UBC Undergraduate Research

Born-Infeld action geometries Hudoba de Badyn, Mathias May 31, 2014

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
52966-HudobaDeBadyn_PHYS_449_2014.pdf [ 2.17MB ]
Metadata
JSON: 52966-1.0085973.json
JSON-LD: 52966-1.0085973-ld.json
RDF/XML (Pretty): 52966-1.0085973-rdf.xml
RDF/JSON: 52966-1.0085973-rdf.json
Turtle: 52966-1.0085973-turtle.txt
N-Triples: 52966-1.0085973-rdf-ntriples.txt
Original Record: 52966-1.0085973-source.json
Full Text
52966-1.0085973-fulltext.txt
Citation
52966-1.0085973.ris

Full Text

The University of British ColumbiaBorn-Infeld Action GeometriesAuthor:Mathias Hudoba de BadynSupervisor:Dr. Joanna KarczmarekA THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OFBACHELOR OF SCIENCECOMBINED HONOURS IN PHYSICS AND MATHEMATICSAuthor DateSupervisor DateBorn-Infeld Action GeometriesMathias Hudoba de Badyn c©2014May 31, 20142AbstractIn this thesis, we present a novel way of studying noncommutative geometries in string theory basedon an effective Hamiltonian given by Berenstein and Dzienkowski![1]. We work in the context of thestudy of two magnetic monopoles deforming a D3 brane as considered by Karczmarek and Sibilia [2].We present numerical evidence that for surfaces defined using n-dimensional generators of the SU(2)algebra in an auxiliary Hilbert space with this effective Hamiltonian, the surface represents a setof eigenvalues for 2n-dimensional eigenvectors that demonstrate the property of splitting into twoparallel n-dimensional sub-eigenvectors. We conjecture that these sub-eigenvectors can then beused to study coherent states in these noncommutative geometries based on the the fact that theannihilation operator appears in block form in the effective Hamiltonian acting on the eigenvector.Lastly, we derive a useful formula for studying the geometric rate of change of these 3-dimensionalsurfaces in 4 dimensions that may prove handy in preparing numerical solutions of the Nahmequation.34Contents1 Introduction 152 Background 192.1 Review of Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 String-Theoretic Mechanics and D-Branes . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Born-Infeld Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Non-Commutative Bion Geometries 293.1 Nahm Equation Solution for the Fuzzy Sphere . . . . . . . . . . . . . . . . . . . . . 314 Analytical Methods 334.1 4D Geometric Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Fuzzy Sphere Geometric Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.1 N = 2 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.2 Arbitrary-N Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2.3 The ‘Lets-not-assume-the-solution’ Case . . . . . . . . . . . . . . . . . . . . . 364.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Numerical Methods 415.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Calculations on an Ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 Conclusion 57Appendices 59A MATLAB Code for calculating eigenvectors/values 61B Mathematica Code for Symbolic D-Brane calculations 67C Mathematica Code for Geometric Rate Calculations 6956 CONTENTSNomenclaturexi Real-valued coordinates.xˆi = xiIn, (n× n)-matrix-valued coordinates.Xˆi Matrix coordinate in terms of angular momentum matrices.σi ith Pauli matrixσ = σ(xi, xj , xk), coordinate of 4th dimensionHeff Berenstein-Dzienkowski effective Hamiltonian.H Determinant of Heff, or Hamiltonian density[xˆj , xˆk] Commutator of matrices xˆj , xˆkIn n× n identity MatrixJi ith n× n angular momentum matrix (N -dimensional irreducible generators of SU(2))78 CONTENTSList of Figures3.1 Schematic of D-brane deformed by the presence of two Bions . . . . . . . . . . . . . 304.1 ‘Fuzzy sphere’ defined by detHeff = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 355.1 Ellipsoid defined by Xˆ1 = J1, Xˆ2 = 0.5J2, Xˆ3 = 3J3 . . . . . . . . . . . . . . . . . 435.2 Intersection of ellipsoid and xy plane, producing the ellipse of eigenvalues for eigen-value problem in Equation 5.14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 Non-commutative ellipsoid intersecting with xy plane. The intersection boundary isa 2-d locus of eigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.4 Non-commutative ellipsoid intersecting with xz plane. The intersection boundary isa 2-d locus of eigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.5 Non-commutative ellipsoid intersecting with yz plane. The intersection boundary isa 2-d locus of eigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.6 Eigenvalue convergence in the xy plane on points selected from the locus depictedin Figure 5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.7 Parallelogram convergence in the xy plane on points selected from the locus depictedin Figure 5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.8 Eigenvalue convergence in the xz plane on points selected from the locus depictedin Figure 5.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.9 Parallelogram convergence in the xz plane on points selected from the locus depictedin Figure 5.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.10 Eigenvalue convergence in the yz plane on points selected from the locus depictedin Figure 5.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.11 Parallelogram convergence in the yz plane on points selected from the locus depictedin Figure 5.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.12 Eigenvalue convergence in the locus of the intersection of the tilted plane and theellipsoid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.13 Parallelogram convergence in the locus of the intersection of the tilted plane and theellipsoid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.14 Best fit line for eigenvalue convergence estimate along z-axis. . . . . . . . . . . . . . 55910 LIST OF FIGURESAcknowledgementsI would like to thank my parents and my grandmother for their unwavering support for my childhoodand education, and also for their decision to name me ‘Math’ias which led me down this careerpath. As always, I would like to thank the laws of physics for existing and allowing me to existand also imbuing the complex biological structure that is me with the ability to comprehend saidlaws (to some non-neglible degree).I would also like to thank my supervisor Joanna Karczmarek for her unwavering patience,guidance and feedback as I went from having little knowledge of string theory, matrix models andsupersymmetry to having a decent amount of knowledge in these areas and being able to contributesome original work presented here. I also thank Ken Yeh and Philippe Sabella Garnier for theirnotes and conversations which made this material easier to go through.I also wish to thank my professors in the Physics and Astronomy, and Mathematics departmentsfor teaching me a lot of things and inspiring me to continue working hard at understanding thereality we live in, and as well as my peers in the Honours Physics program and Honours Mathematicsprogram for making my 4 years at UBC lots of fun. In particular, I must thank Paul Liu and AndrewKuba Karpierz for helping me optimize my sparse matrix handling in my code for this thesis (thisfully complies with any sane definition of ‘fun’ at the undergraduate level1).Lastly I would like to thank Joseph Polchinski for shaking my hand last fall, writing a prettygood set of textbooks and discovering why D-branes are important; otherwise this thesis wouldlikely be on a different topic.1Also at any other level1112 LIST OF FIGURESPrefaceThis thesis contains a section on elementary Lagrangian mechanics similar to a section on La-grangian mechanics in my mathematics thesis [3]. Both sections were written independently fororiginality, but contain the same key elements (namely the basic results of Lagrangian mechanicsfound in the book by Landau and Lifshitz [4]).1314 LIST OF FIGURESChapter 1IntroductionThe goal of modern physics is to consolidate the theories of General Relativity with Quantum FieldTheory, in the sense that physicists would like to have a theory that describes all four FundamentalForces of Nature: strong, weak, electromagnetic and gravitational. The first three have beensomewhat successfully unified under a Grand Unified Theory in which all three forces are relatedby a single coupling constant. This result holds for somewhat smaller