the ‘bottom part’ andis dominated by components with the largest eigenvalues of L3. To achievethis, write Heff as a sum of two parts:Heff(x3) =[0 A†A 0]+[cL3J − x3 (a− ib)(L3J − 1)(a+ ib)(L3J − 1) − (cL3J − x3)]. (3.30)If we focus on vectors whose N -dimensional sub-vectors are dominated bycomponents with large L3 eigenvalues, then the first part can be thought ofas being of order N−1/2 while the second part is of order N−1. Our attemptto find an approximate eigenvector of Heff(x3) will treat the second part asa small perturbation on the first part, suppressed by N−1/2.Consider now a vector—which we will show to be either a zero eigen-vector of A or very close to such, and which will thus be an approximatezero-eigenvector of Heff(x3)—given by[ |α〉0], (3.31)where25|α〉 = 1√KbJc∑m=0ξm√√√√ m∏k=1(2k − 1)(2J − 2k + 2)(2k)(2J − 2k + 1) |J − 2m〉 , (3.32)with ξ is given byξ = −c1 − c4 + ic2c1 + c4 − ic2 . (3.33)25Some standard notation we will use: the ‘floor’ function, bxc = the largest integernot exceeding x; the double factorial, (2n)!! = (2n)(2n − 2) . . . (4)(2) and (2n − 1)!! =(2n− 1)(2n− 3) . . . (3)(1) for n a natural number.593.3. Coherent state and its propertiesThe normalization constant, for which 〈α|α〉 = 1, can be computed in thelarge J limit asK =bJc∑m=0(|ξ|2mm∏k=1(2k − 1)(2J − 2k + 2)(2k)(2J − 2k + 1))(3.34)≈ 1 +∞∑m=1(|ξ|2m (2m− 1)!!(2m)!!)= 1 + 2∞∑m=1|ξ/2|2m (2m− 1)!m!(m− 1)!(3.35)= 1 +|ξ|21− |ξ|2 +√1− |ξ|2 = 1√1− |ξ|2 , (3.36)where it is important that |ξ| < 1, which can be seen from the explicit formin equation (3.33). For completeness, let us state that1− |ξ|2 = 4 detC‖C‖2 + 2 detC , (3.37)or1− |ξ|21 + |ξ|2 =2 detC‖C‖2 . (3.38)Writing |ξ| in terms of rotational invariants of the matrix C gives a cleargeometric interpretation this is quantity: it is a measure of how much themap in equation (3.27) distorts the aspect ratio at the point we are interestedin.With a short calculation26 we see that A|α〉 = 0 for integer spin J , andthat for half-integer spin J , 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〉(3.40)= K−1/2(c1 − c4 + ic2) ξJ+1/2 (2J − 2)!!(2J − 1)!! | − J〉 . (3.41)This is very small: the norm-squared of A|α〉 is bounded above byb(J) :=((c1 − c4)2 + (c2)2) |ξ|2J+1 . (3.42)26 Recall thatL−|k〉 =√(J − k + 1)(J + k) |k − 1〉 , L+|k〉 =√(J − k)(J + k + 1) |k + 1〉 . (3.39)603.3. Coherent state and its propertiesSince |ξ| < 1, the above quantity goes to zero like exp(−(2 ln |ξ|)J) for largeJ . Further,(L3J− 1)|α〉 = − 1√KbJc∑m=02mJξm√√√√ m∏k=1(2k − 1)(2J − 2k + 2)(2k)(2J − 2k + 1) |J−2m〉(3.43)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)), (3.44)which is bounded above by27bJc∑m=0(2mJ)2 (|ξ|2m)< J−2∞∑m=0(2m)2(|ξ|2m):= u(J) . (3.48)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 resultingvector’s norm is, in the large J limit, bounded by√(a2 + b2 + c2)u(J) + b(J),which is itself bounded by a constant times J−1. To summarize,∥∥∥∥Heff(c) [ |α〉0]∥∥∥∥ < C(ci)J , (3.49)27 We need to provide a bound onm∏k=1(2k − 1)(2J − 2k + 2)(2k)(2J − 2k + 1) (3.45)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)!! . (3.46)F (1) = J2J−1 < 1 and F (bJc) can also be easily shown to be less than 1 (we need to considertwo cases, with J integer or half-integer). Finally, we notice that F (m + 1) < F (m) form smaller than roughly J/2 and F (m+ 1) > F (m) for m larger than than. This impliesthat F (m) has a minimum near J/2 and that for 1 < m < bJc it is less than the larger ofF (1) and F (bJc) which are both less than 1. Therefore,m∏k=1(2k − 1)(2J − 2k + 2)(2k)(2J − 2k + 1) < 1 . (3.47)613.3. Coherent state and its propertieswhere C(ci) does not depend on J and therefore on N .It follows that[ |α〉0]is an approximate eigenvector of Heff(c) and wecan place a bound on the corresponding eigenvalue: there exists a vector Λ˜such thatHeff(c) Λ˜ = Λ˜ , with || < C(ci)J. (3.50)One can ask the following question: is Λ˜ close to[ |α〉0]? To answer thisquestion, we examine the argument that guarantees the existence of Λ˜ asabove: consider the length squared of Heff[ |α〉0]as expanded in eigenvec-tors of Heff :HeffΛi = λiΛi , Heff(c)[ |α〉0]=2N∑i=1ciΛi ,∥∥∥∥Heff(c) [ |α〉0]∥∥∥∥2 = 2N∑i=1|ci|2|λi|2 .(3.51)With the bound in equation (3.49), it is clear that at least one of the eigen-values λi must be less than C(ci)/J . Further, if none of the other eigenvaluesare small enough, then the eigenvector corresponding to the unique smalleigenvalue (which we denoted with Λ˜) is very close to[ |α〉0]itself. Forexample, if the next smallest eigenvalue λj of Heff is of order N−1/2 (asnumerical studies suggest), then the corresponding coefficient cj must be oforder N−1/2 as well. Therefore, the difference between Λ˜ and[ |α〉0]haslength 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 , (3.52)with Λ close to Λ˜ and therefore[ |α〉0]. It is possible to argue for this in firstorder perturbation theory: as we deform x3 from c to c− ζ, the eigenvalueof interest changes from  (in equation (3.50)) to 0, while the eigenvectorchanges from Λ˜ to Λ. Since  is of order N−1, ζ should also be of order N−1.Making this analysis rigorous is difficult because, effectively, we are trying to623.3. Coherent state and its propertiesdo perturbation theory in 1/N while taking a large N limit. Since any sumswe take would be over N components, these sums can easily overwhelm any1/N suppression factors. For example, to show that Λ is close to Λ˜, it isagain necessary to bound the remaining spectrum of Heff(c) away from zero.This is the same bound as was necessary above: the remaining eigenvaluesmust be bounded away from zero by at least const/√N , which seems to bethe case when examined numerically.Instead of attempting a rigorous proof, we will obtain some analyticestimates based on the assumption that the 1/N expansion is valid andthen confirm these estimates with numerical analysis.Our idea will be to obtain an analytic result for the leading order con-tribution to x3 − c (which will turn out to be of order 1/N as predictedabove) and confirm its correctness by comparing with with numerical re-sults. We will also confirm that our approximate eigenvector[ |α〉0]is agood approximation to the exact zero eigenvector of Heff(x3). Crucial to thisapproach are two facts: that the eigenvector |α〉 has components which falloff exponentially with m, so that only those components with spin close tothe maximum spin J are appreciable, and that the second term in equation(3.30) is small (of order 1/N) when acting on these components. Furtheranalysis will then reveal that when the first order correction to the approx-imate eigenstate is included:[ |α〉|β〉], the vector |β〉 also has componentswhich fall off exponentially withm. We will interpret this as a ‘quasi-locality’feature of the noncommutative surface.Now, return to our way of writing Heff as a sum of two parts in equation(3.30). Our special vector[ |α〉0]is an approximate zero eigenvector ofthe first of these two operators (and an exact zero eigenvector for odd N).Thinking of the second term in equation (3.30) as a small perturbation infirst order perturbation theory, we obtain, to first order, that the change in633.3. Coherent state and its propertiesthe eigenvalue is equal to[ 〈α| 0 ] [ cL3J − x3 (a− ib) (L3J − 1)(a+ ib)(L3J − 1) − (cL3J − x3)] [ |α〉0](3.53)= 〈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 exponentiallysmall ‘tail’ to the sum, so that the function F (ξ, J) will no longer dependon 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))(3.54)≈ c∞∑m=12m(|ξ|2m (2m− 1)!!(2m)!!)= c|ξ| dKd|ξ| . (3.55)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. (3.56)We have tested the correctness of this formula numerically,28 as can be seenin figure 3.1.Further, we have checked that[ |α〉0]is a good approximation to the ex-act eigenvector. As can be seen in figure 3.2, the magnitude of the differencedecreases as N−1/2.28To facilitate numerical study, it is best to rewrite equation (3.2) in as a genuineeigenvalue equation. Consider the operator σ3Heff . We can rewrite equation (3.2) as(−iσ2 ⊗ (X1 − x1) + iσ1 ⊗ (X2 − x2) + 12 ⊗X3) |Λ〉 = x3|Λ〉. Therefore, to find x3 onthe emergent surface at a given x1 and x2, all we have to do is to solve an eigenvalueproblem. It is important that the operator being diagonalized is no longer hermitian:most (or possibly all) of its eigenvalues are complex. Real eigenvalues (if any) correspondto points on the emergent surface. Since the dimension of the operator is even, there mustbe an even number of real eigenvalues in non-degenerate cases. This naturally correspondsto such points on the emergent surface coming in pairs for a closed surface.643.3. Coherent state and its properties102 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(3.56), which has no free parameters and appears to be an excellent match tothe numerical data. In this figure, (a, b, c) = (1.5, 0.5, 3), c1 = 2, c2 = 5 andc4 = 4. For these values, equation (3.56) implies that c− x3 = 2.71875/J .653.3. Coherent state and its properties102 103 104J10-210-1∆Figure 3.2: Magnitude, ∆, of the difference between the approximate eigen-vector and the exact eigenvector as obtained numerically, for the ellipsoidin figure 3.1. The straight line, shown to guide the eye, is a best fit to thelast few points and corresponds to ∆ = 1.12√J.663.3. Coherent state and its propertiesOnce we understand |α〉, we can