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Aspects of SU(2|4) symmetric field theories and the Lin-Maldacena geometries van Anders, Greg 2008

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Aspects of SU (2|4) Symmetric Field Theories and the Lin-Maldacena Geometries by Greg van Anders B.Sc., The University of Western Ontario, 2003 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Physics) The University Of British Columbia (Vancouver) June, 2008 c© Greg van Anders 2008 Abstract Gauge/gravity duality is an important tool for learning about strongly cou- pled gauge theories. This thesis explores a set of examples of this duality in which the field theories have SU(2|4) supersymmetry and discrete sets of vacuum solutions. Specifically, we use the duality to propose Lagrangian definitions of type IIA Little String Theory on S5 as double-scaling limits of the Plane-Wave Matrix Model, maximally supersymmetric Yang-Mills the- ory on R×S2 and N = 4 supersymmetric Yang-Mills theory on R×S3/Zk. We find the supergravity solutions dual to generic vacua of the Plane-Wave Matrix Model and maximally supersymmetric Yang-Mills theory on R×S2. We use the supergravity duals to calculate new instanton amplitudes for the Plane-Wave Matrix Model at strong coupling. Finally, we study a natural coarse-graining of the vacua, and find that the associated geometries are sin- gular. We define an entropy functional that vanishes for regular geometries, is non-zero for singular geometries, and is maximized by the thermal state. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 AdS/CFT or Gauge/Gravity Duality . . . . . . . . . . . . . 2 1.2 Supergravity and Branes . . . . . . . . . . . . . . . . . . . . 4 1.3 SU(2|4) Symmetric Field Theories . . . . . . . . . . . . . . . 7 1.3.1 Plane-Wave Matrix Model . . . . . . . . . . . . . . . 7 1.3.2 SYM on R× S2 . . . . . . . . . . . . . . . . . . . . . 14 1.3.3 SYM on R× S3/Zk . . . . . . . . . . . . . . . . . . . 15 1.4 Gauge/Gravity Duality for SU(2|4) Theories . . . . . . . . . 16 1.5 Little String Theory . . . . . . . . . . . . . . . . . . . . . . . 22 1.6 Gravitation and Entropy . . . . . . . . . . . . . . . . . . . . 24 1.7 Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 LST from the PWMM . . . . . . . . . . . . . . . . . . . . . . 27 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 Matching Parameters with Gauge Theory . . . . . . . . . . . 39 2.5 Scaling Limits . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5.1 Large d: the D2-brane Limit . . . . . . . . . . . . . . 42 2.5.2 Large V0: the ’t Hooft Limit . . . . . . . . . . . . . . 43 2.5.3 Large R: Little String Theory on S5 . . . . . . . . . . 43 2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.6.1 The Double-Scaling Limit . . . . . . . . . . . . . . . 45 2.6.2 An Infinite Parameter Family of NS5-brane Solutions 47 iii Table of Contents 2.6.3 Energies of Near-BPS States . . . . . . . . . . . . . . 47 2.6.4 Other Definitions of Little String Theory on S5 . . . 48 3 LST from Field Theories . . . . . . . . . . . . . . . . . . . . . 49 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 The Gauge Theories and Their Dual Supergravity Solutions 51 3.3 Little String Theory from SYM on R× S2 . . . . . . . . . . 54 3.4 Little String Theory from N = 4 SYM on R× S3/Zk . . . . 58 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4 General LM Solutions and PWMM Instantons . . . . . . . 66 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2 Supergravity Solutions . . . . . . . . . . . . . . . . . . . . . 67 4.2.1 General Solutions Dual to SYM on R× S2 . . . . . . 68 4.2.2 General Solutions Dual to the PWMM . . . . . . . . 72 4.3 Instanton Calculations . . . . . . . . . . . . . . . . . . . . . 73 4.3.1 Instantons on the Gravity Side . . . . . . . . . . . . . 74 4.3.2 Instantons in the PWMM . . . . . . . . . . . . . . . . 75 4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5 Coarse-Graining the LM Solutions . . . . . . . . . . . . . . . 82 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.2 The SU(2|4) Symmetric Matrix Quantum Mechanics and the Dual Lin-Maldacena Geometries . . . . . . . . . . . . . . . . 85 5.2.1 The Plane-Wave Matrix Model . . . . . . . . . . . . . 85 5.2.2 Electrostatics . . . . . . . . . . . . . . . . . . . . . . 86 5.2.3 Gravity Duals . . . . . . . . . . . . . . . . . . . . . . 87 5.3 Coarse-Graining the Lin-Maldacena Geometries . . . . . . . 88 5.3.1 Macroscopic Variables . . . . . . . . . . . . . . . . . . 88 5.3.2 Typical States . . . . . . . . . . . . . . . . . . . . . . 89 5.3.3 Supergravity Solution for the Average State . . . . . 90 5.3.4 Coarse-Grained Solutions for Large Discs . . . . . . . 92 5.4 An Entropy Functional . . . . . . . . . . . . . . . . . . . . . 94 5.4.1 A Familiar Example . . . . . . . . . . . . . . . . . . . 94 5.4.2 Entropy for Coarse-Grained Matrix Model Vacua . . 95 5.5 Other Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.5.1 Fixed N5 . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.5.2 Fixed N2 or Fixed N2 and N5 . . . . . . . . . . . . . 100 5.6 Higher Dimensional SU(2|4) Symmetric Theories . . . . . . 101 iv Table of Contents 5.6.1 Maximally Supersymmetric Yang-Mills Theory on R× S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.6.2 N = 4 Yang-Mills Theory on R× S3/Zk . . . . . . . 103 5.6.3 Type IIA Little String Theory on S5 . . . . . . . . . 105 6 Conclusion and Future Directions . . . . . . . . . . . . . . . 107 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Appendices A Quantum Mechanical Instantons . . . . . . . . . . . . . . . . 119 B Lin-Maldacena Supergravity Solutions . . . . . . . . . . . . 122 C Simple Electrostatics Solutions . . . . . . . . . . . . . . . . . 124 C.1 Solution of the Electrostatics Problem with Two Discs . . . 124 C.1.1 Limiting Forms of the Solution . . . . . . . . . . . . . 125 C.2 Single disc solution . . . . . . . . . . . . . . . . . . . . . . . 129 C.3 Dual integral equations . . . . . . . . . . . . . . . . . . . . . 131 D More Coarse-Graining . . . . . . . . . . . . . . . . . . . . . . . 132 D.1 Coarse-Graining for Large Discs . . . . . . . . . . . . . . . . 132 D.1.1 The Variational Problem . . . . . . . . . . . . . . . . 132 D.2 Stretched Horizons . . . . . . . . . . . . . . . . . . . . . . . . 135 E Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . 138 F Note on Publication . . . . . . . . . . . . . . . . . . . . . . . . 141 v List of Figures 1.1 Matrix model vacua and Young diagrams . . . . . . . . . . . 12 1.2 Topological cycles in the geometry. . . . . . . . . . . . . . . . 21 2.1 Mapping between vacua, electrostatics and geometry . . . . . 35 2.2 Electrostatics problem for solution with single NS5 throat . . 36 2.3 Plot of q(κ) for PWMM. . . . . . . . . . . . . . . . . . . . . . 40 3.1 Types of electrostatics configurations. . . . . . . . . . . . . . 50 3.2 D2 theory electrostatics problem with single NS5 throat. . . . 56 3.3 Plot of q(κ) for D2 theory. . . . . . . . . . . . . . . . . . . . . 57 3.4 N = 4 SYM on R× S3/Zk electrostatics problem. . . . . . . 59 3.5 Plot of q(κ) for N = 4 SYM on R× S3/Zk . . . . . . . . . . 62 4.1 Example electrostatics configurations. . . . . . . . . . . . . . 69 4.2 Young diagrams for vacua for instanton. . . . . . . . . . . . . 74 4.3 Plot of superpotential. . . . . . . . . . . . . . . . . . . . . . . 77 4.4 Plot of euclidean D2-brane action. . . . . . . . . . . . . . . . 78 5.1 Coarse-graining for large discs. . . . . . . . . . . . . . . . . . 93 5.2 Example electrostatics configuration. . . . . . . . . . . . . . . 