energies, around 1016 GeVor less, but it is currently unknown if the unification holds for much larger energies.Under a Grand Unified Theory, interactions between particles are mediated by what are called‘gauge bosons’. The strong, electromagnetic and weak forces are mediated by the gluon, photonand W±, Z bosons respectively. The strong and weak forces operate on roughly the range of10−13 − 10−15 meters, however the range of the electromagnetic force is essentially infinite.The currently accepted model of particle physics is called the Standard Model (see, for exam-ple, [5]). It uses the three unified forces mentioned above to describe particle interactions, andcertain Lie group theoretic symmetries to describe the actual particles. For example, the simplestGrand Unified Theory which contains the Standard Model is the SU(5) Lie group, in particular itsdecomposition:SU(3)× SU(2)× SU(1) ⊂ SU(5) (1.1)The simplicity of the fact that all currently known particles fit into the group representation aboveis a large motivator for the idea that this Lie group approach will one day yield a complete ‘Theoryof Everything’.One of the main attempts, and certainly the most developed attempt, at unifying gravity withthe other 3 fundamental forces is the study of string theory. In string theory, the point particleapproximation is replaced with the idea that the fundamental constituents of matter abstractlyresemble vibrating strings whose vibrational modes represent different particles. In Section 2.2 wediscuss the basics of string theory, and the respective background relevant in this thesis.In order to better study particle interactions in the framework of string theory, it is foundthat Maxwellian electromagnetism is insufficient for several reasons. Firstly, under this theory itappears that the self-energy of a point charge is infinite [6]. In convenient units, the self energy is1516 CHAPTER 1. INTRODUCTIONgiven byW = mc2 +18pi∫E2dV = mc2 +18pi∫e2r4dV ∼ mc2 +18pi∫e2r2dr. (1.2)where the integral on the right-hand side is over r ∈ [0,∞] which clearly diverges. Classically,this might lead one to believe that the electron must have a finite radius, however in quantummechanics one can prove that the charge density of the electron resembles a delta function, andtherefore must resemble a point particle. Since string theory is inherently quantum, this poses aproblem for trying to study electromagnetic phenomena at the string scale.Some supersymmetric limits of string theory produce an extension of electromagnetism calledBorn-Infeld electromagnetism, in which the self-energy of a point charge is not infinite. We discussthe basic theory of Born-Infeld electromagnetism and how it applies to string theory in Section 2.3.One interesting aspect of non-linear electromagnetic theories is the existence of magnetic monopoles.Classically outlawed by Maxwell’s equations, monopoles have a unique position in string theory asa canonical object of study. In supersymmetric theories, monopoles obey a strict condition relatingcharge and mass of the monopole such that they are linearly proportional.In this thesis, we follow the work of Karczmarek and Sibilia in [2] and [7] in order to study thegeometric properties of extended objects in string theory known as D-branes. Essentially, a D-braneis an object that allows strings to attach to it with Dirichlet boundary conditions. In particular, westudy the geometric properties of D3-branes generated by magnetic monopoles in the Born-Infeldtheory in order to calculate coherent states, or eigenstates of the annihilation operator, on thisD3-brane.In order to pursue this avenue of research, we use of a new result by Berenstein and Dzienkowskiin non-commutative geometry ([1]). They define an effective Hamiltonian with matrix-valued D-brane coordinates Xˆi and real-valued coordinates xi:Heff =3∑i=1(Xˆi − xiIn)⊗ σi (1.3)which defines a real surface when it has a zero eigenvalue. The surface is given by the simpleequation detHeff = 0.In the canonical case of a ‘fuzzy sphere’, where we take Xˆi to be the spin-normalized n-dimensional angular momentum matrices LiJ , this effective Hamiltonian in block form containsthe creation and annihilation operators (without loss of generality, x2, x1 = 0):Heff|~v〉 =(L3J− x3) (L−J)(L+J) (−L3J+ x3)[|α1〉|α2〉], (1.4)where |~v〉 is an eigenvector of this Hamiltonian. Later in this thesis, we explore the conjecture thatthe eigenvectors of this Hamiltonian split into two n-dimensional parallel vectors. Taking advantageof the fact that the annihilation operator exists in a block of this Hamiltonian, elevating the matrixcoordinate Xˆi into this auxiliary Hilbert space seems to have the incredible property that this spliteigenvector will actually allow us to calculate coherent states on the D3-brane defined with the17coordinates Xˆi.The remainder of this thesis will consist of a background section, where we will review La-grangian mechanics, string mechanics and D-branes before discussing aspects of Born-Infeld Elec-tromagnetism and supersymmetry. We will then discuss further the previous work of Karczmarekand Sibilia, and how this thesis builds upon that work. We will conclude with a detailed summaryof the results of this thesis, both analytical and numerical.18 CHAPTER 1. INTRODUCTIONChapter 2Background-When did you become an expert inthermonuclear astrophysics?-Last night.Maria HillTony StarkThe Avengers TMThere is a fair amount of background required in order to successfully read this thesis. Thereader may choose to skip sections that are well-understood. We begin with a review of Lagrangianmechanics in section 2.1, namely the derivation of the Euler-Lagrange equations, which are exten-sively used in quantum field theory and string theory. In section 2.2, we derive the correspondingequations of motion in curved spacetime in the context of string theory using two different actions,and we discuss their symmetries. In section 2.3, we discuss Born-Infeld electromagnetism, whicharises in the context of the work performed by Karczmarek and Sibilia in studying how D-branesare distorted by monopoles. Lastly, for interest we include some background on Supersymmetry insection 2.4.2.1 Review of Lagrangian MechanicsThe Lagrangian mechanics formalism is a way of deriving the equations of motion of a system fromthe potential and kinetic energies of that system, and hence the coordinate representation used inLagrangian mechanics is that of position q and velocity q˙. It is a robust method and widely usedin the study of classical physics and field theory. Most of the results of this section may be foundin the book by Landau and Lifshitz [4].A formalism related to Lagrangian mechanics is Hamiltonian mechanics. In this formalism,the coordinate representation used is that of position q and momentum p. Further, the formalismis defined using the total energy of the system. The advantage of Hamiltonian mechanics overLagrangian mechanics is that momentum is that energy and momentum are typically conserved.In this section, we go over the basics of Lagrangian mechanics in preparation for delving into themechanics of string theory and Born-Infeld Electromagnetism. In section1920 CHAPTER 2. BACKGROUNDWe define the Lagrangian as the function of position and velocity q and q˙ given byL(q, q˙) = T − V, (2.1)where T is the kinetic energy function, and V the potential energy of the system. In order to derivethe equations of motion, we must first define the action functional:S [L(q, q˙), [t0, t1]] =∫ t1t0L(q, q˙)dt. (2.2)In physics, one assumption about the motion of particles in Lagrangian mechanics is that theybehave by the ‘Principle of Least Action’. This is equivalent to stating that the physical value ofthe action is an extremum, or that the functional variation of the action is zero:δS = 0. (2.3)Taking the variation of the action yields the following formula:δS = 0 =∫ t1t0(∂L∂qδq +∂L∂q˙δq˙)dt. (2.4)A fundamental lemma in the Calculus of Variations (see [8]) states that if a differentiable functionf with continuous derivative f ′ is zero at t1 and t2, then if∫ t1t0f(x)g(x)dx = 0, (2.5)g is identically zero on the interval [t1, t2]. So, if we integrate ∂L∂q˙ δq˙ by parts, we getδS = 0 =∫ t1t0(∂L∂qδq +ddt(∂L∂q˙)δq)dt+∂Ldqδq∣∣∣∣t1t0(2.6)where the boundary term is zero, since we assume the variations δq and δq˙ is zero on the boundarydue to the Principle of