ask about the leading correction tothe exact eigenvector of Heff . To next order, the eigenvector has a form[ |α〉+ |∆α〉|β〉], with corrections |β〉 and |∆α〉 that have magnitudes of orderno 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 vector will show a small deviation from this assumeddirection. Finally, Ap ≈√||β||2 − |〈∆α|β〉|2.It is difficult to obtain a closed-form formula for |β〉, and even harder toobtain one for |∆α〉. We should proceed by finding a complete eigenbasisfor the first part of Heff as written in equation (3.30), and then use standardperturbation theory to obtain the desired result. This is beyond the scope ofthis thesis, so we will resort to less complete methods to obtain some insightinto the structure.The formal expression for |β〉 is|β〉 = (A†)−1(cL3J− x3 + p(2)3)|α〉 . (3.57)This expression is formal because A† might not have an inverse when actingon the above operator. However, we notice that since we already know x3,we are able to find, to leading order in N , the first nonzero coefficient of |β〉(which is the coefficient of |J −1〉). To do so, we take our already computedvalue of x3 and solve this equation:A†|β〉 = −(cL3J− x3)|α〉 . (3.58)Once we have the first coefficient, we can substitute it back into the aboveequation and solve for the next coefficient. Repeating this will in principleyield nearly all components of |β〉 (with exception of the component withthe most negative L3 eigenvalue).Explicitly, we obtain that the coefficient of |J − 1〉 in |β〉 isc− x3√K√2Jc1 − c4 − ic2 . (3.59)The magnitude squared of this expression isc22J1detC|ξ|2(1− |ξ|2)1/2 . (3.60)673.3. Coherent state and its propertiesWe need this expression to be small (compared to 1), since we would like‖β‖  ‖α‖. Thus, for nonzero |ξ|, how large J needs to be for our analysis tobe applicable depends, for example, on c. Numerical study confirms equation(3.60); further, it shows that the ratio of the expression in equation (3.60)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 (3.56) and(3.60) are large when N is too small to describe the portion of a given surfacewith a high curvature.At the same order, we also get a correction to |α〉, |∆α〉. A formalexpression, similar to the one for |β〉 above,|∆α〉 = A−1((a+ ib)(L3J− 1))|α〉 , (3.61)does not have a well defined meaning as((a+ ib)(L3J − 1)) |α〉 genericallyhas a significant component parallel to |α〉. It is not possible to solve forcoefficients of |∆α〉 in the same way that we solved for those of |β〉; weneed 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 thecoefficient c determines the local curvature of the surface, the coefficients aand b control how fast the noncommutativity is changing, as we will see insection 3.3.6.As we already mentioned, |∆α〉 is not necessarily orthogonal to |β〉, sowe will now have a correction to the angle of the normal vector,ni ≈ (2 Re (〈∆α|β〉) , 2 Im (〈∆α|β〉) , 1) . (3.62)Numerical work confirms that the angle between the expected normal vectorto the surface (which here points in the x3-direction) and the actual normalvector to the surface scales like N−1 and grows linearly with the coefficientsa 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 ofan ellipsoid. Consider a point of interest such that the normal at this pointis pointing in the positive x3 direction. Let this point lie at x1 = x2 = 0,setting y1 = y2 = 0. Without loss of generality, set y3 equal to zero as well.683.3. Coherent state and its properties102 103 104J10-410-310-2c−x3Figure 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).The line shows equation (3.70).693.3. Coherent state and its propertiesThis allows us to write Heff as a sum of two pieces as before:Heff(x3) =[0 A†A 0](3.63)+[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, theycan be written asp(2)k = dk,1(L+J)2+ dk,2(L−J)2+ dk,3L+L− + L−L+2J2(3.64)= ek,1(L1J)2+ ek,2(L2J)2+ ek,3L1L2 + L2L12J2(3.65)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 higher order polynomials containing at least one powerof L3/J − 1 are either equivalent to polynomials in L1/J and L2/J (fromL21 + L22 + L23 = N2 − 1), or subleading, as we will see in a moment.The vector defined in equation (3.31) together with |α〉 given in equation(3.32) is an approximate zero eigenvector of this more general Heff as well,as we have confirmed numerically. Generically, Ap decreases with large Nlike N−1/2.Analytically, we first compute the following quantities〈α|L−L+J2|α〉 ≈ 2J|ξ|21− |ξ|2 (3.66)〈α|L+L−J2|α〉 ≈ 2J11− |ξ|2 (3.67)〈α|L+L+J2|α〉 ≈ 2Jξ1− |ξ|2 (3.68)〈α|L−L−J2|α〉 ≈ 2Jξ¯1− |ξ|2 . (3.69)These imply that corrections to x3 due to the polynomials p(2)k in equation(3.65) are of order J−1, same as correction in equation (3.56). In fact, wecan compute the new corrections to the eigenvalue 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)).(3.70)703.3. Coherent state and its propertiesFigure 3.3 shows comparison between this approximate result and theexact numerical values. The agreement is excellent.To summarize the size of the various higher order corrections, we noticethat‖ (L3/J − 1)|α〉 ‖ ∼ O(N−1) (3.71)‖ (L1/J)|α〉 ‖ ∼ O(N−1/2) and∥∥ (L1/J)2|α〉 ∥∥ ∼ O(N−1) (3.72)‖ (L2/J)|α〉 ‖ ∼ O(N−1/2) and∥∥ (L2/J)2|α〉 ∥∥ ∼ O(N−1) .(3.73)To go further in our analysis, we could ask how introducing higher-order polynomials affects |β〉 and |∆α〉 (and therefore Ap as well as theangle the actual normal vector makes with its expected direction), or moregenerally, what is the effect of all these terms on the exact eigenvector. Theanalysis parallels one at the end of the previous subsection: coefficients ofthe quadratic terms in p(3)3 enter in the same way that c does and coefficientsof the quadratic terms in p(2)3 and p(2)3 enter in the same way that a and bdo. Thus, again, having a larger curvature on the surface affects ‖β‖2 whilehaving the noncommutativity vary quickly affects ‖∆α‖2 (as we will see).As before, formulas for the first few coefficients of |β〉 can be computedrecursively. The results are too complicated to be illustrative, however, theyare 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 nontrivial way on |ξ|. In contrast to the ellipsoid case, it is possible for‖β‖2 to be nonzero even with zero |ξ|.Finally, even higher order polynomials are proportionately more sup-pressed. For example terms involving (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 . (3.74)3.3.6 Local noncommutativityConsider −i[X1, X2], using the form in equation (3.25). We have−i[X1, X2] = (c1c4 − c2c3)(L3/J2) + terms linear in (L1/J2) and (L2/J2)+ terms with higher powers of Li .(3.75)From the formulas in section 3.3.5, the expectation value of this operator inthe coherent state is justθ12 = 〈α| − i[X1, X2]|α〉 = (c1c4 − c2c3)/J , (3.76)713.3. Coherent state and its properties102 103 104J10-510-410-3θ 23/θ12Figure 3.4: θ23/θ12 for the example in figure 3.3. This ratio appears todecrease like J−1.since the corrections to 〈α|L3/J |α〉 ≈ 1, as well as 〈α|L1/J2|α〉, 〈α|L2/J2|α〉and those terms that are higher order (in Lis), all lead to subleading contri-butions (of order 1/J2 or smaller). It is important to insist that c1c4 − c2c3is nonzero, so the leading contribution above does not vanish.We can examine 〈α| − i[X1, X3]|α〉 and 〈α| − i[X2, X3]|α〉 in a similarway. In this case, only those sub-leading terms are nonzero and we obtainthatθi3 = 〈α| − i[Xi, X3]|α〉 ∼ 1/J2 for i = 1, 2. (3.77)Therefore, we have that θi3/θ12 is of order 1/J , which is well supported byour numerical data (see Figure 3.4). We can then take θ = θ12. A moregeneral, rotationally invariant equation isθ = 〈α| Θ |α〉 , where Θ :=√−∑i 6=j[Xi, Xj ]2 . (3.78)We have introduced a new operator, Θ, which will play an important rolein the next section.Equation (3.76) has a simple geometric interpretation: the local non-commutativity on the round sphere is constant and equal to 1/J . A sin-723.3. Coherent state and its propertiesgle noncommutative ‘cell’ with this area is mapped to an ellipse with area(det C)/J , which is just the noncommutativity in equation (3.76). In otherwords, the local noncommutativity is the volume form on the emergent sur-face divided by the volume form on the sphere, times J−1.The local noncommutativity is not constant on the surface. An explicitcomputation on the ellipsoid in equation (3.27) shows that its derivativesare∂θ∂x=b(c1c3 + c2c4)− a(c23 + c24)(c1c4 − c2c3)J and (3.79)∂θ∂y=a(c1c3 + c2c4)− b(c21 + c22)(c1c4 − c2c3)J . (3.80)If we include higher order polynomials, the appropriate coefficients in p(2)1and p(2)2 enter in the same way as a and b above. Thus, we see that havingthese coefficients larger makes the noncommutativity vary faster, as we havementioned before.3.3.7 Coherent states overlaps, U(1) connection and Fµν ona D2-braneSince coherent states are associated with points, it is important that theoverlap between coherent states corresponding to well-separated points besmall. Consider two points p and p′ on the emergent surface which are withina distance of order 1/√N of each other. For large N ,29 the coefficients ci, a,b, c etc. . . that locally characterize the surface are approximately the same.However, the corresponding pre-images of p and p′ on the unit sphere inw-space are sufficiently far apart that the basis in which equation (3.32) iswritten is completely different. Therefore, the approximate coherent stateat the point p′ can be obtained from the coherent state at the point p by anSU(2) rotation (in the N -dimensional representation). Explicitly,|α′〉 = ei(−D2L1+D1L2) |α〉 , (3.81)where D1 and D2 are small displacements in w-space corresponding to mov-ing from p to p′. Since we have positioned p at the north pole of the unitsphere, there is no displacement in the 3-direction. L1 and L2 can be writtenin terms of A and A† via equation (3.29), and we get that|α′〉 = e i2θ (dA+d¯A†) |α〉 , (3.82)29The question of what constitutes a large enough N is discussed in section 3.4.733.3. Coherent state and its properties0 1 2 3 4 5 6 7 8d/√θ0.