99 vi Acknowledgements I would not have written this thesis without the help of a number of people at various stages. Thanks to my parents for stressing the importance of edu- cation. I am very indebted to my undergraduate professors who sparked my interest in physics and mathematics, particularly Trevor Luke, Volodya Mi- ransky and Gerry McKeon, each of whom had a long lasting impact on my development. I would also like to thank a number of the professors I inter- acted with during graduate school: the members of my committee, Joanna Karczmarek, Douglas Scott and Gordon Semenoff for asking questions and offering helpful feedback; Ian Affleck for providing a model of good teaching, Janis McKenna for her tireless work as Grad Chair (when I was beginning grad school); and Bill Unruh for frequent computing advice and assistance. Hai Lin deserves many thanks for helpful correspondence, especially for pointing out those pesky factors of two. I would also like to thank the string group at UBC for many interesting discussions over the years. I’d particularly like to thank my collaborators Brian Shieh and Henry Ling for the many interesting physics discussions we had. Thanks also to Brian for discussing the daily arXiv offerings on the long bus ride home. I also had enjoyable discussions with the many residents of the theory centre over the years Tami, Bojan, Dastegir, Lara, Mya, and Conan. My other friends in and out of physics, Roz, Rod, Rob, Saul, Celine, Jake, Dan, Dom, Jen, Donovan and Tara all put up with my complaints about the Vancouver’s constant rain. I gratefully acknowledge the financial support I’ve received during my graduate studies from the Natural Sciences and Engineering Council of Canada, the Killam Trusts and a Theodore E. Arnold University Gradu- ate Fellowship. I would like to give special thanks to my advisor Mark Van Raamsdonk for his support and advice and for always being generous with his time, ideas, and travel funds. Finally I would like to thank my partner Sari. You were a constant source of support for me through the last five years, always holding my hand on the proverbial PhD rollercoaster. You were an inspiring example vii Acknowledgements of how to succeed by working hard and persevering. Seeing the way you challenge your own thinking was such a help to me in grad school. I don’t think I could have taken so many years of trudging through the pouring rain without you. I wouldn’t want to live at the top of a hill with anyone else. viii Chapter 1 Introduction The aim of physics is to provide a quantitative description of the phenom- ena we observe. Physics contains many theories and models, and in many cases mathematical techniques exist that allow us to extract detailed pre- dictions from them. Some physical systems can be described using different sets of language, which can be very useful. Some insights are more apparent when the description of a phenomenon is formulated in one language, than in another. An area of physics in which this is particularly true is in un- derstanding Quantum Chromodynamics (QCD), the theory that describes the physics of the strong nuclear interaction. The development of various techniques to study this theory is important because two its most interest- ing aspects, confinement and chiral symmetry breaking, occur in a regime in which the theory is not amenable to perturbative studies [5]. Develop- ing nonperturbative techniques to study this theory is therefore particularly important. In the case of chiral symmetry breaking, a common technique is to use phenomenological theories such as the Nambu–Jona-Lasinio model [6]. This model is an effective theory for the quarks that can be thought of as arising after integrating out the degrees of freedom of the gluons. This theory has a four fermion interaction with a dimensionful coupling constant. Since it does not have gluons, this theory cannot be used to study confinement. However, it does provide a setting for studying chiral symmetry breaking, a phenomenon that is responsible for giving masses to the quarks. For a detailed discussion, see [7]. Confinement, on the other hand can be studied using lattice gauge theory. This approach consists of finding an appropriate discretization of the field theory so that it can be simulated on a computer using Monte-Carlo methods. In the limit that the lattice spacing is taken to zero, the results of the simulations can be compared with experimental results. Lattice gauge theory is particularly well suited to studying gauge theories in thermal equilibrium, but it is challenging to apply it to processes that happen in real time, such as the strongly coupled quark-gluon plasma produced at the Relativistic Heavy Ion Collider (RHIC), as pointed out in, e.g. [8]. It is useful, therefore to look for other approaches to study gauge 1 Chapter 1. Introduction theories non-perturbatively. The approach we will discuss in this thesis is called gauge/gravity du- ality, and comes from string theory. We will meet the duality in the next section. The key ingredients in string theory that lead to the duality pro- posal, supergravity and its connection to string theory, and the existence of branes in string theory will be described in the subsequent section. 1.1 AdS/CFT or Gauge/Gravity Duality Since much of the interesting physics of non-Abelian gauge theories is non- perturbative, it is useful to search for approaches to allow for their study in this regime. One that has proven extremely useful is the so-called AdS/CFT correspondence [9], or more generally the gauge/gravity correspondence. Some time ago, ’t Hooft noticed [10] that a perturbative expansion in a gauge theory with a large number of colours can be organized as an expansion in inverse powers of the number of colours. If the number of colours is N , the diagrams that contribute at each order in 1/N are those that can be drawn on a surface with a fixed number of handles. The leading term in the expansion comes from diagrams that can be drawn on a plane, the next term from diagrams that can be drawn on a torus or donut, and so on. ’t Hooft noticed that this is similar to the diagrammatic expansion in string theory, in which the expansion is also in terms of surfaces with progressively more handles. In string theory, this expansion arises from the world sheets of strings as they join and split when strings scatter from one another. ’t Hooft proposed, therefore, that gauge theories with a large number of colours should have an alternative description in string theory. An explicit realization of this, however, proved elusive for decades. The AdS/CFT correspondence provides such a realization. These ideas arise from considering D-branes, which are extended objects that exist in string theory. In particular, D-branes in string theory can have different descriptions, and it is the existence of these multiple descriptions that motivates the duality. At low energies, D-brane degrees of freedom are described by supersymmetric gauge theories. The gauge theory associated with D3 branes is N = 4 supersymmetric Yang-Mills theory. D-branes can also be associated with solutions of supergravity, and we will describe supergravity and this connection in the next section. In the case of D3 branes, the associated geometry is a solution of type IIB supergravity, which has a region that is the space AdS5×S5. It is the relationship between these pictures that gives rise to the AdS/CFT correspondence. 2 Chapter 1. Introduction In its most studied form, the AdS/CFT correspondence [9], proposed by Maldacena, relates a maximally supersymmetric gauge theory in four- dimensions, N = 4 supersymmetric Yang-Mills theory (N = 4 SYM), with type IIB string theory on the space AdS5 × S5. 