Least Action. Then, by the fundamental lemma, the integrand must beindentically zero on [t1, t2] for the same reason: δq is zero on the boundary. Hence, on this intervalwe obtain the equations∂L∂q−ddt∂L∂q˙= 0. (2.7)These are known as the Euler-Lagrange Equations. Now, if one wishes to add more dimensions(and hence more parameters qi), one only has to see that the previous argument is independentin each qi, since the action treats each qi independently. Hence, the general set of Euler-Lagrangeequations are given byn∑i=1[∂L∂qi−ddt(∂L∂q˙i)]= 0. (2.8)2.2. STRING-THEORETIC MECHANICS AND D-BRANES 212.2 String-Theoretic Mechanics and D-BranesThis chapter serves to familiarize the reader with some of the basic premises and terminology ofstring theory. We begin with a discussion about the two basic types of actions used in string theoryto derive equations of motion for strings, and some of their properties. This will prepare the readerfor an introduction to D-branes, which are an important object of study in string theory and amain topic of this thesis. Much of the discourse in this section can be found in [9] and [10].The simplest example of string mechanics is that of a one-dimensional string of finite lengthmoving in a two-dimensional spacetime. As this string moves forward in time, it sweeps out acertain area which is called the world-sheet. This world-sheet can be described using two parametersXµ(τ, σ), and we must note that any physical quantities in the string theory must be independentof the (τ, σ) parametrization.The simplest parametrization-invariant action is called the Nambo-Gotu action, and it is pro-portional to the area of the world-sheet. In the preceding section, we made the implicit assumptionthat the spacetime is Minkowskian, or flat. In General Relativity, spacetime has some sort ofcurvature which is denoted using an induced metric tensor hab. Therefore, in order to define theNambo-Gotu action successfully, we must have some notion of spacetime curvature and its effecton the world-sheet area. Hence, we can define the induced metrichab = ∂aXµ∂bXµ. (2.9)Then, since the Nambo-Gotu action is proportional to world-sheet area, we can define the actionasSNG =∫M−T√−dethabdτdσ (2.10)where T is the tension of the string, given in terms of the Regge slope parameter αT =12piα, (2.11)where α is related to the string length.The Nambo-Gotu action has two important symmetries, or transformations of the parametriza-tion Xµ(τ, σ) that hold the form of Equation 2.10 invariant. Firstly, it is invariant under the actionof the Poincare´ group in arbitrary dimensions, where the reparametrization is given by(Xµ(τ, σ))′ = ΛµνXν(τ, σ) + aµ (2.12)where Λµν is a Lorentz transformation, and aµ is a linear translation of coordinates. This symmetrycan be directly seen by plugging in Equation 2.12 into Equation 2.9. The aµ term vanishes underthe action of the partial derivatives, and the form of the induced metric is unchanged under theaction of the Lorentz transformation, and thus SNG[X ′µ] = SNG[Xµ].The second symmetry is simply a re-defining of coordinates, called a diffeomorphic invariance.It is clear by choosing new (τ ′, σ′) = (τ ′(τ, σ), σ′(τ, σ)) that X ′µ(τ ′, σ′) = Xµ(τ, σ), and henceSNG[X ′µ] = SNG[Xµ].22 CHAPTER 2. BACKGROUNDWe can define a second action called the Polyakov action by introducing an independent (asopposed to an induced) metric γ(τ, σ) for the world-sheet:SP [X, γ] = −T∫M√−det γγab∂aXµ∂bXµdτdσ. (2.13)In order to vary this action, we need to use the fact that the variation of a determinant is given byδγ = γγabδγab = −γγabδγab (2.14)where we use the shorthand det(γ) ≡ γ. Then, varying the action in Equation 2.13 yieldsδγSP [X, γ] = −T∫m√−det γδγab(hab −12γabγcdhcd)dτdσ = 0 (2.15)and hence hab = 12γabγcdhcd. Dividing this equation by√−h yieldshab√−h=γab√−γ(2.16)which states that γab is proportional to the induced metric. We can now substitute these resultsinto the Polyakov action to obtain:SP [X, γ] =⇒ −12piα∫dτdσ√−h (2.17)which is precisely the form of the Nambo-Gotu action. The original form of the Polyakov actionhas both the symmetries of the Nambo-Gotu action, namely diffeomorphic invariance and Poincare´invariance, but in addition it also has what is called a Weyl invariance: for arbitrary ω(τ, σ),γ′ab(τ, σ) = exp(2ω(τ, σ))γab(τ, σ) (2.18)X ′µ(τ, σ) =Xµ(τ, σ). (2.19)This is sort of a ‘phase’ invariance of the independent metric γab. We would now like to explicitlyderive equations of motion for strings using the Polyakov action. We can do this by varying Xµ inthe action, which yields:∂a[√−γγab∂bXµ] =√−γ∇2Xµ = 0. (2.20)Let us now focus on world-sheets that have a boundary. Following the example in Polchinski,consider a temporal region −∞ < τ < ∞ and spacial region 0 ≤ σ ≤ l. Then varying the actionyields a surface term, from which we can infer boundary conditions:δSp =12piα∫ ∞−∞dτ∫ l0dσ√−γδXµ∇2Xµ −12piα∫ ∞−∞dτ√−γδXµ∂σXµ∣∣∣∣∣σ=lσ=0. (2.21)Then, the boundary term is zero if one of the following three conditions are satisfied:2.3. BORN-INFELD ELECTROMAGNETISM 231. Neumann (open) Boundary Conditions∂σXµ(τ, 0) = ∂σXµ(τ, l) = 02. Periodic (closed) Boundary Conditions∂σXµ(τ, 0) =∂σXµ(τ, l)Xµ(τ, l) =Xµ(τ, 0)γab(τ, l) =γab(τ, 0).3. Dirichlet (open) Boundary ConditionsXµ(τ, 0) = bµ; Xµ(τ, l) = b′µIn Case (2), the periodic boundary conditions cause the string to form a closed loop. In Case (1),we have the case of an open string with free endpoints. In the case where the Dirichlet boundaryconditions are satisfied, or the case where the endpoints of the string are fixed, it turns out thatin order to conserve momentum, we must attach the endpoints to some extended object whichwe call the D-brane, where D stands for Dirichlet. Seminal work by Polchinski in [11] shows thatD-branes are inherently represented via matrix-valued coordinates. One of the goals of this thesisis to characterize D-branes generated via Born-Infeld Electromagnetism in such a way that allowsus to calculate coherent states of strings on those D-branes.2.3 Born-Infeld ElectromagnetismMagnetism, as you recall from physicsclass, is a powerful force that causescertain items to be attracted torefrigerators.Dave BarryIn this section, we discuss Born-Infeld Electromagnetism and how it is related to the mainwork in this thesis. We begin with the derivation of the Born-Infeld Hamiltonian density, therebydirectly giving the Hamiltonian. We then show that the self-energy in the classical case of Born-Infeld electromagnetism (the analogous calculation to the one in the introduction) is finite.In ten dimensions, the Born-Infeld action is given by Callan and Maldacena [12] as:L = −1(2pi)9g∫d10x√det(1 + F ) (2.22)where Fµν is the gauge field strength tensor analogous to the one seen in classical electromagnetism.As noted in the introduction, we align the coordinate system such that we only consider motion in24 CHAPTER 2. BACKGROUNDone transverse direction denoted as X. For a purely electric gauge field, the action reduces toL = −1gp∫dpx√(1− ~E2)(1 + ~∇X2) + ( ~E · ~∇X)2 − X˙2 (2.23)with Lagrangian densityL = −1gp√(1− ~E2)(1 + ~∇X2) + ( ~E · ~∇X)2 − X˙2, (2.24)where gp = (2pi)pg. We now seek to Legendre transform the Lagrangian into a Hamiltonian interms of the canonical momentum for the vector potential ~A and coordinate X. Following thestandard notation, we get the respective momenta:gpP ~A = gp~Π =∂L∂A˙=~E(1 + ~∇X2)− ~∇X( ~E · ~∇X)√(1− ~E2)(1 + ~∇X2) + ( ~E · ~∇X)2 − X˙2(2.25)gpPX = gpP =∂L∂X˙=X˙√(1− ~E2)(1 + ~∇X2) + ( ~E · ~∇X)2 − X˙2(2.26)where by choosing the Coulomb gauge, A˙ = ~E, and thus the derivative ∂L∂A˙= ∂L∂ ~E. Next, we wantto show that the Hamiltonian density H is given byH =A˙~Π + X˙P − L (2.27)= ~E~Π + X˙P − L (2.28)=1gp√(1 + ~∇X2)(1 + g2pP 2) + g2p~Π2 + g2p(~Π · ~∇X)2. (2.29)This is a somewhat involved derivation. We begin by considering H2:H2 =( ~E~Π + X˙P − L)2 (2.30)= ~E2~Π2 + 2 ~E~ΠX˙P − 2 ~E~ΠL+ (X˙P )2 − 2X˙PL+ L2 (2.31)For simplicity, let gp = 1; we will bring this constant back with dimensional analysis in the finalstep. We now expand every term in Equation 2.31, noting that we can write ~Π = X˙L :~E2~Π2 =~E2L2[~E2(1 + ~∇X)2 + (~∇X)2( ~E · ~∇X)2 − 2 ~E(1 + ~∇X2)~∇X( ~E · ~∇X)](2.32)2 ~E~ΠX˙P = 2 ~EX˙2L2[~E(1 + ~∇X2)− ~∇X( ~E · ~∇X)](2.33)−2 ~E~ΠL = −2[~E2(1 + ~∇X2)− ( ~E · ~∇X)2](2.34)(X˙P )2 =X˙4L2(2.35)−2X˙PL = −2X˙LX˙L= −2X˙2 (2.36)L2 = (1− ~E2)(1 + ~∇X2) + ( ~E · ~∇X)2 − X˙2 (2.37)= 1− ~E2 + ~∇X2 − ~E2~∇X2 + ( ~E · ~∇X)2 − X˙2.2.3. BORN-INFELD ELECTROMAGNETISM 25Collecting terms and simplifying a bit yields:H2 =(1 + ~∇X2 +X˙2L2+~∇X2L2X˙2)(2.38)+1L2[~E2(1 + ~∇X2)2 + (~∇X)2( ~E · ~∇X)2 − 2( ~E · ~∇X)2(1 + ~∇X2)]+1L2[~E · ~∇X]2First, note the following:~Π · ~∇X =1L[( ~E · ~∇X)(1 + ~∇X2)− (~∇X)2( ~E · ~∇X)](2.39)=1L[( ~E · ~∇X) + (~∇X)2( ~E · ~∇X)− (~∇X)2( ~E · ~∇X)]=1L[( ~E · ~∇X)]=⇒ (~Π · ~∇X)2 =1L2[( ~E · ~∇X)]2(2.40)Further,(1 + ~∇X2)(1 + P 2) =(1 + ~∇X2 +X˙2L2+~∇X2L2X˙2)(2.41)~Π2 =1L2[~E(1 + ~∇X2)− ~∇X( ~E · ~∇X)]2=1L2[~E2(1 + ~∇X2)2 + (~∇X)2( ~E · ~∇X)2 − 2( ~E · ~∇X)2(1 + ~∇X2)](2.42)Plugging Equations 2.40, 2.41 and 2.42 into each line of Equation 2.38 givesH2 = (1 + ~∇X2)(1 + P 2) + ~Π2 + (~Π · ~∇X)2 (2.43)Dimensional analysis requires that there be a gp coefficient on each P, ~Π term, with an overall factorof g−2p . Hence, we arrive at the final answer:H =1gp√(1 + ~∇X2)(1 + g2pP 2) + g2p~Π2 + g2p(~Π · ~∇X)2. (2.44)This gives a Hamiltonian ofH =∫dpxH (2.45)=1gp∫dpx√(1 + ~∇X2)(1 + g2pP 2) + g2p~Π2 + g2p(~Π · ~∇X)2. (2.46)We now progress by calculating the self-energy of a particle in the Born-Infeld regime, and showthat it is indeed finite. For simplicity, let us write the Lagrangian density in 3 dimensions following26 CHAPTER 2. BACKGROUNDthe example in Zwiebach [10]:L = −b2√1 +E2b2+ b2, (2.47)where b is the critical electric field b = Ecrit = 12piα . Then, the canonical momentum associatedwith the vector potential in the Coulomb gauge is given by~D =∂L∂ ~E=~E√1− E2b2(2.48)=⇒ ~E =~D√1 + D2b2. (2.49)Then, we can rewrite the Lagrangian density asL = −b2√√√√1−D2b2(1 + D2b2) + b2 = b2 −b2√1 + D2b2. (2.50)This gives us a Hamiltonian density ofH = ~E · ~D − L =D2√1 + D2b2+ b21√1 + D2b2− 1 = b2(√1 +D2b2− 1). (2.51)We can now use this to calculate the self-energy of a point charge and show that the integralconverges. First, note that ∇ · ~D = ρ for some charge density ρ, and thus~D =Q4pir2rˆ. (2.52)Thus, the self-energy is equal toU =∫d3xH =b2∫ ∞04pir2dr√1 +(Q4pibr2)2− 1 (2.53)=4pib2∫ ∞0dr√r4 +(Q4pib)2− r2 . (2.54)We now make the substitutionr = x√Q4pib(2.55)which simplifies the self-energy to the expressionU =√Qb4piQ∫ ∞0dx(√1 + x4 − x2) (2.56)which clearly converges.2.4. SUPERSYMMETRY 27We may now proceed to the final background section on supersymmetry.2.4 SupersymmetryA supersymmetric theory is one that has a unique relation between bosons and fermions, in partic-ular that they appear in pairs with equal mass. While this does not happen in Nature (particularly,under the Standard Model this does not hold), symmetry-breaking is well understood and henceone would hope to derive a spontaneous symmetry breaking (ideally one that does not producetachyons) that would lower a supersymmetric theory down to one that we see in Nature.Since Supersymmetry inherently links fermions to bosons, which differ by modulo half integer ofspin, one constructs a supersymmetric theory by considering objects that have spin transformationsymmetries. For example, a supersymmetric extension of the Poincare´ algebra is generated byconsidering the anticommutation relation of two Weyl spinors [13]:{Qα, Qβ˙} = 2(σµ)αβ˙Pµ, (2.57)where σµ are the Pauli Matrices, and Pµ = −i∂µ are the generators of translations. We now showhow one can come up with this algebra. First, consider the Poincare´ group of the isometries ofspacetime. This is a 10-dimensional Lie group given by the semi-direct product of the group ofspatial translations and Lorentz transformationsR1,3 o SO(1, 3), (2.58)where SO(1, 3) is the indefinite orthogonal group that leaves invariant a symmetric bilinear formwith signature (1, 3). A semi-direct product is the generalization of a direct product where a groupis constructed from two subgroups, where one is normal.From this group, we can construct the Poincare´ Lie algebra with the following commutators forthe generator of spatial translations Pµ = −i∂µ and the generator of Lorentz transformations Msatisfying (Mµν)αβ = −ηαµ ηνβ + ηαν ηµβ:[Pµ, Pν ] = 0 (2.59)1i[Mµν , Pγ ] = ηµγPν − ηνγPµ (2.60)1i[Mµν ,Mγσ] = ηµγMνσ − ηµσMνγ − ηνγMµσ + ηνσMµγ (2.61)where η is the Minkowski metric. The supersymmetric Poincare´ algebra consists of an even (com-muting) and an odd (anticommuting) part. The even part contains this Poincare´ algebra, and theodd part is built off of spinors with the anticommutation relation in Equation 2.57. We are nowfree to add in the General Relativistic metric gµν . First, we define the Dirac Gamma matrices{γ0, γ1, γ2, γ3} with the anticommutation{γµ, γν} = γµγν + γνγµ = 2ηµν , (2.62)28 CHAPTER 2. BACKGROUNDand the Pauli spin matrices asσµν =i2[γµ, γν ]. (2.63)Then, the full supersymmetric Poincare´ algebra is given by[Mµν , Qγ ] =12(σµν)βαQβ (2.64)[Qα, Pµ] = 0 (2.65){Qα, Qβ˙} = 2(σµ)αβ˙Pµ. (2.66)In conclusion, supersymmetry is an elegant organization of modern particle physics in terms ofgroup-theoretic structure. It would certainly be elegant of Nature to incorporate supersymmetry inat least some high-energy regime (if only to make life easier for physicists), however supersymmetryin all likelyhood is an approximate, workable model to be built upon with the study of symmetry-breaking.Chapter 3Non-Commutative Bion Geometries-When did you become an expert insupersymmetric string theory?-Last night.The ReaderMathias Hudoba de BadynReal Life TMA non-commutative geometry is one in which position coordinates do not commute, for example[xi, xj ] =√−1θij (3.1)defines a non-commutative geometry for some asymmetric tensor θij with position coordinatesxi, xj . The canonical framework for non-local string theories is such a geometry. From a result bySeiberg and Witten in [14], the non-locality of a D-brane is determined by a non-zero magnetic fluxthrough that D-brane, and hence in this thesis we will study non-commutative D-branes producedfrom the instanton solution of the Born-Infeld action.This thesis follows from the work done by Karczmarek and Sibilia in [2] and [7] in examin-ing a non-commutative system of two parallel, separated D1-branes forming a single D3-brane.This system corresponds to two magnetic monopoles generated from the Born-Infeld electromag-netic theory, known as Bions (Born-Infeld Instantons). They demonstrate a numerical agreementwith the shape of the non-commutative D1-brane configuration with the D3-brane world-volumegenerated by the commutative Born-Infeld action.The single D3-brane constructed from the Born-Infeld action with coordinates chosen such thatthe brane has displacement in only one transverse direction in the embedding 10-dimensional space.The coordinates are explicity Xi = xi for i = 1, . . . , 3 and Xj = 0 for j = 4, . . . , 8 and X9 = σ(xi)is a parameter depending on the three spacial parameters. X0 = τ denotes proper time. The D3-brane with two bions looks like a sheet deformed with two spikes at the location of the monopoles,as seen in the schematic of Figure 3.The non-commutative construction of this system utilizes multiple separated D1-string bundleswhich are describes using matrix transverse scalar fields φi(σ) as coordinates, where i = 1, . . . , 3.These scalar fields are obtained by solving the Nahm equation (equation 17 in [15]) with certain2930 CHAPTER 3. NON-COMMUTATIVE BION GEOMETRIESFigure 3.1: Schematic of D-brane deformed by the presence of two Bions.boundary conditions:∂Xˆi =√−12ijk[Xˆj , Xˆk]. (3.2)This is also the equation that produces the instanton solution in the D3-brane picture.The 3D cross-section (at fixed σ) surface of the D3-brane is computed using the Berenstein-Dzienkowski effective Hamiltonian:Heff =3∑i=1(Xˆi − xi)⊗ σi (3.3)where Xˆi are the matrix-valued coordinates, xi = xiI is the identity times a real-valued coordinatexi, and σi are the 2-dimensional irreducible generators of the SU(2) group satisfying the followingcommutation relation:[J i, J j ] = iijkJk. (3.4)This effective Hamiltonian defines a surface whenever it has a zero eigenvalue, and the surface isgiven by the simple equationdetHeff = 0. (3.5)As discussed in the introduction, one of the goals of this area of research is to be able to calculatecoherent states on this D3-brane using this effective Hamiltonian. The coherent states are definedas the eigenstates of the annihilation operator. For one of the solutions of the Nahm equationcalled the ‘fuzzy sphere’, we can construct the effective Hamiltonian