|〈 α(0)|α(d)〉 |Figure 3.5: Magnitude of the overlap between the eigenstate correspondingto the point p at the north pole and the eigenstate corresponding to a pointp’ a distance |d| away. The green H corresponds to points p’ with x2 = 0,while the blue N corresponds to p’ with x1 = 0. The dashed line correspondsto equation (3.83). Plotted for an ellipsoid with c1=1, c2=0.75, c=12, withN=16,384.743.3. Coherent state and its propertieswhere d = x′2 − ix′1, with x′1, x′2 being the coordinates of point p′. Tocompute 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 , (3.83)since [A,A†] = 2θ(L3/J). As can be seen in figure 3.5, the actual coherentstates have exactly this expected behaviour.Further, we can look at the connection defined (to within a factor of 2)in equation (28) of [67],2viAi = −ivi〈α(xi)|∂i|α(xi)〉 , (3.84)where vi is a tangent vector on the emergent surface. To evaluate it, werewrite equation (3.82) in terms of the small displacements x1 and x2:|α′〉 = e iθ (−x2X1+x1X2) |α〉 . (3.85)Thus, the connection is just (A1, A2) = (−x1/2θ, x2/2θ) and the curvatureis F12 = θ−1. This is exactly the expected result on an emergent D2-brane[42].3.3.8 Nonpolynomial surfacesNot surprisingly, our general conclusions are applicable even when the mapsfrom the sphere to the surface of interest are not polynomial. As long as themaps are smooth enough to be approximated by a Taylor polynomial, thelarge N limit behaviours should be similar. Examples with many desiredproperties can be relatively easily ‘cooked up’. Here we consider two ofconceptual relevance.Our first example using a non-polynomial map is designed to probe intothe role of the parameter ξ. To this end, we examinex1 = w3w1 +√1− w23 w2 , (3.86)x2 = −√1− w23 w1 + w3w2 . (3.87)x3 = w3 . (3.88)This example is designed produce a round sphere with a constant localnoncommutativity θ by ‘shearing’ the original sphere (to preserve the volumeform). We have checked explicitly that θ is constant over the surface and753.3. Coherent state and its properties0 5 10 15 20 25√Nθ′/√θ05101520NφFigure 3.6: Angle φ between the normal vector ~n computed using equation(3.8) and the noncommutativity vector ijkθjk, for the surface in equation(3.89) at a point given by x = 0.5, y = 0. The blue N corresponds to N=3000and the red H to N=12 000; the agreement between plots at different Nshows that the plotted quantities scale with N in the expected way. Onthe horizontal axis we have a derivative of the noncommutativity along thesurface scaled by√θ, which increases as µ is increased in equation (3.89).equal to 1/J in the large N limit. The parameter ξ, however, is not constant,instead, we have ξ = −i sin(φ)/(2 − i sin(φ)). This shows that ξ does notplay a role in the large N limit of the surface: it can be changed by applyinga volume preserving automorphism to the sphere. Another way to look atit is that the three matrices Xi defined by equations (3.86)-(3.88) can beobtained from Li/J by a conjugation (up to some ordering ambiguities). ξcan thus be viewed as a basis-dependent quantity.Another interesting example is given byx1 =w1√w21 + w22 + µ2w23,x2 =w2√w21 + w22 + µ2w23, (3.89)x3 =µw3√w21 + w22 + µ2w23.763.4. Large N limit and the Poisson bracketIn this example, we again get a round sphere, but the local noncommuta-tivity is no longer constant. As we would expect, the actual surface at finiteN differs from a round sphere at order 1/N ; this corresponds to the nor-mal vector deviating from the radial direction at the same order, as givenby equation (3.62). Further, we can compute the noncommutativity vectorijkθjk. Our assertion is that these two vectors should be nearly parallel.Figure 3.6 shows that, indeed, the angle between these two vectors decreasesas 1/N . This angle increases as the coefficient µ is increased, resulting ina more rapidly changing noncommutativity. Interestingly, Ap turns out tobe subleading, of order 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 approachworks for surfaces which are not given by polynomial maps from the sphere.This is not surprising, as our approach should work for any surface whichcan be locally approximated by a polynomial map over the sphere. Relaxingthe polynomial condition allows for just about any smooth surface which istopologically equivalent to a sphere to be studied with our approach.3.4 Large N limit and the Poisson bracketIn the previous section, we have provided a series of examples increasing ingenerality and all sharing the following common features: there existed afamily of matrix triplets Xi labeled by their size N . Each such triplet giverise 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 thenormal to this surface was pointing in the x3 direction was, either exactlyor approximately, of the form [ |α〉0]. (3.90)Where the zero eigenvector was not exactly of this form, the correctionswere small, of order N−1/2.More generally, since a rotation of the coordinate system can be effectedby an SU(2) rotation of the σi matrices in Heff , the zero eigenvector at anarbitrary point p has the form|Λp〉 =[ |α1〉|α2〉]=[a|αp〉b|αp〉]+ O(N−1/2)(3.91)where |a|2 + |b|2 = 1 and where |αp〉 is a unit N -dimensional vector.773.4. Large N limit and the Poisson bracketGiven the two parts of a zero eigenvector of Heff , |α1〉 and |α2〉, atfinite N , we compute |αp〉 as follows: find the normal vector to the surface,ni = 〈Λp|σi|Λp〉. Then, find the SU(2) rotation that brings this vectorto point in the positive x3 direction and apply it to Λp. Then, the topcomponent of of |Λp〉 is |αp〉. Explicitly,|αp〉 = cos(θnˆ/2)eiφnˆ/2|α1〉 + sin(θnˆ/2)e−iφnˆ/2|α2〉 , (3.92)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, wecan define a correspondence between functions on the large-N surface f andoperators (N ×N matrices) Mf throughf(τ) = 〈αp|Mf |αp〉 , (3.93)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 acoherent state to define the symbol is an approach due to Berezin [56]. Theimplied noncommutative star product is(f ? g)(τ) := 〈αp|Mf Mg|αp〉 . (3.94)The star product is not unique, ie it is not fixed by the surface and thenoncommutativity parameter θ alone. There are many different triplets ofmatrices that give the same surface and noncommutativity; different tripletswould lead to different star products. Only the leading order of the com-mutator f ? g − g ? f ≈ θ is universal. For example, the details of the starproduct depend on ξ which we know to be arbitrary. However, the starproduct implies, in the large N limit, a unique antisymmetric bracket,{f, g} := N (f ? g − g ? f) . (3.95)We would like this bracket to give us a Poisson structure on our emergentsurface. It is naturally skew-symmetric and satisfies the Jacobi identity, soit is a Lie bracket. To be a Poisson bracket, it also needs to satisfy theLeibniz Rule:{fg, h} = f{g, h}+ g{f, h} . (3.96)(Notice that these are ordinary multiplications now, not star-products.)Instead of directly proving that the Leibniz Rule holds, we will showthat our definition of a star product is equivalent to{f, g} = 1ρab ∂af ∂bg (3.97)783.4. Large N limit and the Poisson bracketfor some function ρ on the surface. In particular, we will haveρ =√det gNθ, (3.98)where g is the pullback metric on the noncommutative surface and θ is thelocal noncommutativity parameter defined in subsection 3.3.6.Let’s follow our previous approach, and consider not only Xi to be poly-nomials in L1/J , L2/J and L3/J − 1, but also consider operators that arepolynomials 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 operatorM = m(X1, X2, X3) in a coherent state, where m(·, ·, ·) is a polynomialfunction. We can compute 〈αp|M |αp〉 at a point p where the normal pointsstraight up by first writingM as a polynomial in L1/J , L2/J , and (L3−1)/J .Then, from equations (3.71), (3.72) and (3.73), we see that the leading orderpiece (which stays finite as N →∞) is simply the constant term30. Thus,〈αp|M |αp〉 = m(y1, y2, y3) , (3.99)where yi are the coordinates of the surface at point p as defined in equations(3.25).Now that we have shown that the expectation value in a coherent stateat a point of any polynomial (in Xi) operator is exactly what we wouldexpect, let’s think about the expectation value of the commutator of twosuch operators M1 and M2. Consider then two polynomials, m1 and m2 inx1, x2 and x3, and the corresponding operators M1 = m1(X1, X2, X3) andM2 = m2(X1, X2, X3). We have already argued that θ12 is much larger thanθ13 and θ23. A similar argument extended to functions of Xi shows that, aslong as Xis are of the form (3.25), we have〈αp| −i[M1,M2] |αp〉 = θ12(∂m1(y1, y2, y3)∂y1∂m2(y1, y2, y3)∂y2− (m1 ↔ m2)).