1 AdS5 is five-dimensional Anti-De Sitter space, a maximally symmetric space with constant negative curvature, and S5 is a five-dimensional sphere.2 Part of the reasoning for why there should be a correspondence between these particular theories comes from symmetry considerations. N = 4 SYM is a conformal field the- ory with six scalar fields. The conformal group in four dimensions includes the Lorentz group SO(1, 3).3 The generators of the scale transformations and so-called special conformal transformations can be combined with this to form the group SO(2, 4). Furthermore, the scalar fields transform as a vector under SO(6). However, SO(6) is the isometry group of S5, because SO(6) is the group of rotations in six-dimensions, which maps an S5 back to itself. Similarly, SO(2, 4) is the isometry group of AdS5. This is because AdS can be represented as an hyperboloid, and SO(2, 4) is the group of rotations in a space with two time and 4 space dimensions that map a five-dimensional hyperboloid back to itself. A full discussion of these points is given in, e.g. [11]. The fact that these theories have identical symmetries, coupled with their related origin through two different pictures of D3-branes, as described above, suggests that this can provide an explicit realization of ’t Hooft’s pro- posal. Subsequently a dictionary that allowed the comparison of observables on both sides of the duality was found [15, 16], and since, a vast amount of evidence has accumulated to support the duality; an early review is given in [11]. The duality was also extended to field theories that describe the degrees of freedom living on branes of other dimensions [17]. This saw the extension of the duality to field theories without conformal symmetry, and, as result, to spaces that were not AdS. Indeed, ’t Hooft’s original proposal was motivated by general considerations, and so there should be many ex- amples for which such a duality should exist. AdS/CFT has therefore given way to the more general concept of gauge/gravity duality, although the term AdS/CFT is sometimes used in the literature to refer to these dualities in general. This duality has had a number of practical applications. It has been used, for example, to study confinement [18], construct models with both confinement and chiral symmetry breaking [19], suggest a universal bound 1Comprehensive reviews of this subject are given in [11], [12], [13]. 2Throughout this thesis we will denote the n-sphere as Sn. 3We will use the standard notation for simple Lie groups given in, e.g. [14]. 3 Chapter 1. Introduction on the ratio of the shear viscosity to the entropy density of plasmas [20], and more recently to study the strongly coupled quark-gluon plasma at RHIC, e.g. [8]. 1.2 Supergravity and Branes A major development in string theory was the realization that the theory contains not only strings, but also other extended objects called branes [21]. Branes have descriptions in both supergravity and in string theory, and the relationship between these descriptions plays a key role in the gauge/gravity correspondence. In this section we will describe some relevant aspects of supergravity, and then discuss the role of branes in supergravity. We will then discuss the origin of branes in string theory, and the connection to the supergravity description. Supergravity has been a subject that has received considerable interest. One key reason for this is that following the invention of supersymmetry, it was natural to consider a supersymmetric generalization of general rela- tivity. Supergravity is this generalization. Another motivation comes from string theory (see, e.g. [22, 23] for a full discussion). Quantizing the string produces a spectrum of particles; there is a set of massless particles, and an infinite tower of massive particles. One can then write an effective field theory for these particles. If we consider just the massless particles, at very low energies, this field theory is supergravity. In ten dimensions there are two closed oriented superstring theories, the type IIA theory and the type IIB theory. Since they are closed string theories, vibrations can travel along the world sheet of the string in two directions. The supersymmetries of these theories can have two different chiralities. In the type IIA theory, the chiralities of the supersymmetries for the vibrations that travel to the left and the vibrations that travel to the right are different. For the type IIB theory, they are the same. Due to this difference their particle spectra are different. In the type IIA case they produce particles that correspond to antisymmetric tensor fields with an odd number of indices. In the type IIB case they correspond to an even number of indices. Type IIA supergravity, then, contains fields that have odd numbered antisymmetric tensor fields, and type IIB contains even numbered ones. These higher dimensional anti- symmetric tensor fields, though, can be thought of as being like the gauge potential in electromagnetism, but for higher dimensional objects. The reason for this is the following. Consider electromagnetism. A 4 Chapter 1. Introduction particle couples to the electromagnetic field through a term in the action∫ dσAµ dxµ dσ . (1.1) Here Aµ is the electromagnetic potential, and x µ(σ) describes the motion of the particle, and σ parametrizes the path. This could also be written without the explicit dependence on σ as∫ Aµdx µ, (1.2) which expresses the fact that we can choose different parametrizations for the path, on which the result should not depend. To generalize this to higher dimensions it is useful to use the language of differential forms. In this language we can introduce a 1-form A = Aµdx µ so that we can rewrite the interaction of a particle with the electromagnetic field as∫ A. (1.3) Suppose we wanted to generalize this to objects of higher dimension. If we had a two dimensional object, its worldsheet would be described by two coordinates. An analogous expression to (1.1) would be∫ dσdτAµν ∂xµ ∂σ ∂xν ∂τ . (1.4) If we were to parametrize the worldsheet using a different set of coordinates, which we should be free to do, this would introduce a Jacobian factor. Since the Jacobian is antisymmetric (e.g. [24]), the only way for this expression to remain invariant after this Jacobian factor is included is if Aµν is anti- symmetric. We see then that a two dimensional object naturally couples to an antisymmetric tensor field with two indices. This is directly analogous to the coupling of a particle to a field with a single index, because it can be thought of as being extended in one direction, time. The same thing would happen if we were to consider coupling to objects of higher dimension, and therefore higher dimensional antisymmetric tensor fields naturally carry an analogue of the electromagnetic field sourced by higher dimensional objects. Now reconsider the type IIA and IIB supergravities. Within type IIA supergravity are antisymmetric tensor gauge fields with odd numbers of indices. As we have described above, antisymmetric tensor fields are the analogues of the electromagnetic field carried by higher dimensional objects. 5 Chapter 1. Introduction We see then that these can be thought of as the electromagnetic potential sourced by objects that are extended in an odd number of dimensions. If one of these objects is extended in time, like particles are, then it will be extended in even numbers of space dimensions. For example, this theory contains an antisymmetric field with three indices, suggesting the existence of an object extended in three dimensions. If one of these extended dimensions is time, then it will be extended in two space dimensions, and therefore the theory should have objects extended in two spatial dimensions. In the case of the type IIB theory, the tensor fields have even numbers of indices, so they can be thought of as carrying the electromagnetic potential of objects extended in an odd number of space dimensions. Indeed, corresponding supergravity solutions that describe extended objects of the correct dimension have been known for some time [25]. These are called black p-brane solutions, because they are like higher dimensional black holes. These p-branes, however, were later recognized to be of great impor- tance in string theory [21], as a connection was understood to exist between these supergravity solutions and D-branes in string theory. Comprehensive treatments of D-branes can be found in [26–28]; we will summarize some of the relevant details here. Open strings have endpoints and therefore some boundary conditions must be imposed at these endpoints. Natural boundary conditions to impose are that the string endpoints are free to move, which, mathematically, are Neumann boundary conditions4 (e.g. [22]). String theo- ries have a symmetry called T-duality (e.g. [22]) that arises when considering a string theory on a space with a compact direction. A closed string can wrap around a compact direction in the same way that an elastic band can wrap around a pole. T-duality is a symmetry of string theory that exchanges a string wrapping a compact direction of size R with another string carrying momentum around a compact direction of size l2s/R, where ls is a parameter called the string length. These two different pictures of the string are completely equivalent. For open strings, however, this has another effect, which is that it exchanges Neumann boundary conditions for Dirichlet boundary conditions. Dirichlet boundary conditions, in turn, mean that the string endpoints are fixed, rather than free to move. Dirichlet boundary conditions are therefore also natural, and by imposing them, we are creating a region in space for the endpoints of open strings to live. These regions can be thought of as objects in their own right, called D-branes after 4Neumann boundary conditions are conditions imposed on the value of the derivative of a function, whereas Dirichlet conditions are imposed on the value of the function itself. See e.g. [29]. 