so that in block form, one cansee the creation and annihilation operators:Heff|~v〉 =(L3J− x3) (L−J)(L+J) (−L3J+ x3)[|α1〉|α2〉]. (3.6)3.1. NAHM EQUATION SOLUTION FOR THE FUZZY SPHERE 31The numerical work of this thesis is to show that the 2n-dimensional eigenvectors of this Hamil-tonian split into two parallel n-dimensional ‘eigenvectors’. Since the annihilation operators appeardirectly in the Hamiltonian in block form, one can see that this will indeed give the eigenstate ofthe annihilation operator.The analytical work of this thesis is to study how this D3-brane picture evolves by changing theparameter X9 = σ(xi). This will allow us to study the the entire 4-dimensional world-volume of theD3-brane instead of just its 3-dimensional cross-sections. This will have applications in numericalsolutions of the Nahm equation, as one can use the results in this section to set the behaviour ofthe Nahm equation solutions at the boundaries.We first begin by solving the Nahm equation for the fuzzy sphere solution. Afterwards, wediscuss analytical methods for determining how the shape of the D3-brane changes by varying σ(xi).Finally, we conclude by summarizing the numerical methods of studying the parallel eigenvectorsplitting conjecture.3.1 Nahm Equation Solution for the Fuzzy SphereWe now solve the Nahm equations for the fuzzy sphere solution. Recall the definition of the Nahmequation [15]:∂Xˆi =√−12ijk[Xˆj , Xˆk]. (3.7)Begin with the ansatz Xˆi = 2R(σ)J i, where R(σ) is a parameter dependent on X9 = σ(xi). Then,∂Xˆi = ∂(2R(σ)J i) (3.8)=i2ijk[Xˆj , Xˆk]=i2ijk[4R2J jJk − 4R2JkJ j ]=4i2R2ijk[JjJk − JkJ j ]= 2iR2ijk[Jj , Jk]= 2i2R2ijkjklJl (3.9)Note thatεijkεlmn =∣∣∣∣∣∣∣δil δim δinδjl δjm δjnδkl δkm δkn∣∣∣∣∣∣∣(3.10)= δil (δjmδkn − δjnδkm)− δim (δjlδkn − δjnδkl) + δin (δjlδkm − δjmδkl) . (3.11)and hence ijkjkl = −ijkjlk = ijkljk = 2δil. Plugging this into Equation 3.9 yields2dRdσJ i = −2R2ijkjklJl = −2R2(2δilJl) = −4R2J i,32 CHAPTER 3. NON-COMMUTATIVE BION GEOMETRIESwhich gives a final result ofdRdσ= −2R2, (3.12)which can be integrated by separating variables:∫dRR2= −2∫dσ1R= 2σ + CR =12σ + C. (3.13)We can choose to use the solution with C = 0, which gives the solutionXˆi =1σJ i. (3.14)This is the ‘fuzzy sphere’ with constant radius 1σ .Chapter 4Analytical MethodsIn this section, we discuss the geometric variation of the D3-brane geometry in the σ(xi) parameter.This is an important quantity, as the boundary conditions of Xˆi(σ) at σ = 0 and as σ → ∞ arenecessary to discern behaviour of the solutions of the Nahm equation.In particular, Karczmarek and Sililia in [2] and [7] consider such boundary conditions para-materized by σ such that the behaviour of the Nahm equation solution at σ = 0 produces anirreducible representation of SU(2), giving a geometric interpretation of a single D3-brane; and asσ → ∞, the solution desired was one that produced separated bundles of D1-branes. This wouldcorrespond to a block-diagonal matrixXˆi(σ →∞) ∼ Diag(xin1 + 2RJin1 , . . . , xinm + 2RJinm) (4.1)where xinj = xiInj and of course Jinj is the nj-dimensional irreducible representation of SU(2).4.1 4D Geometric Rate of ChangeThe goal of this calculation is to find a formula for ∂σ∂xi . We do this by computing partial derivativesof det(Heff), using the shorthand:H = det(Heff). (4.2)∂H∂x1∣∣∣∣∣x2x3=∂H∂x1∣∣∣∣∣ σx1x2+∑i 6=1a,b∂H∂Xˆi∂Xˆi∂σ∂σ∂x1+∂H∂Xˆ1∂Xˆ1∂σ∂σ∂x1(4.3)We will make use of the following formula. Let A be an invertible, finite dimensional squarematrix. Then, det(A) 6= 0. Further, the derivative of the determinant with respect to xi is givenby∂ detA∂xi= det(A)Tr[A−1∂A∂xi]. (4.4)3334 CHAPTER 4. ANALYTICAL METHODS4.2 Fuzzy Sphere Geometric Rate of ChangeNow, we apply the methodology introduced in the previous section to the study of the fuzzy sphereto test its validity. We first do this for angular momentum matrices of size n = 2, and then moveon to arbitrary n. Lastly, we examine the case when we assume the explicit solution Xˆi to theNahm equation is not known, and thus we will have to use only matrix calculus to compute theresult.4.2.1 N = 2 CaseSince we know that the fuzzy sphere solution is represented using the parameter σ:σ =1R=1√x2 + y2 + x2, (4.5)we wish to show, using the previously worked-out methodology, that∂σ∂xi= −xi(x2 + y2 + x2)3/2= −xiR3. (4.6)Let the fuzzy sphere of radius R(σ) be given by Xˆi = R(σ)j Ji, for n = 2. Then,Heff =3∑i=1(Xˆi − xˆi)⊗ σi =R(σ)− z −x+ iy 0 0−x− iy z −R(σ) 2R(σ) 00 2R(σ) −R(σ)− z −x+ iy0 0 −x− iy R(σ) + z(4.7)and so taking the determinant yieldsdet(Heff) = H =(x2 + y2 + z2 −R(σ)2) (3R(σ)2 + x2 + y2 + z2)= 0. (4.8)Notice that this is indeed a sphere of radius R(σ), since(x2 + y2 + z2 −R(σ)2)= 0 only whenx2 + y2 + z2 = R(σ)2, and that(3R(σ)2 + x2 + y2 + z2)is never zero. A plot of this surface isshown in Figure 4.1.We use the ‘less expanded’ derivative form of Equation 4.3:∂H∂x1∣∣∣∣∣x2x3=∂H∂x1∣∣∣∣∣x2x3σ+∂H∂σ∣∣∣∣∣x1x2x3∂σ∂x1(4.9)Computing the right-hand-side gives∂H∂x1∣∣∣∣∣x2x3σ+∂H∂σ∣∣∣∣∣x1x2x3∂σ∂x1= −12R3∂R∂σ∂σ∂x1+4R∂R∂σ∂σ∂x1(x21 + x22 + x23)+4x1(x21 +x22 +x23 +R2). (4.10)The left-hand-side is zero, since we are on the surface det(Heff) = 0. Grouping terms with∂σ∂x1on4.2. FUZZY SPHERE GEOMETRIC RATE OF CHANGE 35-1.0-0.50.00.51.0x-1.0-0.50.00.51.0y-1.0-0.50.00.51.0zFigure 4.1: ‘Fuzzy sphere’ defined by detHeff = 0one side and substituting R2 = x2 + y2 + z2 and ∂R∂σ = −1σ2 = −R2 gives∂R∂σ∂σ∂x1(−8R3)= −8x(x2 + y2 + z2) (4.11)= −8xR2=⇒∂σ∂x1(8R5) = −8xR2=⇒∂σ∂x1= −xR3(4.12)as desired.4.2.2 Arbitrary-N CaseHere, we repeat the calculation of the preceding section, however we do not impose a restriction onthe size of the angular momentum matrices. Since the surface detHeff = 0 converges to the large-Nlimit for n as low as 2, we see no problems taking this result and extending the limit as n → ∞.From Berenstein and Dzienkowski [1], for N ≥ 2 the determinant of the effective Hamiltonian isgiven bydet(Heff) =(x21 + x22 + x23 −R(σ)2) ∗ F (x1, x2, z3, R(σ)) (4.13)for some function F (x1, x2, z3, R(σ)). Again, we can use the ‘less expanded’ derivative form ofEquation 4.3:∂H∂x1∣∣∣∣∣x2x3=∂H∂x1∣∣∣∣∣x2x3σ+∂H∂σ∣∣∣∣∣x1x2x3∂σ∂x1. (4.14)36 CHAPTER 4. ANALYTICAL METHODSComputing:∂H∂x1∣∣∣∣∣x2x3σ= 2xF (x1, x2, x3, R(σ)) +∂F∂x(x21 + x22 + x23 −R(σ)2) (4.15)∂H∂σ∣∣∣∣∣x1x2x3∂σ∂x1=∂σ∂x1[(x21 + x22 + x23 −R(σ)2) ∂F∂R∂R∂σ− 2RF∂R∂σ](4.16)Again, ∂H∂x1∣∣∣∣∣x2x3= 0, and so collecting terms yields:2xF +∂F∂x(x21 + x22 + x23 −R(σ)2) = −∂σ∂x1[(x21 + x22 + x23 −R(σ)2) ∂F∂R∂R∂σ− 2RF∂R∂σ]. (4.17)Setting (x21 + x22 + x23 = R(σ)2 and ∂R∂σ = −1σ2 = −R2 gives2xF =− 2R3∂σ∂x1F (4.18)=⇒∂σ∂x1=−xR3(4.19)as desired.4.2.3 The ‘Lets-not-assume-the-solution’ CaseFinally, we must discuss the case where we do not assume we know the solution to the Nahmequation in explicit form. Of course, for the fuzzy sphere we do know the solution, however ingeneral one cannot necessarily write an explicit solution to the Nahm equation. Thus, we mustuse a methodology in which we do not make the substitution Xˆi = R(σ)Jij until the very end. Toproceed, consider for n = 2 :∂H∂x1∣∣∣∣∣x2x3=∂H∂x1∣∣∣∣∣x2x3σ+∂H∂σ∣∣∣∣∣x1x2x3∂σ∂x1. (4.20)Again, the LHS= 0 and we know that∂H∂x1∣∣∣∣∣x2x3σ= 4x1(x21 + x22 + x23 +R2) = 8x1R2, (4.21)since σ is fixed and we do not need to consider all possible solutions. Thus, all that is needed is tocompute the second term in Equation 4.20:∂H∂σ∣∣∣∣∣x1x2x3∂σ∂x1= det(Heff)Tr[Heff−1∂Heff∂σ]∂σ∂x1(4.22)= det(Heff)Tr[Heff−1 ∂∂σ(3∑i=1(Xˆi − xi)⊗ σi)]∂σ∂x1. (4.23)4.2. FUZZY SPHERE GEOMETRIC RATE OF CHANGE 37The only terms that depend on σ in the formula for Heff are the matrix coordinates Xˆi. Thus, wecan write Equation 4.23 as= det(Heff)Tr[Heff−1(3∑i=1∂Xˆi∂σ⊗ σi)]∂σ∂x1(4.24)= det(Heff)Tr[Heff−1(3∑i=1(√−12ijk)[Xˆj , Xˆk]⊗ σi)]∂σ∂x1(4.25)where we have substituted the defintion of the Nahm Equation (Equation 3.2):∂Xˆi∂σ=√−12ijk[Xˆj , Xˆk].Next, let us finally add in the information we know for the fuzzy sphere. We can make thesubstitution [Xˆi, Xˆj ] = XˆiXˆj − XˆjXˆi = 2iR2J i, where J i is spin-normalized. Thus, we get forarbitrary n:∂H∂x1∣∣∣∣∣x2x3= −R2 det(Heff)Tr[Heff−1(3∑i=1J in ⊗ σi)]∂σ∂x1. (4.26)For n = 2, J i = σi, and this reduces down to∂H∂x1∣∣∣∣∣x2x3= −R2 det(Heff)Tr[Heff−1(3∑i=1σi ⊗ σi)]∂σ∂x1. (4.27)In order to compute the trace, we need to know what Heff−1 looks like. In particular, the trace willhave every element of Heff−1 selected by the Kronecker Product multiplied by det(Heff), so