(3.100)Thus, for the two functions on the noncommutative surface given as restric-tions of the polynomials ma: fa(σ) = ma(xi(σ)), the bracket is{f1, f2} = N〈αp| [M1,M2] |αp〉 (3.101)= Nθ(∂σa∂x1∂σb∂x2− ∂σa∂x2∂σb∂x1)∂f1∂σa∂f2∂σb= Nθab√det g∂f1∂σa∂f2∂σb,30Any ambiguities due to the fact that L21 + L22 + L23 = N2 − 1 are subleading in N.793.4. Large N limit and the Poisson bracketin agreement with equations (3.97) and (3.98).To summarize, we have proven that our emergent surface is equippedwith natural Poisson bracket which satisfies the correspondence principle{·, ·} ↔ − iN [·, ·] . (3.102)Essential for our argument to work was the noncommutativity vectorijkθjk being nearly parallel to the normal vector ni, as shown in Figure 3.6.If this was not the case, the bracket we defined would fail to be a Poissonbracket.For the remainder of this section, we will answer the following question:given a genus-zero surface embedded in three dimensions and a Poissonstructure on this surface, does there exist a matrix description that approx-imates this surface?Our construction gives a positive answer to this question, and providesrestrictions on the surface and on Nθ for the approximation to be good.We focus on Nθ (rather than θ itself) as this is a finite quantity in thelarge N limit and determines the Poisson structure through equation (3.98).Given a surface and a function Nθ on this surface, we can always define amap from the unit sphere to this surface such that the ratio of the volumeform on the surface to the volume form on the sphere is Nθ (see equation(3.26)), up to corrections subleading in N .31 In fact, we can find manysuch functions. Which we pick will affect ξ and the higher orders of the starproduct, but not the overall noncommutative structure. Note, however, thatit is not possible to set ξ to zero everywhere for a generic noncommutativesurface. ξ is zero if the metric on the emergent surface is proportional to themetric on the sphere, while the coefficient of this proportionality must be thenoncommutativity θ, which is fixed. These two requirements would fix (upto diffeomormisms) the metric on the emergent surface, which is alreadyfixed by the embedding. To view this in a different way, the freedom inchoosing a map from the sphere to the emergent surface is the freedom topick two functions on the sphere. One of these functions is fixed by requiringa particular noncommutativity θ. The remaining function can be used tochange ξ. However, ξ is a complex function, so requiring it to vanish over-constrains the problem.Given a map from the sphere to the desired surface, we need only replacethe rectilinear coordinates on the sphere with some SU(2) generators Li31 The correspondence proposed in equation (3.102) is only expected to hold to leadingorder in N . The non-integer part of the Poisson symplectic form will be subleading in Nand can therefore be neglected.803.5. Area and minimal area surfacesand we obtain a triplet of matrices Xi which lead us to the appropriatenoncommutative structure. Here, again, there is ambiguity in the orderingof the operators. Its effects are suppressed by powers of 1/N and it affectshigher order terms in the star product (but not the leading order term).For this construction to work, the surface we start with must be suf-ficiently smooth. Alternatively, we could say that we need to pick anirrep of SU(2) large enough to accommodate a rapidly varying surface.Two conditions seem necessary: that the curvature radii of the surfaceat any point be much larger than the diameter of a noncommutative ‘cell’(Rcurvature √θ ∼ N−1/2) and that θ change slowly. Let θ′ be a derivativeof θ in some tangent direction. Then, the change in noncommutativity overa single cell (which has an approximate diameter of√θ),√θθ′, should bebe small when compared with θ itself: θ′/√θ  1 (θ′/√θ ∼ N−1/2). As wehave already discussed, in equation (3.63)—which was was the basis for ourperturbative definition of a general surface near some point—the coefficientsin 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 (3.79) and (3.80)). Further, as we have discussed, large ‘cur-vature coefficients’ lead to large |β〉 while large ‘theta variability coefficients’lead to large |∆α〉. The larger these coefficients are, the larger N must beto compensate, or higher order terms would spoil the correspondence withthe classical limit we have built up. Generally speaking, the factorizationof eigenstate property in equation (3.91) fails when curvatures are too largeat a given N (since |β〉 becomes large). On the other hand, when the non-commutativity varies too quickly, the Poisson brackets involving it (such as{Nθ, f}) will turn out to be too large.Since the arguments offered in this section are in some sense local, it isplausible that they can be extended to higher-genus surfaces. As a demon-stration, following the same prescription we were able to explicitely definematrices corresponding to a noncommutative torus embedded in three di-mensions in section Area and minimal area surfacesIn equation (3.78), we introduced an operator whose expectation value ina coherent state is the local noncommutativity θ. The noncommutativity θhas units of length-squared, and it can be interpreted as the area of a singlenoncommutative ‘cell’. This is similar to thinking of phase space as madeup of elementary cells whose area is ~. In string theory, where a noncom-813.5. Area and minimal area surfaces102 103J10-310-2Relative errorFigure 3.7: Relative error in the noncommutative area as given in equation(3.103) compared to the classical area, for an ellipsoid with major axes 6, 3and 1. The error falls off with J like J−1; a best fit line, 1.02/J , is shownto guide the eye.mutative surface is made up of lower dimensional D-branes ‘dissolved’ inthe surface, we can think of θ as the area occupied by a single D-brane, or,equivalently, the inverse of the D-brane density. If we divide the surface intoN noncommutative cells, adding up the areas of all these cells we shouldget the total area of the surface. This is in fact borne out here, as the op-erator Θ introduced in equation (3.78) has a second role: its trace seems tocorrespond to the area of the surface32A = 2pi Tr Θ = 2pi Tr√−∑i,j[Xi, Xj ]2 . (3.103)Numerical evidence that this formula holds in is shown in figure 3.7.Consider now minimal area surfaces. If we parametrize our emergent32Factor of 2pi can be arrived at by considering the round sphere. Since our matricesXi are the SU(2) generators scaled by J, the more usual factor of 4pi/N is multiplied byJ ≈ N/2.823.5. Area and minimal area surfacessurface with coordinates σa and define the pullback metric on this surface:gab =3∑i=1∂xi∂σa∂xi∂σb, (3.104)(locally) minimal surfaces are solutions to the equations∆xk(σa) = 0 , k = 1 . . . 3 , (3.105)where the Laplacian is, as usual∆ =1√g∂∂σa√ggab∂∂σb, (3.106)and where g is the determinant of the metric gab.It is easy to check that these minimal surface equations can be writtenin terms of the Poisson bracket (3.97) as333∑i=1{xi, {xi, xk}} − 123∑i=1ρ2g{xi,gρ2}{xi, xk} = 0 . (3.107)Let’s now rewrite this equation in terms of θ (using equation (3.98)):3∑i=1{xi, {xi, xk}} − 123∑i=1θ−2{xi, θ2} {xi, xk} = (3.108)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 . (3.109)33This approach was used to study matrix models for minimal area surfaces in [79]. Inthat work, an assumption was made that ρ =√g. This assumption, combined with anidentification of g with the pullback metric from the embedding space, restricts the localnoncommutativity to be constant. This should be contrasted with our approach whereρ 6= √g in general, and with the approach of [73] where the simplification of the Laplacianoperator due to ρ =√g is made possible by assuming the effective metric to be a Weylrescaling of the pullback metric.833.5. Area and minimal area surfacesThis should be compared with the variation of our expression for the areaof the noncommutative surface (3.103):∂A∂X1=12( [X2,Θ−1[X2, X1] + [X2, X1]Θ−1]+ (2→ 3) ) = 0 .(3.110)Taking an expectation value of equation (3.110) w.r.t. a coherent state, weobtain equation (3.109), confirming that the area of the noncommutativesurface is indeed given by equation (3.103).Notice that this equation differs from that for a static configuration in ageneric matrix model (such as BFSS or IKKT), which is[Xi, [Xi, Xk]] = 0 . (3.111)This is because the Lagrangian for these matrix models contain a term ofthe form [Xi, Xj ]2 which is the square of our operator Θ. When consideringminimum area surfaces in matrix models, when the noncommutativity variesover the surface, the appropriate equation is not (3.111), but (3.110), or moregenerallyΘ−1[Xi, [Xi, Xk]]+[Xi, [Xi, Xk]]Θ−1+[Xi,Θ−1][Xi, Xk]+[Xi, Xk][Xi,Θ−1] = 0 ,(3.112)which, in the large N limit where ordering issues can be ignored, can besimplified to[Xi, [Xi, Xk]] + Θ[Xi,Θ−1][Xi, Xk] = 0 (3.113)or[Xi, [Xi, Xk]] − 12Θ−2[Xi,Θ2][Xi, Xk] = 0 . (3.114)This last equation matches the original equation (3.107). It is importantto notice that the second term in the above equation (3.114) has the sameN-scaling as the first term: both are proportional to N−2. Thus, this termcannot be neglected even in the large N limit.To gain more insight into the formula for the area of the surface, we canexamine the formula for the area in terms of the Poisson bracket:A =∫d2σ√gNθ√∑i,j{xi, xj} →∫d2σ√gθ√−[Xi, Xj ]2 . (3.115)The formula in equation (3.115) is essentially the bosonic part of the