6 Chapter 1. Introduction the Dirichlet boundary conditions. Indeed, the open string spectrum in the presence of a D-brane contains states that encode fluctuations of the brane, implying that it is an object with its own dynamics. Due to the connection between string theory and supergravity, however, these extended objects should have some description in supergravity, and were identified as the the p-brane supergravity solutions [21]. These complementary points of view are very satisfying. We began this section by considering supergravity, which arose as an effective theory of the massless particles in the string spectrum. This theory had higher di- mensional black hole solutions corresponding to higher dimensional objects. These higher dimensional objects provided the source for the higher form gauge fields in the supergravity theory. We also considered the spectrum of open strings. This led to the consideration of strings with ends fixed to some higher dimensional objects. These higher dimensional objects have su- pergravity descriptions themselves, which are the higher dimensional black holes we had met previously. 1.3 SU(2|4) Symmetric Field Theories We have mentioned that the gauge/gravity correspondence provides a con- nection between N = 4 SYM and supergravity on AdS5×S5 via the different descriptions of D3 branes. However, there are other interesting gauge the- ories for which it would be interesting to have a similar dualities, and the work in this thesis considers a particular set, field theories that have SU(2|4) supersymmetry. To motivate the consideration of this particular set, we will describe each of theories, discuss some ways in which they can be derived, and list some of their properties. We will show that an interesting property of each of these theories is the existence of discrete sets of vacua. 1.3.1 Plane-Wave Matrix Model A very interesting development in string theory was the understanding that as the strength of the coupling in type IIA string theory is increased, it appears to pass from a ten-dimensional theory into an eleven-dimensional description, which was named M-theory (e.g. [23]). The Plane-Wave Matrix Model was first proposed in [30] as a microscopic description of this theory on a plane-wave space-time. It is similar to the Matrix theory proposal [31], which provided a microscopic description of M-theory in flat space. In this section we will describe the Plane-Wave Matrix Model briefly. 7 Chapter 1. Introduction The plane-wave solution of eleven dimensional supergravity can be found by taking an appropriate limit of either the AdS7×S4 or AdS4×S7 solutions, so that the metric and field strength are given by ds2 =− 4dx−dx+ + d~x2−((µ 3 )2 (x21 + x 2 2 + x 2 3) + (µ 6 )2 (x24 + · · ·+ x29) ) (dx+)2, F+123 =µ. (1.5) Here x± are lightcone coordinates, xi, i = 1, 2, 3 and xa, a = 4, . . . , 9, are coordinates in the transverse directions, and µ is a mass parameter. A first approach to the Plane-Wave Matrix Model action then is to consider the action for a superparticle in this background. This action is most tractable in the light-cone gauge in which it takes the form S = ∫ dt ( (Ẋi)2 + (Ẋa)2 − (µ 3 )2 (Xi)2 − (µ 6 )2 (Xa)2 + ψT ψ̇ − µ 4 ψTγ123ψ ) (1.6) where Xi, i = 1, 2, 3, and Xa, a = 4, . . . , 9, are bosonic scalars, Ψ is fermionic, µ is the same mass parameter above, dots denote time deriva- tives, and γ123 is a product of Dirac matrices. The bosonic part of the action here is similar to that of an ordinary particle of unit mass attached to a spring with unit spring constant S = 1 2 ∫ dt ( ẋ2 − x2) (1.7) We can think of the bosonic part of (1.6) as being the action of a particle in a 9-dimensional anisotropic harmonic oscillator, with the fermionic part added to make it supersymmetric. Equation 1.6 is the action for a single D0 brane. To find the action for multiple D0 branes it was argued in [30] that an extension could be found by ensuring supersymmetry. The resulting action is, with some rescaling, S = ∫ dtTr (1 2 (Ẋi)2 + 1 2 (Ẋa)2 − 1 2 (µ 3 )2 (Xi)2 − 1 2 (µ 6 )2 (Xa)2 + ψT ψ̇ − µ 4 ψTγ123ψ − iψT γi[Xi, ψ]− iψT γa[Xa, ψ] + 1 4 [Xi,Xj ]2 + 1 4 [Xa,Xb]2 + 1 2 [Xi,Xa]2 − iµ 3 ǫijkX iXjXk ) , (1.8) 8 Chapter 1. Introduction where Xi, Xa and ψ have become N × N matrices and Tr denotes the corresponding trace. Another approach to this model came from considering membranes in the plane-wave background. Since membranes are fundamental objects in M-theory, a microscopic description of M-theory in the plane-wave back- ground should also arise from considering the action for membranes in that background. It was shown in [32] that the Plane-Wave Matrix Model action arises as a regularization of the membrane action. Thus the same action arises as a description of M-theory in a plane-wave background from two completely different perspectives. This is also the case for M-theory in flat space [31, 33]. One additional way to find the Plane-Wave Matrix Model is to consider N = 4 SYM on R × S3 and truncate the theory to the modes that are constant on S3 [34]. The equations of motion for these modes are equivalent to the equations of motion that derived from the Plane-Wave Matrix Model action. A convenient way of expressing this model that will help to obviate some of its properties is the form [32] S = ∫ dtTr (1 2 (D0X i)2 + 1 2 (D0X a)2 + iψ†IαD0ψIα − 1 2 (1 3 )2 (Xi)2 − 1 2 (1 6 )2 (Xa)2 − 1 4 ψ†IαψIα − i 3 gǫijkX iXjXk − gψ†Iασiαβ[Xi, ψIβ ] + 1 2 gǫαβψ †αI g a IJ [X a, ψ†βJ ]− 1 2 gǫαβψαI(g a†)IJ [Xa, ψαJ ] + 1 4 g2[Xi,Xj ]2 + 1 4 g2[Xa,Xb]2 + 1 2 g2[Xi,Xa]2 ) , (1.9) where now i = 1, 2, 3, a = 1, . . . , 6, I = 1, . . . , 4, α = 1, 2, and g is a constant that determines the strength of interaction or coupling between the matrices. The model contains three types of bosonic fields, three scalar fields Xi, six scalar fields Xa, and a gauge field A0 that enters through the gauge covariant derivative D0 = ∂t − i[A0, ]. (1.10) The model also contains fermions ψαI . In addition to gauge symmetry, there are other symmetries of the matrix model. Let us concentrate first on the bosonic terms in the action. If we think of the Xi matrices as being 9 Chapter 1. Introduction like a vector in three dimensions, then terms in the action that have (Xi)2 are like the lengths of vectors in three dimensions. However, the length of a vector is preserved by rotations, and so such terms are invariant under transformations that preserve the lengths of vectors in three dimensions, which is the group called SO(3). Similarly the matrices Xa are like vectors in six dimensions, and terms that have the form (Xa)2 are invariant under SO(6) transformations. Indeed, all the bosonic terms in the action are invariant under the group SO(3) × SO(6) because they all take the form of lengths of three- and/or six-dimensional vectors. In the usual language, each term is a singlet. Now consider the fermionic terms. In ordinary quantum mechanics, fermions do not transform under the rotations SO(3), but instead the cover- ing group SU(2), which allows for them to have half-integer spins. Likewise, in the Plane-Wave Matrix Model, the fermions carry an index α for the spin 1 2 representation of SU(2), rather than an SO(3) index. In six dimensions, the group SU(4) plays the same role for SO(6) that SU(2) does for SO(3) in three