we cancompute the slightly simpler expression:det(Heff)Heff−1 =−3R3 − 4zR2 +(R2 − 2z2)R −(x− iy)(R2 − z2 + α2)−2R(x− iy)α −2R(x− iy)2−(x+ iy)(R2 − z2 + α2)−(R2 − z2 + α2)β −2Rαβ −2R(x− iy)β−2R(x+ iy)α −2Rαβ −α(R2 − z2 + β2)−(x− iy)(R2 − z2 + β2)−2R(x+ iy)2 −2R(x+ iy)β −(x+ iy)(R2 − z2 + β2)−3R3 + 4zR2 +(R2 − 2z2)Rwhere the following substitutions were made for brevity:x2 + y2 → R2 − z2R+ z → αR− z → βNext, using the notation Heff−1[i,j] to denote the ijth element of Heff−1, we have thatTr[Heff−1(3∑i=1σi ⊗ σi)]= Heff−1[1,1] −Heff−1[2,2] + 2Heff−1[2,3] + 2Heff−1[3,2] −Heff−1[3,3] +Heff−1[4,4], (4.28)38 CHAPTER 4. ANALYTICAL METHODSsince3∑i=1σi ⊗ σi =1 0 0 00 −1 2 00 2 −1 00 0 0 1. (4.29)Therefore,∂H∂x1∣∣∣∣∣x2x3=−R2 det(Heff−1)(Heff−1[1,1] −Heff−1[2,2] + 2Heff−1[2,3] + 2Heff−1[3,2] −Heff−1[3,3] +Heff−1[4,4])(4.30)=4R3σ′(3R2 − x2 − y2 − z2)=8R5∂σ∂x1(4.31)Thus, we get the final result∂H∂x1∣∣∣∣∣x2x3σ+∂H∂σ∣∣∣∣∣x1x2x3∂σ∂x1= 0 (4.32)=⇒∂σ∂x1(8R5) = −8xR2=⇒∂σ∂x1= −xR3(4.33)as expected. To do this in general for arbitrary n will require a proof of the a more general versionof the result by Berenstein and Dzienkowski [1] thatdet(Heff) =(x21 + x22 + x23 −R(σ)2) ∗ F (x1, x2, z3, R(σ)) (4.34)and hence∂H∂σ∣∣∣∣∣x1x2x3∂σ∂x1= −R2 det(Heff)Tr[Heff−1(3∑i=1J in ⊗ σi)]∂σ∂x1(4.35)=∂σ∂x1[(x21 + x22 + x23 −R(σ)2) ∂F∂R∂R∂σ− 2RF∂R∂σ](4.36)using only the matrix calculus approach utilized in this subsection. The difficulty of calculatingHeff−1 for arbitrary n makes this proof challenging, and is an area of further work. However,Appendix C contains Mathematica code that will calculate∂H∂x1∣∣∣∣∣x2x3σ+∂H∂σ∣∣∣∣∣x1x2x3∂σ∂x1= 0for any n, and one can see that for n up to 6, the result holds. For n > 6, one needs a ratherpowerful (by 2014 standards) computer to calculate Heff−1.4.3. CONCLUSION 394.3 ConclusionThe most useful conclusion of this chapter is the formula for the geometric rate of change. Thishas applications in the study of numerical solutions to the Nahm equations, particularly in settingboundary conditions to give the desired properties of the solution. To give a final statement of thegeometric rate of change, recall∂H∂x1∣∣∣∣∣x2x3=∂H∂x1∣∣∣∣∣ σx1x2+∂H∂σ∣∣∣∣∣x1x2x3∂σ∂x1= 0 (4.37)which gives∂H∂σ∣∣∣∣∣x1x2x3∂σ∂x1=−∂H∂x1∣∣∣∣∣ σx1x2(4.38)det(Heff)Tr[Heff−1(3∑i=1∂Xˆi∂σ⊗ σi)]∂σ∂x1= det(Heff)TrHeff−1 ∂∂x1∣∣∣∣∣ σx1x2(3∑i=1(Xˆi − xi)⊗ σi)(4.39)det(Heff)Tr[Heff−1(3∑i=1∂Xˆi∂σ⊗ σi)]∂σ∂x1= det(Heff)Tr[Heff−1(I⊗ σ1)](4.40)which gives∂σ∂x1=det(Heff)Tr[Heff−1(I⊗ σ1)]det(Heff)Tr[Heff−1(∑3i=1∂Xˆi∂σ ⊗ σi)] . (4.41)The det(Heff) factors are kept in the numerator and denominator since det(Heff)Heff−1 is oftensimpler to write out by hand. Of course, this equation generalizes for derivatives with respect toany xj :∂σ∂xj=det(Heff)Tr[Heff−1(I⊗ σj)]det(Heff)Tr[Heff−1(∑3i=1∂Xˆi∂σ ⊗ σi)] . (4.42)40 CHAPTER 4. ANALYTICAL METHODSChapter 5Numerical Methods5.1 IntroductionThis section seeks to summarize the numerical work done to answer the questions regarding eigen-value convergence as well as the parallel eigenvector splitting. Recall the definition of the effectiveBerenstein-Dzienkowski Hamiltonian:Heff =3∑i=1(Xˆi − xi)⊗ σi. (5.1)When the determinant of this is equal to zero, we have shown that this defines a surface. We expectthat as the size n of Xˆi and xi go to infinity, the surface defined by this equation converges tosome ‘large n’ limit surface. The exact nature of this convergence is yet to be determined, howeverwe do have an idea of what this surface looks like. In the analytical section, we derived the ‘fuzzysphere’ that actually converges completely at n = 2 in the sense that Heff = 0 at n = 2 is indeedthe limiting surface of a sphere of radius 1σ .In general, the properties of the surfaces generated by Heff = 0 are difficult to ascertain, so welimit our conjectures to the study of those surfaces generated by finite polynomials of the irreduciblegenerators of the n-dimensional SU(2) algebra, namely the (spin normalized) angular momentummatrices. Using the fact that the spin-normalized angular momentum matrices satisfyJ21 + J22 + J23 = 1, (5.2)we can use the three equations defined using angular momentum matrices to solve for Ji as afunction of Xˆ1R(σ) ,Xˆ2R(σ) , andXˆ3R(σ) . Then, Equation 5.2 gives us our large n surface if we identify thematrix Xˆi to the real coordinate xi. For example, consider the noncommutative ellipsoid definedby:1R(σ)Xˆ1 = a1J1 (5.3)1R(σ)Xˆ2 = a2J2 (5.4)1R(σ)Xˆ3 = a3J3. (5.5)4142 CHAPTER 5. NUMERICAL METHODSUsing Equation 5.2, we getJ21 + J22 + J33 =(Xˆ1R(σ)a1)2+(Xˆ2R(σ)a2)2+(Xˆ3R(σ)a3)2= 1 (5.6)which gives the equation of an ellipsoid of ‘radius’ R(σ) and semi-major axes a1, a2 and a3:(Xˆ1a1)2+(Xˆ2a2)2+(Xˆ3a3)2= R(σ)2. (5.7)Notice that this equation is still in matrix coordinates. We must now make our identificationXˆi → xi:(x1a1)2+(x2a2)2+(x3a3)2= R(σ)2. (5.8)Now we have a true, real-valued expression for the large n surface of the noncommutative ellipsoid.By following this procedure, one can construct such a surface for any set of polynomials of angularmomentum matrices.5.2 Calculations on an EllipsoidIn this section, we discuss the parallel eigenvector splitting conjecture. In particular, we define aneigenvalue problem which generates eigenvectors perpendicular to the surface given by detHeff = 0.We show numerical evidence that the surface determined by this equation is actually a surface ofeigenvalues for these eigenvectors. Convergence of the eigenvectors and eigenvalues is also discussedas we take the large-n limit of the size of the J i matrices used to define the surface.Since we are interested in studying eigenvectors along a certain direction on the surface, we mustcome up with a general eigenvalue problem. First, recall the definition of the effective Hamiltonian:Heff =3∑i=1(Xˆi − xiI)⊗ σi (5.9)Consider the case where x1, x2 = 0. When Heff has a zero eigenvalue, then we have the surfacedefined by detHeff = 0. Hence, we convert this to the eigenvalue problem:Heff|v〉 = 0|v〉. (5.10)Since we want to look at the eigenvector along x3 in this example, we have to right-multiply by σ3,which is the unit basis vector pointing in the σ3 direction in Pauli matrix space, analogous to thex3 direction in real space. We then get:Heffσ3|v〉 =3∑i=1(Xˆi − xiI)⊗ σiσ3|v〉 = 0|v〉 (5.11)=(−iXˆ1 ⊗ σ1 + iXˆ2 ⊗ σ2 + (Xˆ3 − x3)I2n)|v〉 = 0|v〉 (5.12)5.2. CALCULATIONS ON AN ELLIPSOID 43-202x-202y-202zFigure 5.1: Ellipsoid defined by Xˆ1 = J1, Xˆ2 = 0.5J2, Xˆ3 = 3J3which we can turn into the eigenvalue problem=(−iXˆ1 ⊗ σ1 + iXˆ2 ⊗ σ2 + Xˆ3I2n)|v〉 = x3I2n|v〉. (5.13)This equation suggests an interesting thing: since we are on the point (0, 0, x3) on the surface, andwe have oriented our eigenvector to the σ3 direction in the Pauli matrix auxiliary Hilbert space,it seems like the point x3 should actually be the eigenvalue of this eigenvector. This leads to thegeometrically elegant conjecture that the surface defined by detHeff = 0 is actually a surface ofeigenvalues of the eigenvectors pointing in the direction of that eigenvalue on the surface.Let us give another example to illustrate this idea. Consider an ellipsoid given by Xˆ1 =J1, Xˆ2 = 0.5J2, Xˆ3 = 3J3. The surface defined by this is given in Figure 5.1. Let us look atthe Pauli space vector given by σˆ = 〈σ1 cos θ, σ2 cos θ, 0〉. What we are conjecturing is that theeigenvalues λ of(3∑i=1Xˆi ⊗ σi)σˆ|v〉 = λ|v〉 (5.14)are the actual value of the surface of the point given by (x1, x2, x3) = (cos θ, sin θ, 0), in this casethe intersection of the ellipsoid with the xy plane defined by the ellipse(x1)2 +(x22)2= 1. (5.15)44 CHAPTER 5. NUMERICAL METHODS-1.0-0.50.00.51.0x-1.0-0.50.00.51.0y-1.0-0.50.00.51.0z-2 -1 0 1 2-2-1012xyFigure 5.2: Intersection of ellipsoid and xy plane, producing the ellipse of eigenvalues for eigenvalueproblem in Equation 5.14.Figure 5.1 shows the resulting ellipse of large-n eigenvalues. The remainder of this section isconcerned with the numerical calculation of these eigenvalues, and their convergence to the large-n‘classical’ surface.Following the numerical methods of Baker in [16], the code written for this thesis in Appen-dices A, B and C defines the J i matrices as follows:J3n = {J3(i, i) = j − i+ 1} (5.16)J1n = {J1(i, i+ 1) =√i(n− i)2}+ {J1(i+ 1, i) =√i(n− i)2} (5.17)J2n = −i(J1nJ3n − J3nJ1n). (5.18)We explicitly write the eigenvalue problem as in Equation 5.14. Since for large n, these matricesare sparse and so we use the sparse matrix handling available in Matlab to solve the eigenvalueproblem. We ‘guess’ the eigenvalue using the large-n equation evaluated at the point given by thereal-valued analogue of the vector σˆ, and then plot log(λn=∞−λn) versus log(n) for different valuesof large n, where λn=∞ is the large-n eigenvalue we conjecture convergence towards.We also would like to see if the eigenvectors of this problem indeed split into two paralleln-dimensional components(3∑i=1Xˆi ⊗ σi)σˆ|v〉 = λ|v〉 =⇒ |v〉 =(|v1〉|v2〉). (5.19)To check this, we compute the area of the parallelogram spanned by |v1〉, |v2〉 given byA =√〈v1|v1〉〈v2|v2〉 − 〈v1|v2〉〈v2|v1〉 (5.20)and make sure that it goes to zero. We plot logA versus log n. Results for this ellipsoid are com-puted for 10 evenly distributed directions along ellipses given by the intersection of the ellipsoidwith the xy plane (see Figures 5.6, 5.7), the yx plane (see Figures 5.8, 5.9), the xz plane (see Fig-ures 5.10, 5.11), and a tilted plane defined by the normalized vector [cos(4pi/3) sin(θ), sin(θ) sin(4pi/3), cos(θ)](see Figures 5.12, 5.13). Geometric visualizations of these eigenvalue intersections are shown in Fig-5.2. CALCULATIONS ON AN ELLIPSOID 45ures 5.3, 5.4 and 5.5.