Nambu-Goto action for a string worldsheet. This action is classically equivalent to843.6. The torusthe Schild action [80], whose quantization via matrix regularization givesthe IKKT model [81]. Equivalence of these two actions is proven by thestandard method involving an auxiliary field the inclusion of which removesthe square root from the action [82] (for a review, see [83]). In the case ofthe correspondence between the Nambu-Goto and the Polyakov action, thisauxiliary field is the worldsheet metric. Here, its role seems to be linked tothe local noncommutativity θ. This is not surprising: if the matrix modelis to be viewed as a quantization of the surface, we should be free to pickany local noncommutativity we chose, so it can play the role of an auxiliaryfield. This point of view provides a physical interpretation to the quantumequivalence of the IKKT and the nonabelian Born-Infeld model.Finally, our computation allows us to write down the noncommutativeLaplacian on our emergent surface; it is, ignoring higher 1/N -corrections∆ = Θ−2[Xi, [Xi, · ]] − 12Θ−4[Xi,Θ2][Xi, · ] . (3.116)This equation could be the starting point for a study of the effects of varyingnoncommutativity on noncommutative field theory.3.6 The torusOur construction has a natural extension to a toroidal surface embeddedin flat three space. Just as surfaces topologically equivalent to a spherewere build by considering maps from the noncommutative sphere algebra,to make a torus we use maps from the appropriate algebra.Consider a surface given byx1 = (R+ r cosu) cos v , (3.117)x2 = (R+ r cosu) sin v , (3.118)x3 = r sinu , (3.119)where u, v ∈ [0, 2pi] and r < R. Now, consider the standard clock-and-shift matrices U and V that are usually used to define the noncommutativetwo-torus:UV = e2pii/NV U , (3.120)Ukl = δkle2pii(k/N) , (3.121)Vkl = δkmodN ,(l+1)modN . (3.122)853.7. Open questions and future workIn the noncommutative torus, operators of the form UnV m are associatedwith functions on the torus of the form einueinv. To define the noncommuta-tive torus embedded in R3 we thus simply substitute eiu → U and eiv → Vin equations (3.117)-(3.119), symmetrizing when necessary to obtain hermi-tian matrices. Numerical analysis shows that the resulting toroidal surfaceis smooth and has the appropriate large N limit (with Ap decreasing forlarge N as N−1/2, the surface approaching the classical shape and the areaof the surface well approximated by equation (3.103)).Once we have obtained this particular toroidal surface, any other surfacewith this topology (including surfaces with the same shape but different localnoncommutativity, for example uniform one) can be obtained by smoothmaps in a way that parallels our discussion of spherical surfaces. It wouldbe interesting to consider a deformation which connects the torus and thesphere and to examine what happens at the point of topological transitionin detail.3.7 Open questions and future workThere are many questions which our work does not address.For example, one can ask if equation (3.103) can be proven analytically,starting with the definition of the surface from Heff . A reasonable start forsuch a proof might be equation (3.115). If we assume that1NTr · = 12pi∫d2σ√gNθ〈α(σ)| · |α(σ)〉 , (3.123)we recover equation (3.103). Equation (3.123) is equivalent to12pi∫d2σ√gθ|α(σ)〉〈α(σ)| = 1N . (3.124)Above equation implies a relationship between the trace and the integral ofthe noncommutative surface1NTr ↔ 12pi∫d2σ√gNθ. (3.125)A completeness relationship such as (3.124) is necessary for the symbolmap from operators to functions on the emergent surface to have a uniqueinverse, which in turn is necessary for the definition of the star product tomake sense. In principle, it should be possible to prove such a completenessrelationship starting with equation (3.1).863.7. Open questions and future workIn subsection 3.3.7, we briefly addressed the question of the U(1) con-nection on the emergent D2-brane. Extending this approach should allowus to prove the equivalence of the nonabelian effective action for D0-branesand the abelian effective action for a D2-brane. More simply, it should bepossible to show the equivalence of the BPS conditions in these two scenar-ios.It would be interesting to see how our set up could be extended tosurfaces which are not topologically equivalent to a sphere or a torus. Itshould be possible, for example, to find matrix triplets Xi which correspondto emergent surfaces with a larger number of handles—and for which thelarge N limit we describe holds. One could check, for example, whether thenoncommutative surfaces given in [77] have a large N limit in the sense inwhich we define it here. Further, it would be interesting to see how ourtoroidal construction in section 3.6 is related to that in [77].Finally, there are many generalizations of equation (3.1) that would beinteresting to explore, including generalizations to higher dimensions (bothof the embedding space and the emergent surface) and those to curved em-bedding space. One could also consider Lorentzian signature models, whichwould be useful in the context of recent progress in cosmology arising frommatrix models, as in [84].87Chapter 4Energy conditions fromentanglement inequalities4.1 IntroductionAs discussed in the first chapter of this thesis, the Ryu-Takayanagi formulacreates a direct connection between bulk geometry (and therefore gravity)and boundary entanglement. Entanglement is subject to restrictions comingfrom the axioms of quantum mechanics, notably unitarity. In this chapter,we translate some of these restrictions, expressed in the form inequalitiesinvolving entanglement entropy, into constraints on which geometries canhave holographic field theory duals. More specifically, we will establishnecessary (but not always sufficient) geometrical conditions for the existenceof a field theory dual to asymptotically AdS space-times. Assuming theEinstein field equations, these conditions take the form of energy conditions.For (2 + 1)-dimensional bulks, the positivity of relative entropy of statesdescribed by metrics that are translationally-invariant in the field theorydirections with respect to a thermal state will be used to obtain asymptoticconstraints that are equivalent to the weak energy condition for vectors in thefield theory directions. For higher-dimensional bulks, we will use the dualitybetween canonical energy and Fisher information to constrain a particularintegral of the bulk energy and examine that constraint asymptotically forcertain forms of the metric.4.1.1 Relative entropy and Fisher informationFirst, let us define a quantity called the relative entropy of two densitymatrices ρ and σ:S(ρ|σ) = Tr (ρ log ρ)− Tr (ρ log σ) . (4.1)Note that this definition is not symmetric in ρ and σ. Nonetheless, relativeentropy has an attractive property [85]: positivity. The relative entropy of884.1. Introductionany two density matrices ρ and σ is non-negative, and is zero if and only ifρ = σ:S(ρ|σ) ≥ 0 . (4.2)Now, considerρ = σ + λδσ(1) + λ2δσ(2) +O(λ3) , (4.3)with both ρ and σ being proper density matrices. One can show [86] thatS(ρ|σ) = λ22Tr(δσ(1)ddλlog ρ∣∣∣∣λ=0)+O(λ3) . (4.4)Equation 4.4 is a properly ordered version of the naive resultS(ρ|σ) = λ22Tr(δσ(1)σ−1δσ(1))+O(λ3) . (4.5)The statement that the first-order variation of relative entropy vanishes isknown as the first law of entanglement entropy, for reasons that will soonbecome clear. By the positivity of relative entropy, 4.4 must be positive 34and is zero if and only if δσ(1) = 0. Note that the vanishing of the firstderivative of relative entropy and the positivity of the second derivative atλ = 0 is equivalent to saying that relative entropy is minimal at λ = 0, afact that also follows from positivity of relative entropy.We define〈δσ, δσ〉 = d2dλ2S(σ + λδσ|σ)|λ=0 , (4.6)〈δσ1, δσ2〉 = 12(〈δσ1 + δσ2, δσ1 + δσ2〉 − 〈δσ1, δσ1〉 − 〈δσ2, δσ2〉) . (4.7)4.6 is known as Fisher information. It is easy to see that 4.7 is symmetric inits arguments, linear in the first argument and positive-definite. Therefore,it forms an inner product on the space of first order perturbations to σ.4.1.2 The first law of entanglement entropyWe now return to the issue of the vanishing of the first derivative of relativeentropy. For any density matrix ρ, we can formally define a matrix Hρ suchthatρ = e−Hρ . (4.8)34Alternately, σ has positive eigenvalues and the expression is quadratic in δσ(1), whichalso shows positivity.894.1. IntroductionHρ is referred to as the modular Hamiltonian of ρ. If ρ is a thermal densitymatrix, then Hρ corresponds to the actual Hamiltonian of the system (upto a constant corresponding to temperature). Entanglement entropy can bewritten asS = −Tr (ρ log ρ) = 〈Hρ〉 (4.9)and relative entropy asS(ρ|σ) = Tr (ρ log ρ)− Tr (σ log σ) + Tr (σ log σ)− Tr (ρ log σ) (4.10)= −∆S + ∆〈Hσ〉 . (4.11)Therefore, positivity of relative entropy can be expressed as∆〈Hσ〉 ≥ ∆S , (4.12)and, for small variations ρ = σ+λδσ (i.e. to first order in λ), we have thatδ〈Hσ〉 = δS . (4.13)The parallel with the first law of thermodynamics (dU = TdS, if no work isdone) is now clear.4.1.3 The modular Hamiltonian for CFTs and the EinsteinequationWhile the first law of entanglement entropy nicely parallels thermodynamics,so far we have done nothing more than a change of variables. However, wecan extract some