dimensions (e.g. [35]). The fermions, therefore, also carry an index I in SU(4). Like the bosonic part of the model, the terms in the action involving fermions also form singlets, in this case under the group SU(2)× SU(4). The symmetries discussed so far are all examples of ‘bosonic symmetries’, even though they involve fermions, because they are symmetries that rotate bosons into bosons, and fermions into fermions. This model also has super- symmetries that rotate bosons into fermions and fermions into bosons [30], and turn out to be important for understanding a number of aspects of the model [36]. In considering systems with symmetries in quantum mechanics, it is useful to group states by how they transform under those symmetries. This is what is done, for example, when considering the hydrogen atom. A similar strategy can be applied here, and gives useful results [36]. We will return to this shortly. The model itself has a number of very interesting properties. One in- teresting feature of this model, as first noticed in [30], is that the model possesses a large number of discrete vacua. To see why this is the case consider the potential for the scalars that transform under SO(3), which is proportional to V ∼ Tr((Xi + 3igǫijkXjXk)2). (1.11) here Xi are again the three matrices that transform under SO(3) and ǫijk is the totally antisymmetric tensor with three indices, and g is the coupling constant. Vacua of the matrix model are field configurations that minimize 10 Chapter 1. Introduction this potential, and it was noticed in [30] that the minima occur when Xi = 1 3g J i, (1.12) where the matrices J i satisfy the algebra of SU(2):[ J i, Jj ] = iǫijkJk. (1.13) These matrices are N ×N , and so there will be vacua of the matrix model that are given by any N×N dimensional representation of SU(2). Different representations can be formed by taking the matrices to be block diagonal with blocks of different sizes as Xi = 1 3g   J iN1×N1 . . . J iNk×Nk   , (1.14) where the sizes of the blocks satisfy∑ j Nj = N. (1.15) Supersymmetry has important consequences for these vacua. The bosonic and supersymmetries for the matrix model were shown in [36] to combine into a particular supergroup, called SU(2|4), and an analysis of the rep- resentations of states in the model was made. The vacua of the matrix model were shown to sit in the trivial representation of this supersymmetry algebra. The analysis in [36] showed that there were certain states in the matrix model whose energies are the same, regardless of the strength of the coupling constant. The vacua are examples of these states, and therefore the classical vacua (1.12) are vacua of the full quantum theory. What is the interpretation of these vacua? The matrix model should provide a description of M-theory in a plane-wave background. The vacua of the model, therefore, should correspond to objects in M-theory: M2 and M5 branes. To understand how these arise it is useful to introduce a short- hand notation for describing the vacua. Vacua are given by different block diagonal combinations of representations of SU(2). To keep track of the vacua, we can record the number of times that a block of a given size oc- curs on the diagonal, and introduce a Young diagram to do so.5 In this 5To avoid potential confusion, the Young diagrams represent the partition of N into a sum of integers, not the symmetrization and antisymmetrization of indices in a represen- tation of SU(2). For Young diagrams for partitions, see, e.g. [37]. Contrast this with the treatment for SU(2) give in, e.g. [38]. 11 Chapter 1. Introduction N1N2 Nk · · · X i = 1 3g   JiN1 ... JiNk   Figure 1.1: Correspondence between matrix model vacua and Young dia- grams. notation, each time an Ni-dimensional representation of SU(2) appears, we add a column to the Young diagram with Ni boxes. For example, suppose the matrices were 3 × 3. The vacua would be: three copies of the trivial (singlet, or one-dimensional) representation; one copy of the doublet (two- dimensional) representation and a copy of the singlet representation; or one copy of the triplet (three-dimensional) representation. This expresses the fact that 3 = 1 + 1+ 1, 3 = 2 + 1 or simply 3 = 3, where the corresponding Young diagrams are , , . (1.16) The general case is depicted in figure 1.1. The matrix model provides a description of M-theory in a limit in which N →∞. In the Young diagram notation, this is a limit in which the number of boxes is taken to infinity, and the interpretation of the vacua in terms of M2 or M5 branes depends on the way in which this limit is taken [39]. For N finite, the vacua can be interpreted as collections of fuzzy spheres. The reason for this is that if we think again of the matrices Xi as being vectors, the length of the vector (Xi)2 ∼ (J i)2. However, from the form of SU(2), (J i)2 is a matrix that commutes with each of the J i, so it must be proportional to the identity. In a sense, then, (Xi)2 is like a collection of pure numbers (one for each block) that gives the physical radii of a set of spheres. However, the Xi cannot be ordinary coordinates, since they are matrices. Whereas ordinary coordinates commute, matrices do not. In quantum mechanics the fact that position and momentum do not commute leads to the Heisenberg uncer- tainty relation, which implies that phase space is fuzzy: we cannot perform 12 Chapter 1. Introduction physical experiments to localize a particle in an arbitrarily small volume in phase space. When two coordinates do not commute, this leads to a similar uncertainty relation, implying that space itself is fuzzy. Therefore the spheres that are described by our vacua are called ‘fuzzy spheres’, since we cannot resolve arbitrarily small angles on them. The resolution on the sphere, however, is inversely proportional to Ni, the dimensions of the rep- resentations of SU(2) [39]. If we take the large N limit that gives M-theory by allowing the Ni → ∞, i.e. by allowing the Young diagram in figure 1.1 to get very tall, then the fuzziness of the spheres goes away, and we can interpret the vacua as spherical M2 branes. The picture for M5 branes is more subtle. Since all of the vacua can be interpreted as fuzzy spheres, it was initially unclear how to represent M5 branes. This interpretation was provided in [39]. The limit that was taken to find M2 branes was a large N limit that arose by taking the dimensions of the representations of SU(2) or the height of the Young diagram to infinity. In the Young diagram notation, however, a large N limit simply means taking the number of boxes to infinity, and we could imagine doing this by taking the width of the Young diagram to infinity instead. How do we interpret this in terms of representations of SU(2)? Since in the Young diagram notation we add a column of fixed length for each representation of SU(2) of a given dimension, to get a Young diagram that is very wide we see that we need to add a large number of representations of a given dimension. This is in contrast to what was done for M2 branes, which was to fix the number of times a representation occurred, but take the sizes to infinity. Support for this proposal of taking a large number of representations was provided in [39]. The analysis of [36] showed that in addition to the vacua, there are other states of the matrix model that have energies that are protected from perturbative corrections by supersymmetry. In particular, the spectrum of excitations about the putative M5-brane vacua are protected, and were shown to correspond precisely to the spectrum of fluctuations of the M5 branes [39]. We will mention one further limit of this theory here, which will lead us naturally into a discussion of the other SU(2|4) symmetric field theories, although we will defer a full discussion until Chapter 2. We have learned so far that the Plane-Wave Matrix Model provides a regularization of the field theory on spherical M2 branes. The limit of the matrix model that gives the M2 branes is a large N limit, but it is also a limit in which the coupling constant is taken to infinity in a controlled way. Since this is a field theory that comes from considering D0 branes in type IIA string theory, it is natural to consider whether there is an appropriate limit in which this theory gives 13 Chapter 1. Introduction a regularization of the field theory on spherical D2 branes. This prescription was given in [39]. For the theory of k D2 branes, one can expand the Plane- Wave Matrix Model about a vacuum with N copies of the k dimensional representation and take N to infinity while keeping gYM2 = gYM0√ N (1.17) fixed. This provides a direct link between the Plane-Wave Matrix Model and the field theory on spherical D2 branes, which is maximally supersymmetric Yang-Mills theory on R× S2. 