-202x-202y-202zFigure 5.3: Non-commutative ellipsoid intersecting with xy plane. The intersection boundary is a2-d locus of eigenvalues.One thing to notice is that the convergence of eigenvalues seems to go as an for sufficiently largen, irregardless of the direction of the eigenvector, and the convergence of the parallelogram area tozero seems to go as a√N. Unpublished notes by Joanna Karczmarek have analytically proven thatfor the eigenvector along the σ3 direction for the ellipsoid defined byXˆ1 = a1J1, Xˆ2 = a2J2, Xˆ3 = a3J3 (5.21)the convergence for large n is estimated bya3 − x3 =a3(a1 − a2)22a1a21n. (5.22)46 CHAPTER 5. NUMERICAL METHODS-202x-202y-202zFigure 5.4: Non-commutative ellipsoid intersecting with xz plane. The intersection boundary is a2-d locus of eigenvalues.5.2. CALCULATIONS ON AN ELLIPSOID 47-202x-202y-202zFigure 5.5: Non-commutative ellipsoid intersecting with yz plane. The intersection boundary is a2-d locus of eigenvalues.48 CHAPTER 5. NUMERICAL METHODSFigure 5.6: Eigenvalue convergence in the xy plane on points selected from the locus depicted inFigure 5.3.5.2. CALCULATIONS ON AN ELLIPSOID 49Figure 5.7: Parallelogram convergence in the xy plane on points selected from the locus depictedin Figure 5.3.50 CHAPTER 5. NUMERICAL METHODSFigure 5.8: Eigenvalue convergence in the xz plane on points selected from the locus depicted inFigure 5.4.5.2. CALCULATIONS ON AN ELLIPSOID 51Figure 5.9: Parallelogram convergence in the xz plane on points selected from the locus depictedin Figure 5.4.52 CHAPTER 5. NUMERICAL METHODSFigure 5.10: Eigenvalue convergence in the yz plane on points selected from the locus depicted inFigure 5.5.5.2. CALCULATIONS ON AN ELLIPSOID 53Figure 5.11: Parallelogram convergence in the yz plane on points selected from the locus depictedin Figure 5.5.Figure 5.12: Eigenvalue convergence in the locus of the intersection of the tilted plane and theellipsoid.54 CHAPTER 5. NUMERICAL METHODSFigure 5.13: Parallelogram convergence in the locus of the intersection of the tilted plane and theellipsoid.For our ellipsoid, this estimate is a3−x3 = 34n−1. As seen in Figure 5.14, for reasonably large n wefind a estimate of e0.3354n−1 = 0.7165n−1, which indeed shows this is a good estimate. Further workhas yielded a similar estimate for rotations about this vector in the direction of σ3. In particular,defining the ellipsoid withXˆi = AijJ j (5.23)where A is the rotation matrixA =c1 c2 ac3 c4 b0 0 1 (5.24)yields the estimate1− x3 =(c1 − c4)2 + c222c1c4n−1. (5.25)5.2. CALCULATIONS ON AN ELLIPSOID 55Figure 5.14: Best fit line for eigenvalue convergence estimate along z-axis.56 CHAPTER 5. NUMERICAL METHODSChapter 6ConclusionIn this thesis, we have presented a self-contained introduction to matrix models, supersymmetryand Born-Infeld electromagnetism. We presented work furthering that by Karczmarek and Sibiliain understanding the connection between the abelian D1-brane construction and non-abelian D3-brane construction of the multicore bion picture. Our main results suggest that for the effectiveBerenstein-Dzienkowski HamiltonianHeff =3∑i=1(Xˆi − xiI)⊗ σi, (6.1)the surface defined by the determinant of this equation detHeff = 0 is actually a surface of eigen-values given by the eigenvector problem(3∑i=1Xˆi ⊗ σi)σˆ|v〉 = λ|v〉, (6.2)and that these eigenvectors split into two parallel components from which we can calculate coherentstates of the annihilation operator of the large-n SU(2) algebra. We also presented some analyticevidence aiming towards a proof that these two conjectures are true.We also derived and verified an equation for the geometric rate of change in σ for these surfaces:∂σ∂xj=det(Heff)Tr[Heff−1(I⊗ σj)]det(Heff)Tr[Heff−1(∑3i=1∂Xˆi∂σ ⊗ σi)] . (6.3)This is a useful equation for studying and setting the boundary conditions when attempting tosolve the Nahm equation for various configurations of the embedding space.Further work stemming from this thesis involves analyzing the properties of higher-order polyno-mials of J i and showing that the same convergence rules apply for the eigenvectors and eigenvaluesof these surfaces. We conjecture that all polynomials of J i matrices should have this parallel eigen-vector splitting property, and hence one can use this methodology to study coherent states in thesegeometries.5758 CHAPTER 6. CONCLUSIONAppendices59Appendix AMATLAB Code for calculatingeigenvectors/values1 % ANGULAR MOMENTUM MATRICES23 % setup f i g u r e s45 f i g u r e (1 )6 f i g u r e (2 )78 % Loop over matrix s i z e s910 ParSp l i t = ze ro s (20 ,2 ) ;11 EigOut = ze ro s (20 ,2 ) ;12 Times = ze ro s (20 ,2 ) ;13 out = 0 ;14 BoundCheck = ze ro s (20 ,5 ) ;15 % Def ine and f i l l Paul i Matr ices1617 S1 = spar s e (2 , 2 ) ;18 S2 = spar s e (2 , 2 ) ;19 S3 = spar s e (2 , 2 ) ;2021 S1 (1 , 2 ) = 1 ;22 S1 (2 , 1 ) = 1 ;2324 S2 (1 , 2 ) = −s q r t (−1) ;25 S2 (2 , 1 ) = s q r t (−1) ;2627 S3 (1 , 1 ) = 1 ;28 S3 (2 , 2 ) = −1;6162 APPENDIX A. MATLAB CODE FOR CALCULATING EIGENVECTORS/VALUES2930 opts . t o l = 1e−15;31 f o r k=1:1032 %k coresponds to a d i r e c t i o n in the y−z plane3334 thet = (k−1)∗( p i /2) /10 ;3536 %guess the e i g enva lue37 nvec = [ cos ( thet ) ∗ s i n ( p i /2) , s i n ( thet ) ∗ s i n ( p i /2) , cos ( thet ) ] ;38 %nvec = [ 0 , 0 , 1 ] ;39 nvec = nvec/norm( nvec ) ;40 guess = 1/ s q r t ( ( nvec (1 ) /1) ˆ2 + ( nvec (2 ) / 0 . 5 ) ˆ2 + ( nvec (3 ) /3) ˆ2) ;41 out = 0 ;4243 f o r j =18:1944 t i c45 out = out +1;46 % F i r s t d e f i n e s i z e o f matrix4748 N = 2ˆ j ;49 sp in = (N−1) /2 ;5051 % Al lo ca t e and f i l l matr i ce s5253 L1 = spar s e (N,N) ;54 L2 = spar s e (N,N) ;55 L3 = spar s e (N,N) ;56 t e s t v e c=spar s e (2∗N, 1 ) ;5758 f o r i =1:N59 L3( i , i ) = sp in − i +1;60 end6162 f o r i =1:(N−1)63 L1( i , i +1)=s q r t ( i ∗(N−i ) ) /2 ;64 L1( i +1, i ) = L1( i , i +1) ;65 end6667 L2 = −s q r t (−1) ∗(L3∗L1−L1∗L3) ;6869 % Def ine Sur face Coordinates7071 X = L1/ sp in ;6372 Y = 0.5∗L2/ sp in ;73 Z = 3∗L3/ sp in ;7475 %Def ine d i r e c t i o n a l vec to r767778 Sn = S1∗nvec (1 ) + S2∗nvec (2 ) + S3∗nvec (3 ) ;7980 % t h i s i s r e a l l y Hef f ∗ s igma 381 %Hef f = −s q r t (−1)∗kron (X, S2 ) + s q r t (−1)∗kron (Y, S1 ) + kron (Z , eye (2 ) ) ;82 Hef f = kron (X, S1∗Sn) + kron (Y, S2∗Sn) + kron (Z , S3∗Sn) ;8384 % P r e a l l o c a t e the matrix from which we order the e i g e n v a l u e s8586 Order = ze ro s (2∗N, 2 ) ;8788 % s o l v e the e i g enva lue problem . V i s a matrix with e i g e n v e c t o r columns ,D i s89 %a matrix with e i g e n v a l u e s a long the d iagona l .9091 opts . t o l = 1e−5;9293 [V, Val ] = e i g s ( Heff , 1 , guess , opts ) ;94 %Rayle igh Method9596 % Put the e i g e n v a l u e s in to a s i n g l e column979899 %This i s f o r use without the Rayle igh method100 % f o r i =1:2%∗N i f us ing ’ s i ’ f l a g in e i g s , then only two r e a l va lue sw i l l be chosen101 % Val ( i )=D( i , i ) ;102 % end103 Lambda = V( : , 1 ) ;104105 % s p l i t the e i g e n v e c t o r in to the two components106 lamb1 = ze ro s (N, 1 ) ;107 lamb2 = ze ro s (N, 1 ) ;108109 f o r i =1:N110 lamb1 ( i )=Lambda(2∗ i −1) ;111 lamb2 ( i )=Lambda(2∗ i ) ;112 end64 APPENDIX A. MATLAB CODE FOR CALCULATING EIGENVECTORS/VALUES113114 lamb1 = lamb1 ’ ;115 lamb2 = lamb2 ’ ;116117 % c a l c u l a t e para l l e l og ram area ( e i g e n v e c t o r s are118 %normal ized by d e f a u l t )119120 %ParMag = norm( lamb1 ) ∗norm( lamb2 ) ;121 %ParMag = s q r t (norm( lamb1 ) ˆ2 + norm( lamb2 ) ˆ2 − norm( dot ( lamb1 , lamb2 ) ) )/(norm( lamb1 ) ˆ2 + norm( lamb2 ) ˆ2) ;122 ParMag = s q r t ( dot ( lamb1 , lamb1 ) ∗dot ( lamb2 , lamb2 )−dot ( lamb1 , lamb2 ) ∗dot (lamb2 , lamb1 ) ) ;123124125 % record 3−e i g enva lue126127 OutVal = r e a l ( guess − Val (1 ) ) ;128129 EigOut ( out , 1 ) = N;130 EigOut ( out , 2 ) = OutVal ;131132 ParSp l i t ( out , 1 ) = N;133 ParSp l i t ( out , 2 ) = ParMag ;134 Times ( j , 1 )=N;135 Times ( j , 2 )=toc ;136137138139 end % of loop ing over n ( matrix s i z e )140141142 %record Bound Set by Joanna143144 % f i r s t th ree components are Nvec ( d i r e c t i o n in p a u l i b a s i s )145 % M ( the s l ope in l o g l o g space ) and lambda , the l a r g e n e i g enva lue146147 % compute Slope148149 Slope = ( log10 ( EigOut (2 , 2 ) )−l og10 ( EigOut (1 , 2 ) ) ) / ( log10 ( EigOut (2 , 1 ) )−l og10 ( EigOut (1 , 1 ) ) ) ;150 Bound = log10 ( EigOut (1 , 2 ) ) − l og10 ( EigOut (1 , 1 ) ) ∗Slope ;151 logBound = 10ˆBound ;15265153 BoundCheck (k , 1 ) = nvec (1 ) ;154 BoundCheck (k , 2 ) = nvec (2 ) ;155 BoundCheck (k , 3 ) = nvec (3 ) ;156 BoundCheck (k , 4 ) = logBound ;157 BoundCheck (k , 5 ) = guess ;158159 % f i g u r e160 % l o g l o g ( EigOut ( : , 1 ) , EigOut ( : , 2 ) )161 % t i t l e ( ’ Eigenvalue convergence in y−z plane ’ )162 % x l a b e l ( ’ l og (n) ’ )163 % y l a b e l ( ’ l og (x−l ) ’ )164165 % f i g u r e166 % t i t l e ( ’ Para l l e logram area convergence in y−z plane ’ )167 % x l a b e l ( ’ l og (n) ’ )168 % y l a b e l ( ’ l og ( Area ) ’ )169 % l o g l o g ( ParSp l i t ( : , 1 ) , ParSp l i t ( : , 2 ) )170171 f i g u r e (1 )172173 l o g l o g ( EigOut ( : , 1 ) , EigOut ( : , 2 ) )174 hold on175 f i g u r e (2 )176177 l o g l o g ( ParSp l i t ( : , 1 ) , ParSp l i t ( : , 2 ) )178 hold on179180181 end % end o f l oop ing over l i n e s in the y−z plane182183 % f i g u r e (1 )184 % t i t l e ( ’ Eigenvalue convergence along [ cos ( x ) s i n ( p i /2) , s i n ( x ) s i n ( p i /2) ,cos ( x ) ] ’ )185 % x l a b e l ( ’ l og (n) ’ )186 % y l a b e l ( ’ l og (x−l ) ’ )187 % f i g u r e (2 )188 % t i t l e ( ’ Para l l e logram area convergence [ cos ( x ) s i n ( p i /2) , s i n ( x ) s i n ( p i/2) , cos ( x ) ] ’ )189 % x l a b e l ( ’ l og (n) ’ )190 % y l a b e l ( ’ l og ( Area ) ’ )66 APPENDIX A. MATLAB CODE FOR CALCULATING EIGENVECTORS/VALUESAppendix BMathematica Code for SymbolicD-Brane calculations1 (∗This Code Sets up the Matr ices and s t u f f ∗)2 n = 23 j = (n − 1) /24 id = Ident i tyMatr ix [ n ] ;5 L1 = ConstantArray [ 0 , {n , n } ] ;6 L2 = ConstantArray [ 0 , {n , n } ] ;7 L3 = ConstantArray [ 0 , {n , n } ] ;8 Do [ L3 [ [ i , i ] ] = j − i + 1 , { i , n } ]9 Do [ L1 [ [ i , i + 1 ] ] = 1/2∗ Sqrt [ i ∗(n − i ) ] , { i , n − 1} ]10 Do [ L1 [ [ i + 1 , i ] ] = L1 [ [ i , i + 1 ] ] , { i , n − 1} ]11 L1 ;12 L2 = −I ∗(L3 . L1 − L1 . L3) ;13 L3 ;14 X = (R/ j ) ∗L115 Y = (R/( j ) ) ∗L216 Z = (R/ j ) ∗L317 (∗ Paul i Matr ices ∗)18 Sx = {{0 , 1} , {1 , 0}} ;19 Sy = {{0 , −I } , { I , 0}} ;20 Sz = {{1 , 0} , {0 , −1}};21 Hef f [ x , y , z ] =22 KroneckerProduct [X − x∗ id , Sx ] + KroneckerProduct [Y − y∗ id , Sy ] +23 KroneckerProduct [ Z − z∗ id , Sz ]24 (∗ This Express ion Agrees with Daniel ’ s n=2 case ∗)25 Sigma =26 Factor [ Expand [ Det [ Hef f [ x , y , z ] ] ] ]27 ContourPlot3D [ Sigma == 0 , {x , −1, 1} , {y , −1, 1} , {z , −1, 1} ,28 Axes −> True , Boxed −> True , AxesLabel −> Automatic ]29 S imp l i f y [D[ Sigma , x ] ] ;6768 APPENDIX B. MATHEMATICA CODE FOR SYMBOLIC D-BRANE CALCULATIONS30 S imp l i f y [D[ Sigma , y ] ] ;31 S imp l i f y [D[ Sigma , z ] ] ;32 Hin = F u l l S i m p l i f y [ Inve r s e [ Hef f [ x , y , z ] ] ] ;3334 MatrixForm [X]35 MatrixForm [Y]36 MatrixForm [ Z ]Appendix CMathematica Code for GeometricRate Calculations1 n = 22 j = (n − 1) /23 id = Ident i tyMatr ix [ n ] ;4 L1 = ConstantArray [ 0 , {n , n } ] ;5 L2 = ConstantArray [ 0 , {n , n } ] ;6 L3 = ConstantArray [ 0 , {n , n } ] ;7 Do [ L3 [ [ i , i ] ] = j − i + 1 , { i , n } ] ;8 Do [ L1 [ [ i , i + 1 ] ] = 1/2∗ Sqrt [ i ∗(n − i ) ] , { i , n − 1 } ] ;9 Do [ L1 [ [ i + 1 , i ] ] = L1 [ [ i , i + 1 ] ] , { i , n − 1 } ] ;10 L1 ;11 L2 = −I ∗(L3 . L1 − L1 . L3) ;12 L3 ;13 X = (R/ j ) ∗L1 ;14 Y = (R/( j ) ) ∗L2 ;15 Z = (R/ j ) ∗L3 ;16 (∗ Paul i Matr ices ∗)17 Sx = {{0 , 1} , {1 , 0}} ;18 Sy = {{0 , −I } , { I , 0}} ;19 Sz = {{1 , 0} , {0 , −1}};20 Hef f [ x , y , z ] =21 KroneckerProduct [X − x∗ id , Sx ] + KroneckerProduct [Y − y∗ id , Sy ] +22 KroneckerProduct [ Z − z∗ id , Sz ] ;23 (∗ This Express ion Agrees with Daniel ’ s n=2 case ∗)2425 Sigma = Factor [ Expand [ Det [ Hef f [ x , y , z ] ] ] ] ;26 ContourPlot3D [{ Sigma == 0} , {x , −1, 1} , {y , −1, 1} , {z , −1, 1} ,27 Axes −> True , Boxed −> True , AxesLabel −> Automatic ,28 ContourStyle −> Opacity [ 0 . 5 ] ]29 S imp l i f y [D[ Sigma , x ] ] ;6970 APPENDIX C. MATHEMATICA CODE FOR GEOMETRIC RATE CALCULATIONS30 S imp l i f y [D[ Sigma , y ] ] ;31 S imp l i f y [D[ Sigma , z ] ] ;32 Hin = F u l l S i m p l i f y [ Inve r s e [ Hef f [ x , y , z ] ] ] ;3334 MatrixForm [X ] ;35 MatrixForm [Y ] ;36 MatrixForm [ Z ] ;37 Summing =38 KroneckerProduct [X, Sx ] + KroneckerProduct [Y, Sy ] +39 KroneckerProduct [ Z , Sz ] ;40 F u l l S i m p l i f y [41 D[ Sigma , x ] − R∗\ [ Sigma ] ’∗ Factor [ F u l l S i m p l i f y [ Sigma∗Tr [ Hin . Summing ] ] ] ]42 (∗Use Alpha4 to v i s u a l i z e i n d i c e s ∗)4344 Alpha4 = Table [Hˆ{−1}[ i , j ] , { i , 1 , 2∗n} , { j , 1 , 2∗n } ] ;45 Factor [ Tr [ Alpha4 . Summing ] ] ;46 MatrixForm [ Summing ] ;47 F u l l S i m p l i f y [D[ Sigma , x ] ] ;48 F u l l S i m p l i f y [−R∗\ [ Sigma ] ’∗ Factor [ F u l l S i m p l i f y [ Sigma∗Tr [ Hin . Summing ] ] ] ]For n = 2, . . . , 6 the output of this code gives the desired geometric rate ∂σ∂x1 = −xR3:Bibliography[1] David Berenstein and Eric Dzienkowski. Matrix embeddings on flat R3 and the geometry ofmembranes. Physical Review D, 86(8):086001, 2012.[2] Joanna L Karczmarek and Ariel Sibilia. Noncommutative geometry of multicore bions. Journalof High Energy Physics, 2013(1):1–14, 2013.[3] Mathias Hudoba de Badyn. On the hilbert-polya´ and pair correlation conjectures. Undergrad-uate honours thesis, University of British Columbia, 2013.[4] Lev Landau and Evgeny Lifshitz. Mechanics: Volume 1 (Course Of Theoretical Physics).Butterworth-Heinemann, 1976.[5] David Jeffrey Griffiths. Introduction to elementary particles. John Wiley & Sons, 2008.[6] David Jeffrey Griffiths and Reed College. Introduction to electrodynamics, volume 3. prenticeHall Upper Saddle River, NJ, 1999.[7] Ariel Sibilia. Noncommutative geometry of multicore bions: numerical solution to the borninfeld action for d1-branes. Master’s thesis, University of British Columbia, 2013.[8] Izrail Moiseevitch Gelfand and Sergej V Fomin. Calculus of variations. Courier Dover Publi-cations, 2000.[9] Joseph Gerard Polchinski. An introduction to the bosonic string. Cambridge University Press,2005.[10] Barton Zwiebach. A first course in string theory. Cambridge University Press, 2004.[11] Joseph Polchinski. Dirichlet branes and ramond-ramond charges. Physical Review Letters,75(26):4724, 1995.[12] Curtis G Callan Jr and Juan M Maldacena. Brane dynamics from the born-infeld action.Nuclear Physics B, 513(1):198–212, 1998.[13] Julius Wess. Supersymmetry and supergravity. Princeton University Press, 1992.[14] Nathan Seiberg and Edward Witten. String theory and noncommutative geometry. Journalof High Energy Physics, 1999(09):032, 1999.7172 BIBLIOGRAPHY[15] W. Nahm. All self-dual multimonopoles for arbitrary gauge groups. In J. Honerkamp,K. Pohlmeyer, and H. Rmer, editors, Structural Elements in Particle Physics and Statisti-cal Mechanics, volume 82 of NATO Advanced Study Institutes Series, pages 301–310. SpringerUS, 1983.[16] Daniel Thomas Baker. Comparison of methods for determining d-brane geometry. Undergrad-uate honours thesis, University of British Columbia, 2013.

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.52966.1-0085973/manifest

Comment

Related Items