physics from it by knowing that, for a ball-shaped regionof size R in the vacuum of a conformal field theory on Rd−1,1, the modularHamiltonian takes the form [87]H = 2pi∫|x|<Rdd−1xR2 − |x|22RTCFT00 . (4.14)TCFT00 is the 00 component of the CFT stress tensor. Therefore, for CFTs themodular Hamiltonian is related to the physical Hamiltonian of the theory.Using the Ryu-Takayanagi formula, we have thatδS =14GNδA . (4.15)As seen in Chapter 1, the expectation value of the CFT stress tensor is givenby the coefficient of the lowest-order perturbation to the AdS metric writtenin Fefferman-Graham coordinates, so that:〈TCFT00 〉 =dL16piGNh(d)00 . (4.16)904.1. IntroductionThis relationship, expressed in different frames of reference, was used in[6, 7] to derive the Einstein field equations in an AdS background.4.1.4 Canonical energyIn [86], it was shown that Fisher information for a density matrix dual toa ball-shaped subregion and a vacuum reference state is dual to a gravita-tional quantity called canonical energy. In this subsection, we summarizethe definition of canonical energy given in that paper.35 Consider a familyof asymptotically-AdS metrics g(µ) 36 parametrized by µ, such that g(0) ispure AdSd+1 on the Poincare´ patch in spherical coordinates:37ds2∣∣µ=0=L2r2 cos2 θ(−dt2 + dr2 + r2dΩ2d−1) (4.17)At µ = 0, the minimal surface anchored on a sphere of radius R on theboundary is simplyt = 0 , r = R . (4.18)The modular Hamiltonian for the reduced density matrix of the ball-shapedregion B on the boundary generates a flow (analogous to time evolution forthe physical Hamiltonian) on the causal diamond of B. This flow can benaturally extended to the Rindler wedge in the bulk: it follows the KillingvectorK = − piR(−R2 + t2 + r2) ∂t − 2piRrt∂r . (4.19)It is always possible to pick a coordinate system such that for every µ theminimal surface remains at the same coordinate position 4.18 and the vectorwith coordinate definition 4.19 continues to obey the Killing equation onthe extremal surface [88]. Let V be the vector that enforces that choice ofcoordinates, that is to say, g(µ)+LV g(µ) is in the proper coordinate system.We are interested in perturbations h to AdS that obey the linearizedEinstein equations, so we takeg(µ) = gAdS + µh . (4.20)For the rest of this subsection, we will simply write g for gAdS. The canonicalenergy associated with the AdS metric, the perturbation h and the minimal35Canonical energy was first defined in [88] for asymptotically-flat spacetimes36To lighten the text, we omit space-time indices as they should be clear from context37This is not to be confused with the global patch of AdS, in which the boundary is asphere. Here, we just use coordinates such that the boundary is located at θ = pi/2, butthe boundary is still a plane.914.1. Introductionsurface B˜ anchored on the edge of the boundary ball B of radius R isE =∫Σω(g, h,LKh) +∫B˜ρ(h, V ) (4.21)where Σ is the bulk region bounded by B and B˜ (i.e. r ≤ R) andω(g, γ1, γ2) =116piaPabcdef(γ2bc∇dγ1ef − γ1bc∇dγ2ef),ρ(h, V ) = χ(h+ LV g, [K,V ])− χ(LKh, V ) , (4.22)with the following auxiliary quantities defined to lighten notation:P abcdef = gaegfbgcd − 12(gadgbegfc + gabgcdgef + gbcgaegfd − gbcgadgef),c1···ck =1(d− k + 1)!√−gc1···ckak+1···ad+1dxak+1 ∧ · · · ∧ dxad+1 ,χ(γ,X) =116piab(γac∇cXb − 12γ cc ∇aXb +∇bγacXc −∇cγacXb +∇aγccXb).To find the gauge-fixing vector V , let us split our coordinates Xa into Xi =σi and XA = XA0 , with X0 the position of the extremal surface and σ thecoordinates along that surface. The gauge-fixed perturbation isγab = (h+ LV g)ab . (4.23)The condition that under the perturbation γ the extremal surface remainsextremal is (∇iγiA −12∇Aγii)B˜= 0 (4.24)and the condition that the vector K continues to obey the Killing equationwhen evaluated on the extremal surface can be written as(γiA)B˜= 0 ,(γAD −12δADγCC)B˜= 0 . (4.25)For a given perturbation h, inserting 4.23 into 4.24 and 4.25 produces a setof differential equations that V must obey on the surface B˜. Solving theseequations gives us V , from which we can then find the canonical energyusing 4.21.Canonical energy has many interesting properties in terms of analyzingthe dynamical and thermodynamic stability of space-times under small per-turbations. However, for our purposes it suffices that it is a well-defined924.1. Introduction(albeit complicated) function of on-shell metric perturbations that is dualto the Fisher information of the corresponding perturbations to the vacuumstate on the boundary:d2dµ2S(g(µ)|g)∣∣∣∣µ=0= E . (4.26)4.1.5 Energy conditionsRecall that the Einstein field equations take the formRab − 12Rgab = 8piGNTab , (4.27)where Rab is the Ricci tensor, R its trace and Tab the stress-energy tensor(which can acomodate a cosmological constant term). In principle, generalrelativity puts no restriction on the right-hand side of that equation: anymetric is a solution to the Einstein equation for a stress tensor defined by4.27. Nonetheless, we would like to be able to make general statements aboutmetrics. For example, singularity theorems guarantee that singularities area generic feature in a large class of metrics and not a result of the highdegree of symmetry of metrics that are actually studied analytically [89]. Inorder to prove these singularity theorems,38 constraints on the curvature aretranslated to constraints on the stress tensor using 4.27. These constraintsare called energy conditions. The weak energy condition states thatTabuaub ≥ 0 , (4.28)where ua is any timelike vector. If ua is instead a null vector, 4.28 is calledthe null energy condition. Another condition is the strong energy condition:Tabuaub ≥ −12T aa , (4.29)where ua is any unit timelike vector. If we can diagonalize Tab, then usingan orthonormal basis (ta, x(i)a ), with i = 1 · · · d for a (d + 1)-dimensionalspacetime we can writeTab = ρtatb +∑ipix(i)a x(i)b . (4.30)38as well as statements about the existence of wormholes and the possibility of timetravel by advanced alien civilizations [90]934.2. Asymptotic constraints from positivity of relative entropyρ is then interpreted as the energy density of the matter, with pi the pressuresalong the ith direction. The weak energy condition is then the statementthat [89]ρ ≥ 0 , ρ+ pi ≥ 0 ∀i . (4.31)The null energy condition allows energy density to be negative as long asthe other condition is respected. The strong energy condition translates to39ρ+∑ipi ≥ 0 , ρ+ pi ≥ 0 ∀i . (4.32)In this chapter, we will obtain analogues of these energy conditions from thepositivity of relative entropy in a dual CFT.4.2 Asymptotic constraints from positivity ofrelative entropy4.2.1 Warm-up: spatial interval with the vacuum as areference stateLet us first restrict our attention to (2+1)-dimensional bulks, with a metricof the formds2 =L2z2(−g(z)dt2 + f(z)dx2 + dz2) . (4.33)Conformal invariance lets us write an asymptotic expansion near z = 0:f(z) = 1 + f2z2 + f3z3 + f4z4 + · · ·g(z) = 1− f2z2 + g3z3 + g4z4 + · · · , (4.34)where the same coefficient is used in both expansions at second order toensure the tracelessness of the CFT stress tensor. These bulks are dual tostates in a (1 + 1)-dimensional CFT. As a warm-up to our main calcula-tion, we consider the region {x ∈ [−R,R] , t = 0} as our subsystem andthe vacuum as our reference state. We aim to calculateS(ρ|σ0) = ∆〈H0〉 −∆S . (4.35)First, we use 4.14 and 4.16 to find that∆〈H0〉 = R2L6GNf2 . (4.36)39Note that despite its name, the strong energy condition does not imply the weakenergy condition.944.2. Asymptotic constraints from positivity of relative entropyTo evaluate the entanglement entropy, we parametrize the minimal curve asx(z), with z0 the point of deepest penetration in the bulk (which we cantake to be at x = 0 by symmetry). For the perturbed metric, the length ofthe minimal curve isL = 2L∫ z00dzz√1 + f(z)x˙2 , (4.37)where z0 is now the point of deepest penetration in the perturbed metric(still at x = 0). The Euler-Lagrange equation gives us2Lzx˙f(z)√1 + f(z)x˙2= K (4.38)with K an integration constant. By noticing that x˙ diverges at z = z0, wefind thatK =√f(z0)2Lz0. (4.39)Thus,x˙ =√f(z0)z/z0f(z)1√1− f(z0)f(z) (z/z0)2. (4.40)We can integrate 4.40 to get thatR =∫ z00dz√f(z0)z/z0f(z)1√1− f(z0)z2f(z)z20. (4.41)For pure AdS, f(z) = 1, and soR =∫ z00dzz/z0√1− z2/z20= z0 , (4.42)as we would expect. Using 4.40, we also find thatL = 2L∫ z00dzz1√1− z2f(z0)z20f(z). (4.43)This again reduces to the known result for pure AdS. Of course, this ex-pression is divergent because of contributions near z = 0. However, weknow that near z = 0 the metric approaches AdS, therefore the difference in954.2. Asymptotic constraints from positivity of relative entropylengths should be finite. To see this, let’s impose a cutoff at z =  for pureAdS. We get thatLAdS = lim→02L∫ Rdzz1√1− z2/R2= lim→02L log(R+√R2 − 2)= lim→02L(log(2Rz0)+∫ z0dz1z). (4.44)This allows us to isolate the divergent part of LAdS in such a way that itwill manifestly cancel the divergence in the perturbed geometry. Using thesame regulator for both geometries, we obtain∆S =L2GN∫ z00dzz 1√1− z2f(z0)z20f(z)− 1− log(2Rz0) . (4.45)Therefore, the relative entropy for a state reduced to an interval of size 2Rthat is dual to a metric of the form 4.33 compared to the vacuum state isS(ρ|σ0) = L2GNR23f2 + log(2Rz0)−∫ z00dzz 1√1− z2f(z0)z20f(z)− 1 .