1.3.2 SYM on R× S2 Another interesting gauge theory with SU(2|4) supersymmetry is maximally supersymmetric Yang-Mils theory on R×S2. The action for this theory can be written as [39] S = 1 g2YM ∫ dt dΩ2 µ2 Tr ( − 1 4 FµνF µν − 1 2 (DµX a)2 − 1 2 (Dµφ) 2 + i 2 ψ†D0ψ − i 2 ǫijkψ†γixjDkψ + 1 2 ψ†γixi[φ,ψ] + 1 2 ψ†γa[Xa, ψ] + 1 4 [Xa,Xb]2 + 1 2 [φ,Xa]2 − µ 2 8 (Xa)2 − µ 2 2 φ2 − 3iµ 8 ψ†γ123ψ + µ 2 φǫijkxiFjk ) , (1.18) where xi is a unit vector on S2, dΩ2 measures the solid angle of an S 2 whose radius is inversely proportional to µ. Also, Xa, a = 1, . . . , 6, and φ are scalar fields, F is the gauge field strength, ψ is fermionic, D is the gauge covariant derivative and γi are Dirac matrices. In addition to the limiting procedure mentioned in the previous subsection, this theory can be found by truncating N = 4 SYM on R× S3 [40]. Fields on S3 fall in representations of SO(4), which is isomorphic to SU(2) × SU(2). If we take fields that are invariant under a U(1) subgroup of one of the SU(2) factors, the resulting theory is maximally supersymmetric Yang-Mills on R × S2. More recently, this theory has been derived by directly dimensionally reducing N = 1 SYM in 10 dimensions to R× S2 [41]. This theory, like the Plane-Wave Matrix Model, has SU(2|4) symmetry. The Xa are six scalar fields that transform as a vector under SO(6), but here the SO(3) is the group of rotations of S2. The theory also has an 14 Chapter 1. Introduction additional scalar field, as well as fermions. Also, like the Plane-Wave Matrix Model, it has a discrete set of vacuum solutions [39]. These can be found by considering the potential for the spatial components of the gauge field and the scalar field φ. This potential can be written in the form V ∼ (Fjk − µφxiǫijk)2. (1.19) Solutions, which cause this potential to vanish, will simply be Fjk = µφx iǫijk. (1.20) However, not all F ’s are allowed. Having the spatial components of F 6= 0, introduces a magnetic flux through the S2, that is∫ S2 Tr(F ). (1.21) This magnetic flux must be quantized, meaning Tr(F ) must be an integer, and since F is a gauge field strength, independent configurations will be given by F having different sets of integers along its diagonal. The limiting procedure of the Plane-Wave Matrix Model that yields this theory also allows for an interpretation of the vacua of this theory as limits of vacua of the Plane-Wave Matrix Model. We will see, when we discuss the gravity duals of these field theories, that this connection manifests itself nicely in the dual gravity picture. 1.3.3 SYM on R× S3/Zk The final SU(2|4) symmetric field theory we will discuss at this stage is maximally supersymmetric Yang-Mills theory on R × S3/Zk [40], which is a close relative of the theory on R × S2. In the previous section, we have seen that SYM on R × S2 can be derived by considering N = 4 SYM on R× S3/Zk, and keeping a set of states invariant under some U(1) subgroup of SO(4). Suppose that instead of keeping states that are invariant under U(1), which are rotations of any amount around a circle, we keep states that are invariant under rotation by some fraction of the whole circle. This group of rotations by 2π/k is known as Zk. This is the group of integers modulo k, since k rotations of 2π/k, correspond to a rotation of 2π. We will note briefly here that in the limit k → ∞, arbitrarily small rotations are allowed. This means that SYM on R× S3/Zk when k →∞ is just SYM on R × S2. This relation is also manifest in the dual geometries as we will discuss later. 15 Chapter 1. Introduction The vacua of this theory must be gauge configurations for which the field strength vanishes. Unlike the theory on S3, however, the theory on S3/Zk may have nontrivial vacua. Consider taking coordinates on S3 in which the metric takes the form dΩ23 = 1 4 ( dθ2 + sin2 θdφ2 + (2dψ + cos θdφ)2 ) (1.22) where θ and φ are the usual coordinates on S2, and ψ is an angular co- ordinate with period 2π. The space S3/Zk is obtained by identifying ψ ≡ ψ + 2π/k. If we consider gauge configurations of the form A = −diag(n1, n2, . . . , nN )dψ, (1.23) the holonomy of the gauge field along the path 0 ≤ ψ < 2π W = ei R A (1.24) will be trivial if the numbers ni are suitably chosen. We might similarly consider the holonomy of the gauge field along the 0 ≤ ψ < 2π/k, which, due to the identification, is a topologically closed but non-trivial cycle. Non- trivial vacua of this field theory will be those that have non-trivial holonomy around the cycle 0 ≤ ψ < 2π/k, but that have trivial holonomy around the topologically trivial cycle 0 ≤ ψ < 2π. This means that the ni must be integers so that e2piini/k are kth roots of unity. Since the holonomy is a gauge invariant quantity, these configurations will be gauge inequivalent for each unique set of integers {ni}, and so we will have a discrete set of vacua parametrized by these sets of integers. 1.4 Gauge/Gravity Duality for SU(2|4) Theories The SU(2|4) symmetric field theories, as we have described, are an interest- ing set of supersymmetric non-Abelian gauge theories with discrete vacua. Since gauge/gravity duality provides a way of studying field theories non- perturbatively, to better understand these theories it is useful to search for their gravity duals. One factor that both complicates this search, and also makes it more interesting, is that the field theories have discrete sets of vacua. Before describing the supergravity duals in detail, it is useful to consider what we expect their properties to be. In the case of the Plane-Wave Ma- trix Model, this theory is a massive deformation of the theory describing D0 branes, and therefore we would expect that supergravity duals to this field 16 Chapter 1. Introduction theory would have an asymptotic region of the geometry that looks like the gravity dual for D0 branes. On the other hand, in the case of maximally supersymmetric Yang-Mills theory on R × S2, since this is the field theory describing spherical D2 branes, we would expect the dual geometries to have an asymptotic region that looks like the gravity dual for D2 branes. Also, as described in section 1.3, the representation theory of SU(2|4) leads to a discrete set of protected vacuum states. This supersymmetric protection implies that the vacua are vacua of the interacting theory, and therefore if we are interested in finding a gravity dual, there should be a gravity solution dual to each vacuum. A key ingredient in such a search for gravity duals, as in the case described in section 1.1, is the symmetry of the field theory. Therefore dual gravity solutions should have the same SU(2|4) symmetry as the field theories themselves. From these ingredients, we would expect that there should be discrete families of supergravity solutions identified with vacua of the SU(2|4) symmetric field theories, and the discrete families as- sociated with the Plane-Wave Matrix Model and maximally supersymmetric YM on R×S2 should differ in their asymptotics. This is what Lin and Mal- dacena did [40], and we will now discuss their work. Lin and Maldacena [40] looked for all SU(2|4) symmetric solutions of type IIA supergravity. These solutions are a special case of SU(2|4) solu- tions of 11D SUGRA, which they sought in earlier work with Lunin [42]. The special case arises because type IIA supergravity can be found by compact- ifying 11D SUGRA on a small circle, and so solutions of 11D SUGRA with a translational symmetry are also solutions of type IIA SUGRA. Finding 11D solutions was difficult because the supergravity equations reduced to a non-linear