(4.46)This quantity needs to be positive for the corresponding metric to have avalid field theory dual. This naturally excludes some f(z), but not in a veryilluminating way. To make more sense of this constraint, we can use theTaylor expansion 4.34 and expand our expression for small z0 (that is tosay, we consider the constraint for surfaces that do not go very deep intothe bulk). We get thatR = z0 +16f2z30 +(9pi32− 12)f3z40 +(1730f4 − 11120f22)z50 +O(z60) ,∫ z00dzz 1√1− z2f(z0)z20f(z)− 1 = log 2 + 12f2z20 +(3pi8− 12)f3z30+(56f4 − 112f22)z40 +O(z50) . (4.47)964.2. Asymptotic constraints from positivity of relative entropyUsing these expansions, the relative entropy isS(ρ|σ0) = L4GN[−3pi16f3z30 +1645(f22 − 3f4)z40 +O(z50)]. (4.48)The positivity of relative entropy therefore imposes the condition thatf3 ≤ 0 ; f4 ≤ 13f22 (if f3 = 0) . (4.49)The bulk stress tensor for our perturbed metric has the following compo-nents:Ttt =g(z)4z(2f ′(z)f(z)+ zf ′(z)2f(z)2− 2z f′′(z)f(z)),Txx = −f(z)4z(2g′(z)g(z)+ zg′(z)2g(z)2− 2z g′′(z)g(z)),Tzz = − 14z(2g′(z)g(z)+ 2f ′(z)f(z)− z f′(z)g′(z)f(z)g(z)). (4.50)Expanding Ttt for small z = z0 givesTtt = −32f3z0 +(f22 − 4f4)z20 +O(z0)3 . (4.51)The condition that energy be positive is f3 ≤ 0, with f4 ≤ 14f22 if f3 = 0.This is stricter than the positivity of relative entropy. It should also benoted that Ttt ≥ 0 is not a covariant condition. The latter issue can befixed by looking at space-like intervals that have some extension in the timedirection as well (in other words, boosted intervals).4.2.2 Finding a stricter reference stateWe can try to find a reference state that leads to a stricter constraint thanthe vacuum state. For a given perturbation, this would mean making therelative entropy as small as possible. Unfortunately, there are only a fewstates that have a simple, known modular Hamiltonian. An interesting classof such states are thermal states. For a thermal state of inverse temperatureβ on a line, the modular Hamiltonian of an interval x ∈ [−R,R] is [91]Hβ = 2βcsch(2piRβ)∫ R−Rdx sinh(pi(R− x)β)sinh(pi(R+ x)β)TCFT00 (x) .(4.52)974.2. Asymptotic constraints from positivity of relative entropyA thermal state is dual to a black hole in the bulk. In our coordinate system,this corresponds tof(z) =(1 +z2z2h), g(z) =(1− z2z2h)2. (4.53)Since we are dealing with CFT states that are translationally invariant, wecan calculate ∆〈Hβ〉:∆〈Hβ〉 = β2pi(2piRβcoth(2piRβ)− 1)∆〈TCFT00 〉≡ Kβ∆〈TCFT00 〉 . (4.54)The entanglement entropy of a thermal state is [91]Sβ =c3log(βpisinh(2piRβ)). (4.55)The above result can be calculated from both the Ryu-Takayanagi prescrip-tion and ordinary CFT methods. We note that both these methods matchif and only if zh = β/pi, a result that can also be calculated using the usualtechnique by Hawking.Armed with these results about thermal states, we now try to find thethermal state σβ with a temperature such that the relative entropy with ourstate is minimal. To do so, start by writingS(ρ|σβ) = Kβ∆〈TCFT00 〉 − (Sρ − Sβ) , (4.56)where Sρ is the entanglement entropy for our state. Extremizing the relativeentropy is equivalent to∂S(ρ|σβ)∂β=∂Kβ∂β∆〈TCFT00 〉+Kβ∂〈∆TCFT00 〉∂β+∂Sβ∂β= 0 . (4.57)The last two terms cancel and∂Kβ∂β 6= 0, therefore the thermal state leadingto an extremal relative entropy is the one with a temperature such that〈∆TCFT00 〉 = 0 . (4.58)Since we know that relative entropy is bounded below and since there isonly one extremal value for β, it must correspond to a minimum. Therefore,we conclude that for thermal reference states, the strictest constraint com-ing from positivity of relative entropy occurs when the reference state hasthe same stress tensor expectation value as the state under consideration.In such circumstances, the difference in expectation values of the modularHamiltonians is zero, and relative entropy is simply the difference betweenthe entanglement entropies.984.2. Asymptotic constraints from positivity of relative entropy4.2.3 Tilted intervals with a thermal reference stateWe have found a way to set our thermal reference state’s temperature tooptimize our constraint coming from positivity of relative entropy, but westill need to make our constraint covariant. While we previously dealt withintervals only in the x direction, we can in fact look at space-like intervalswith all possible boosts. Consider our boundary interval to be{(x, t) |x ∈ [−Rx, Rx] , t ∈ [−Rt, Rt]} , (4.59)with v = Rt/Rx, −1 ≤ v ≤ 1. The length of the minimal curve is nowL =∫ z00dzz√1− g(z)t˙2 + f(z)x˙2 , (4.60)where a dot is again a derivative with respect to z and z0 is the maximumvalue of z reached by the surface. Defines0 =√g(z0)f(z0)dtdz∣∣∣∣z=z0, (4.61)which is essentially the “tilt” of the curve at z = z0. For spacelike intervals,|s0| ≤ 1. Following the same extremization procedure as for the purelyspatial interval, we get thatx˙2 =z2f(z0)z20f(z)21[1− z2f(z0)z20f(z)]− s20[1− z2g(z0)z20g(z)] ,t˙2 = s20z2g(z0)z20g(z)21[1− z2f(z0)z20f(z)]− s20[1− z2g(z0)z20g(z)] . (4.62)Rx and Rt can be obtained by integrating x˙ and t˙ respectively, and thereforewe can write v as a series in z0 and s0. At fixed order in z0, we can invertthat series to obtain s0 in terms of v. Finally, we get that the relativeentropy isS(ρ|σβ) = 3pi16f3 − g3v2v2 − 1 z30 −215(−4f4 + 4g4v2 − f22 (v2 − 1))v2 − 1 z40 +O(z50) .(4.63)There are a few things to note here. First, consider v = 0, i.e. the casewhere we only consider a purely spatial interval. The constraints we getfrom the positivity of relative entropy aref3 ≤ 0 ; f4 ≤ 14f22 (if f3 = 0) . (4.64)994.3. Canonical energy: higher-dimensional constraintsThis is a stricter condition than the one we got when using the vacuumas a reference state, as expected. In fact, it is the same constraint as thepositivity of Ttt. If we allow the full range of values of v, the constraint weget isf3 ≤ g3 , f3 ≤ 0 ;f4 ≤ g4 , f4 ≤ 14f22 (if f3 = g3 = 0) . (4.65)To interpret the full constraint, consider a vector uµ with:ut = cosh ξ , ux = sinh ξ , uz = 0 . (4.66)This is a unit-norm vector always pointing in a time-like field theory direc-tion. We have thatTµνuµuν =32cosh2 ξ(−f3 + g3 tanh2 ξ) z0+ 4 cosh2 ξ((14f22 − f4)−(14f22 − g4)tanh2 ξ)z20 +O(z0)3 .(4.67)We can see that the weak energy condition for field theory directions,Tµνuµuν ≥ 0 ∀ ξ ∈ R , (4.68)is asymptotically equivalent to the positivity of relative entropy.4.3 Canonical energy: higher-dimensionalconstraintsIn the previous section, we focused on energy conditions in three-dimensionalgravity. This was out of necessity: the minimal surfaces we needed to findcorrespond to curves, which greatly simplified our calculations. In the moreinteresting case of higher-dimensional spacetimes, the dimensionality of thesurfaces increases. This makes the direct generalization of the calculations inthe previous section to higher-dimensional space-times close to intractable.On the other hand, the formalism surrounding canonical energy givesus a powerful way to calculate the second-order change to relative entropy,provided that we know both the unperturbed surface and the Killing vectorgenerating the modular flow in the unperturbed space-time and that wecan find the gauge-fixing vector V . In [86], it was shown using 4.26 that1004.3. Canonical energy: higher-dimensional constraintsthe positivity of Fisher information implied the following constraint for aperturbation to the vacuum that obeys the vacuum Einstein equations tofirst order:∫~x2+z2<R2dzdd−1xpiLd−1(R2 − z2 − ~x2)Rzd−1T(2)00 ≥ −E , (4.69)where E is the canonical energy calculated using 4.21 and T (2)00 is the sec-ond derivative of the matter stress tensor with respect to the perturba-tion parameter. This was used to derive an integrated energy condition for(2 + 1)-dimensional bulks. We will generalize that computation to higherdimensions.In (3 + 1)-dimensions, the planar black hole metric takes the form [87]ds2 =L2z2(−(1− µz3)dt2 + dx21 + dx22 +11− µz3dz2)=L2z2(−dt2 + dx21 + dx22 + dz2)+ µL2z (dt2 + dz2)+O(µ2) . (4.70)For the computation of canonical energy, we are considering a boundaryregion x21 +x22 ≤ R2. Therefore it is natural to work in spherical coordinates:ds2 =L2r2 cos2 θ(−dt2 + dr2 + r2(dθ2 + sin2 θdφ2))+ µL2r cos θ(dt2 + cos2 θdr2 + r2 sin2 θdθ2 − 2r cos θ sin θdrdθ)+O(µ2) .(4.71)The minimal surface ist = 0 , r = R , θ ∈ [0, pi/2] , φ ∈ [0, 2pi] . (4.72)There are 8 gauge-fixing equations that must be obeyed by V on the surface.If we take the ansatzV µ = µL2(0, V r(r, θ), V θ(r, θ), 0), (4.73)then five of the equations are trivially satisfied. We are left with16V r(r, θ) + r(r(7 cos θ + cos 3θ) + 8∂V r(r, θ)∂r)= 0 (4.74)r2 cos2 θ sin θ + 2 tan θV r(r, θ) =∂V r(r, θ)∂θ+∂V θ(r, θ)∂r(4.75)cos2 θ(r2 cos θ(1 + 7 cos 2θ) + 16V r(r, θ)− 4∂2V r(r, θ)∂θ2+ 4 tan θ∂V θ(r, θ)∂r)= 4 cos 3θcscθ∂V r(r, θ)∂θ. (4.76)1014.3. Canonical energy: higher-dimensional constraintsIt can be checked thatV r(r, θ) = − r232(7 cos θ + cos 3θ)V θ(r, θ) = −r38sin3 θ (4.77)is a solution to these equations. The canonical energy can then be calculated.The result isE = pi280L2R6 . (4.78)Using 4.69, we can obtain an integrated constraint on our matter stresstensor:∫x21+x22+z2<R2dzdx1dx2R2 − z2 − x21 − x22z2T(2)00 ≥ −1280R7 . (4.79)We can get more explicit constraints on the geometry if we assume that themetric for a (d+ 1)-dimensional bulk takes the formds2 = ds2∣∣AdS+ µL2zd−2(dt2 + dz2) + µ2(hαβ(z)dxαdxβ), (4.80)where α, β run over all the bulk directions. If T(2)00 is independent of ~x, wecan perform some of the integrals in 4.69 to get a general constraint:∫ R0dz(R2 − z2)(d+1)/2zd−1T(2)00 (z) ≥ −R(d2 − 1)Γ(d−12 )2Ld−1pi(d+1)/2E . (4.81)For the metric 4.80 with d = 3, we have thatT00(z) =µ28piL2[hxx(z) + hyy(z) + hzz(z) + z(3L2z3−∂z(hxx(z) + hyy(z) + hzz(z))− 12∂2z (hxx(z) + hyy(z)))]+O(µ)3 .(4.82)T00(z) is second-order in µ, which was expected since it is necessary for thevacuum Einstein equations to be obeyed to first order. We can integrate byparts to get rid of the z derivatives, simplifying the condition to40∫ R0dz[2(R2 − z2)hzz(z) +(R2 + z2)(hxx(z) + hyy(z))] ≤ R7L27. (4.83)40Recall that T(2)00 =d2T00dµ2∣∣∣µ=0.1024.3. Canonical energy: higher-dimensional constraintsThis constraint differs from the one we found in the previous section inseveral ways. Instead of an asymptotic constraint valid only close to theboundary, we now have an integrated constraint that must hold everywherein the bulk. Furthermore, this constraint was obtained by considering onlyboundary regions at t = 0, therefore we do not expect it to be covariant.Finally, this constraint comes from examining the relative entropy with re-spect to the vacuum state, as opposed to a thermal state. This last pointneeds to be emphasized: our perturbations are part of a class of pertur-bations to the vacuum state that satisfy the vacuum Einstein equations tofirst order, hence the first-order perturbation to the AdS metric matchesthat of a planar black hole. The planar black hole is simply a particularperturbation in that class, namely the one withhzz(z) = L2z4 , (4.84)with the other hαβ vanishing. For this particular perturbation, the con-straint reduces to4R7L235≤ R7L27. (4.85)While this inequality of course holds, it is not saturated. We can obtaina particularly compact constraint by picking hzz(z) =54L2z4 and hxx(z) =hyy(z) ≡ h(z): ∫ R0dz(R2 + z2)h(z) ≤ 0 . (4.86)It is also possible to obtain asymptotic constraints by expanding hαβ(z)in z near the boundary, where small z is equivalent to small R. For a (d+1)-dimensional bulk, the lowest power of z allowed is d − 2, since the scalingdimension of the stress tensor is d. In our case, this meanshαβ(z) =∞∑n=1h(n)αβ zn . (4.87)1034.3. Canonical energy: higher-dimensional constraintsThe constraints are, order by order:h(1)xx + h(1)yy +23h(1)zz ≤ 0 ,h(2)xx + h(2)yy +12h(2)zz ≤ 0 ,h(3)xx + h(3)yy +25h(3)zz ≤ 0 ,h(4)xx + h(4)yy +13h(4)zz ≤5L212,h(5)xx + h(5)yy +27h(5)zz ≤ 0 ,· · ·h(n)xx + h(n)yy +2(n+ 2)h(n)zz ≤ 0 . (4.88)Each of these constraints is only applicable if the previous one is saturated.Notice that the fourth-order one is qualitatively different than the others.This is because the lower-order expansion coefficients end up being mul-tiplied by R to a power lower than seven when the integral is performed(as can easily be seen by dimensional analysis). However, at order z4 thepowers of R are matched on both sides of the inequality. At higher orders,if the fourth-order inequality was saturated then the right-hand side of theinequality is zero, leading to the general form.The calculations in this section can be repeated for higher dimensionswithout much additional difficulty: the black hole metric generalizes easilyand the gauge-fixing vector V can always be taken to depend only on rand θ (with additional angular directions dropping out), leading to threenon-trivial differential equations. We find thatE5D = 4pi525L3R8E6D = 8pi22079L4R10E7D = 320pi263063L5R12 . (4.89)1044.3. Canonical energy: higher-dimensional constraintsThe constraints we get are then∫ R0dz(R2 − z2)1/2z[3(R2 − z2)hzz(z) + (2R2 + z2)3∑i=1hxixi(z)]≤ 64525L2R9∫ R0dz(R2 − z2)2/2z2[4(R2 − z2)hzz(z) +(3R2 + z2) 4∑i=1hxixi(z)]≤ 32297L2R11∫ R0dz(R2 − z2)3/2z3[5(R2 − z2)hzz(z) + (4R2 + z2)5∑i=1hxixi(z)]≤ 7207007L2R13 ,(4.90)for five-dimensional, six-dimensional and seven-dimensional bulks, respec-tively. We can see a clear pattern in the integrands, which also applied tofour dimensions. In higher dimensions, the bulk and boundary do not decou-ple and there is therefore no straightforward realization of the gauge/gravityduality[17].105Chapter 5ConclusionWe must admit with humilitythat, while number is purely aproduct of our minds, space hasa reality outside our minds, sothat we cannot completelyprescribe its properties a priori.Carl Friedrich Gauss, 1830In this thesis, we have explored different ways in which properties ofspace can be understood in terms of inherently quantum-mechanical con-cepts, namely entanglement entropy and noncommutativity.In Chapter 2, we explored the geometrical distribution of degrees offreedom of a free scalar field using the entanglement entropy between dif-ferent regions. We did so on a noncommutative sphere, a geometrical ob-ject constructed from something familiar to every undergraduate quantummechanics student: spin matrices. Using a symbol map, we found whichentries in the matrix representing the scalar field correspond to values ofthe field on a polar cap of varying size. Using this map, we were able toevaluate the ground state entanglement entropy between polar caps andtheir complement. We discovered that the entanglement entropy obeyed anapproximate volume law for small polar caps and did not converge to theexpected area law even in the limit of vanishing noncommutativity. This di-rect computation qualitatively agrees with known results obtained throughthe gauge/gravity correspondence. To probe the IR structure of the theory,we also computed the mutual information between two regions on the fuzzysphere separated by an annulus. We matched that result to the commutativetheory. The mutual information in the commutative theory was computednumerically in general and checked by matching it to known analytic resultsin the special case of a conformal theory. Our results for mutual informationare qualitatively different than results obtained through holography, leadingus to conclude that strong coupling or a large number of fields affects thelow-energy distribution of degrees of freedom more than non-locality does.106Chapter 5. ConclusionThe techniques we developed in that chapter could be used to study thetime dependence of entanglement in noncommutative space-times. In [92],the spread of entanglement as a function of time following a global quenchwas studied and modeled by the propagation of free particles at the speedof light. It would be interesting to study this “entanglement tsunami” innoncommutative theories and contrast its behavior to commutative theories,given the infinite speed of light in noncommutative geometry. On the fuzzysphere, this could improve our understanding of the relationship between thematrix degrees of freedom (which behave as harmonic oscillators with localcoupling) and the nonlocal field theory. A holographic calculation could alsotest the importance of locality for the propagation of entanglement.In Chapter 3, we studied noncommutative surfaces in more detail. Start-ing from a complicated polynomial equation describing the surface associ-ated with any three matrices, we understood the properties of a large class ofnoncommutative membranes in terms of a generalization of coherent states.We formulated conditions for the existence of smooth commutative limits ofthese noncommutative surfaces and derived the particular Poisson structurethat the surfaces must have in that limit. We constructed explicit exam-ples, focusing on smooth deformations of noncommutative spheres. Ourtreatment was a mix of analytical and numerical: we found general prop-erties analytically and confirmed them numerically when explicit proof wasunavailable.The construction in that chapter has been generalized to codimensionone surfaces embedded in odd-dimensional spaces, but the generalization toeven-dimensional space remains challenging because the surface itself wouldbe odd-dimensional. Odd-dimensional surfaces are not compatible withPoisson structures, which makes the na¨ıve generalization of this work un-tenable. Nonetheless, we physically expect that some generalization shouldexist, and it would be interesting to study it further. Another issue that wasignored is that of time, which was reduced to a simple parameter. Unlikein the BFSS model, time in the IKKT model is dynamical. This raises thepossibility of studying noncommutative cosmology with a similar approach.In Chapter 4, we used the positivity of relative entropy to derive con-straints on the geometry of space-times dual to conformal field theories.These constraints must hold based on very basic assumptions: that theaxioms of quantum mechanics hold and that general relativity is the lowest-order approximation to quantum gravity. We compared these constraintsto known energy conditions, which are usually imposed by fiat to provesingularity theorems. These results were obtained analytically, with three-dimensional constraints being more strict because of the relative simplicity107Chapter 5. Conclusionof computations in lower dimensions.It should be possible to obtain the canonical energy for boosted balls,which we expect to lead to covariant constraints. As we saw in the three-dimensional case, such constraints should be tighter than those obtained bysimply considering balls living on a fixed time slice. 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