partial differential equation. However, by taking an additional symmetry, it was found that the supergravity equations can be reduced to a simple elliptic linear partial differential equation. Let us outline how this comes about. SU(2|4) is a supergroup, meaning that it consists of taking a group with generators that satisfy a set of commutation relations, and adding in a set of generators that satisfy anti-commutation relations, in such a way that all (anti-) commutation relations among the generators close amongst them- selves. In the case of SU(2|4) this supergroup is an extension of the bosonic group R × SO(3) × SO(6), or more precisely its universal covering group R × SU(2) × SU(4), as we have mentioned in section 1.3. A starting point for looking for SU(2|4) supergravity solutions, is then to search for solutions in which the isometry group of the solutions is R × SO(3) × SO(6). This means that the metric in the space will have one translational symmetry, as well as the symmetries of two- and five-dimensional spheres. Together 17 Chapter 1. Introduction these symmetries mean that, in the case of 11D SUGRA, the metric can only depend on the three remaining coordinates, or in the case of type IIA supergravity, just two coordinates. This is already a drastic simplification of the supergravity equations. The remaining simplification comes from apply- ing the fermionic part of the supergroup. In supergravity, supersymmetric solutions are sought by setting the fermionic fields to vanish. Because su- persymmetry transformations mix the bosonic and fermionic fields, however, to consistently set the fermionic fields to zero we must also make sure that supersymmetry variations of the bosonic fields do not change the fermionic fields. To ensure that this is the case it is necessary to find what is called a “Killing spinor” (see, e.g. [43]). In differential geometry, when a space has a symmetry, there is a “Killing vector” associated with the symmetry. This is loosely analogous to the Noether theorem in classical mechanics re- lating symmetries with conserved quantities. A Killing vector satisfies the relation6 ∇avb +∇bva = 0. (1.25) A Killing spinor in supergravity satisfies an equation that schematically takes the form ∇aη + (ΓF )aη = 0, (1.26) where Γ is an antisymmetrized product of Dirac matrices and F is some an- tisymmetric tensor supergravity field. Killing spinors are particularly useful in finding supersymmetric supergravity solutions because they can be com- bined with various Dirac matrices to form bilinears that can then be used to constrain the form of the metric in the remaining dimensions. Certain of these bilinears may be Killing vectors; hence the terminology Killing spinor. In the present case, the analysis of [42] found the supergravity metric to have the form ds211 =− 4e2λ(1 + y2e−6λ)(dt + Vidxi)2 + e−4λ 1 + y2e−6λ (dy2 + eD(dx21 + dx 2 2)) + 4e2λdΩ25 + y 2e−4λdΩ22, (1.27) where λ and Vi depend on a function D, which also appears explicitly. The solution is completely determined by this function D, which was shown to 6The defining condition for a Killing vector is that the Lie derivative of the metric along the vector field vanishes. That is equivalent to the form presented here if ∇ is the covariant derivative associated with the metric. We present the equation in this form so that the similarity with the Killing spinor is more apparent. 18 Chapter 1. Introduction satisfy the differential equation (∂21 + ∂ 2 2)D + ∂ 2 ye D = 0. (1.28) We will be interested in solutions that can be reduced to type IIA super- gravity, which must have an additional isometry direction. By taking D to be independent of x1, the change of variables [42] y = ρ∂ρV, x2 = ∂ηV, e D = ρ2, (1.29) can be used to transform 1.28 into a simple linear form 1 ρ ∂ρ(ρ∂ρV ) + ∂ 2 ηV = 0. (1.30) With this change of variables, the full supergravity solution now has the form ds210 = ( V̈ − 2V̇ −V ′′ ) 1 2 [ −4V̈ V̈ − 2V̇ dt 2 − 2V ′′ V̇ (dρ2 + dη2) + 4dΩ25 + 2 V ′′V̇ ∆ dΩ22 ] , e4Φ = 4(V̈ − 2V̇ )3 −V ′′V̇ 2∆2 , C1 = − 2V̇ ′ V̇ V̈ − 2V̇ dt, F4 = dC3, C3 = −4 V̇ 2V ′′ ∆ dt ∧ d2Ω, H3 = dB2, B2 = 2 ( V̇ V̇ ′ ∆ + η ) d2Ω, ∆ ≡ (V̈ − 2V̇ )V ′′ − (V̇ ′)2 . (1.31) The SU(2|4) symmetric solutions of type IIA supergravity are then specified by the unknown function V that satisfies the simple equation (1.30). Some remarks about the supergravity solutions are necessary. One nice feature of equation (1.30) is that it is the Laplace equation for a function in three dimensional coordinates that does not depend on the angle of the axis about the cylinder. However, in three dimensional electrostatics the potential satisfies the Laplace equation, so supergravity solutions will be solutions to axially symmetric electrostatics problems. If we are interested in supergravity solutions that are dual to the Plane-Wave Matrix Model 19 Chapter 1. Introduction and maximally supersymmetric YM on R×S2, we have already pointed out that since these theories are related to the theories describing the degrees of freedom of D0 and D2 branes, respectively, we should look for solutions V that give these asymptotic forms. Lin and Maldacena noted [40] that for this to occur the solutions dual to the Plane-Wave Matrix Model must have, for large ρ, η, the form of an electric dipole plus V∞ = ρ2η − 2 3 η3, (1.32) and in the case of SYM on R× S2, an electric monopole plus V∞ = ρ2 − 2η2. (1.33) It should also be noted that not all functions V will give rise to regular supergravity solutions. The solution (1.31) contains S2 and S5 factors that may shrink to zero size, and avoiding conical singularities requires imposing additional conditions. Lin and Maldacena showed [40] that in the case of the S2 this meant that when the size of the S2 shrinks, ∂ρV = 0 with η fixed, and that when the size of the S5 shrinks, V must be regular at ρ = 0. In electrostatics ∂ρV is just the electric field, and a vanishing electric field implies the presence of a conductor. Taking these facts together, regular supergravity solutions then correspond to regular electrostatics solutions with a set of coaxial circular conducting discs. One additional condition must be imposed at this stage. The change of variables included x2 = ∂ηV ; for this to be well defined, we must also require that at the edge of the conducting discs ∂ηV = 0, so that x2 does not change discontinuously. In the electrostatics interpretation, this means that at the edge of the conducting disc there can be no charge. Although the problem of finding the supergravity duals to the SU(2|4) symmetric field theories of interest has been reduced to an electrostatics problem, there is still the task of matching the solutions to the vacua of the field theories. One remaining issue in doing this is that there appears to be a continuum of supergravity solutions, but discrete vacua. Lin and Maldacena noted, however, that the presence of the shrinking S2s and S5s in the geometry gives rise to topological cycles. See figure 1.2. In analogy with the usual Dirac quantization condition, this implies that some of the higher form fluxes in the geometry have to be quantized. The number of units, N5, of H3 flux on a 6-cycle 7 was found [40] to give, in the electrostatics language, the number of units of electrostatic charge on the conducting disc 7In electromagnetism the spatial components of the field strength F give the mag- 20 Chapter 1. Introduction z r S2 S5 Σ6 Σ3 Figure 1.2: Topological cycles in the regular supergravity solutions. A path connecting two S2s defines a 3-cycle Σ3, and a path connecting two S 5s de- fines a 6-cycle Σ6. We have drawn here only the (r, z)-plane and suppressed the other coordinates. it surrounds. On the other hand it was found that the number of units, N2, of ⋆F4 on a 3-cycle gives the distance between the conducting discs in the electrostatics picture that it connects. By quantizing both the charges on the discs and their relative positions we find that on the gravity side there is also a discrete set of solutions. The fluxes H3 and ⋆F4 count the charges of NS5 and D2 branes, respectively, which is reflected in the subscripts N5 and N2. To match the discrete set of supergravity solutions to the field theory vacua, note that solutions that have the same asymptotics as the D0-brane geometry had to have an asymptotic dipole charge, and so have no monopole charge. To implement this we can consider configurations of discs sitting above some infinite conducting plane. By the method of images, each charge above the plane will have a negative opposite charge below the plane, and will look like a dipole asymptotically, and these configurations will then be dual to vacua of the Plane-Wave Matrix Model. On the other hand we may consider configurations that have a monopole moment, which should netic field (e.g. [44]), so that R F over a surface counts magnetic charge. There is an operation denoted ⋆ that maps the field strength to another anti-symmetric tensor whose spatial components are the electric field (e.g. [35]) such that R ⋆F counts electric charge. The charges we are speaking of here are simply higher dimensional analogues of those in electromagnetism. 21 Chapter 1. Introduction correspond to some isolated set of discs. Such configurations will then cor- respond to vacua of SYM on R × S2. Finally, there may be configurations that asymptotically look like a line of charge along the axis. Lin and Mal- dacena [40] showed that configurations with periodic charge distributions correspond to solutions dual to N = 4 SYM on R× S3/Zk. In the previous section we have pointed out that the SU(2|4) symmetric field theories are related to one another. Let us now show how these rela- tions come about in the dual supergravity picture. SYM on R×S2 arises by taking a particular large N limit of the Plane-Wave Matrix Model in which the number of copies of the representations SU(2) in the vacuum is taken to infinity. From the electrostatics picture of the gravity duals, we see that the analogous limit is to start with a set of discs above the infinite conducting plane and then take a limit which the set of discs moves infinitely far away from the plane. Similarly, SYM on R × S2 is related to N = 4 SYM on R × S3/Zk in the limit k → ∞. In the electrostatics picture, the configu- rations dual to N = 4 SYM on R × S3/Zk are periodic, so to get isolated configurations dual to SYM on R× S2 we take the period to infinity. Now that we have described the SU(2|4) symmetric field theories and their supergravity duals, we will introduce two other topics that will be brought into the context of gauge/gravity duality for SU(2|4) symmetric theories in the later chapters. 1.5 Little String Theory Part of the goal of this thesis will be to provide a connection between the field theories we have just described, and one called Little String Theory. We will now turn to a description of some of the salient aspects of this theory. As we described above, the massless degrees of freedom living on branes in string theory are described by gauge theories. For lower dimensional branes the descriptions in terms of gauge theories are sensible, since the gauge theories are renormalizable (see, e.g. [17] for a discussion). This is not true for gauge theories on higher dimensional branes. To see why this is the case, consider a pure gauge theory in d dimensions. The action for such a theory can be written as S ∼ 1 g2 ∫ ddxTr(F 2). (1.34) The Tr(F 2) term in the action has dimensions of length−4 so that for the action to be dimensionless, the coupling constant g must have dimensions 22 Chapter 1. Introduction of length(d−4)/2. If we define the coupling constant at an energy scale, Λ, to be gΛ = gΛ (d−4)/2, (1.35) then as we go to higher energies, the coupling becomes stronger if d > 4. In this case the gauge theory cannot be renormalized by adding a finite number of counter terms, implying that the theory does not make sense by itself and requires something else to be well understood at higher energies. This is the situation that occurs for the gauge theories that describe the degrees of freedom living on higher dimensional branes. An interesting class of these theories are called Little String Theories. A review of these theories is given in [45]. Type IIA Little String Theory is the field theory that describes the degrees of freedom living on NS5 branes in type IIA string theory. NS5 branes are in some respects similar to D-branes. Like D-branes NS5 branes have corresponding supergravity solutions. However, D-branes couple to antisymmetric tensors that come from the Ramond- Ramond sector. The Ramond-Ramond sector is composed of bosonic states for which the left- and right-moving parts of the closed string state are fermionic (e.g. [23]). Conversely, the NS5 brane couples magnetically to the Neveu-Schwarz two-form. The NS 2-form is a rank two antisymmetric tensor field that comes from the sector of string states that are created using only bosonic operators (e.g. [23]). The type IIA and IIB theories have the same bosonic sector so they both have NS5 branes. This is in contrast to the states in the Ramond-Ramond sector, which are created using fermionic operators. The IIA and IIB theory have different fermionic sectors and this results in tensor fields of different rank, and therefore D-branes of different dimensions. The other difference is that NS5 branes and D-branes couple differently to the dilaton. 8 The degrees of freedom that live on NS5 branes should be described by a gauge theory in 5+ 1 dimensions. By the above discussion, however, such a gauge theory is not well defined. But Little String Theory turns out to be even more peculiar. Type IIA or IIB Little String Theories are defined by considering the gs → 0 limit of a stack of NS5 branes. One interesting fact about Little String Theories is that even in the limit that the string coupling goes to zero, they have non-trivial dynamics. This is unusual, because usually the field theory describing the degrees of freedom living on a stack of branes 8The dilaton, denoted by Φ in (1.31), is a field that controls the strength of the coupling between strings (e.g. [22]). D-branes include a coupling to the dilaton e−Φ, whereas NS5 branes include a coupling e−2Φ. 23 Chapter 1. Introduction becomes a free field theory in the limit that the string coupling is taken to zero with fixed α′. To see why this does not happen for Little String Theory, consider the case of type IIB Little String Theory. The following argument is due to Seiberg [46]. Consider NS5 branes in type IIB string theory. By S-duality, these are related to D5 branes. The coupling constant for the theory on the D5 branes goes like g2D5 ∼ gsα′. (1.36) S-duality takes the combination gsα ′ → α′ so that the coupling constant for the theory on the NS5 branes goes like g2NS5 ∼ α′. (1.37) Now, whereas keeping α′ fixed and taking gs → 0 in the D5-brane theory implies that the theory becomes weakly coupled, we see that in the case of the NS5-brane theory that is not true. Therefore, even when the string coupling vanishes, the theory of the NS5-brane degrees of freedom remains non-trivial. Another interesting fact comes from T-duality. The action of T-duality maps NS5 branes in type IIB string theory to NS5 branes in type IIA string theory, and vice-versa. The action of the T-duality sends a circle with radius R to a circle with radius R′ ∼ α ′ R . (1.38) Note that this does not involve gs, so that T-duality should persist even in the limit that gs → 0 that defines Little String Theory. Field theories are usually thought of as theories of particles, which are not capable of winding around compact directions. Yet Little String Theory, the putative field theory describing the degrees of freedom living on NS5-branes, exhibits T-duality, a symmetry intrinsically tied to winding states. This is a peculiar feature of Little String Theory that means it is a rather interesting theory. 1.6 Gravitation and Entropy One of the successes of string theory is the ability to account for the mi- crostates of black holes. General considerations of black holes led to a com- parison of the evolution of classical black holes to the laws of thermody- namics; for a summary see e.g. [47]. Part of this comparison required that the area of the horizon of a black hole be proportional to the thermody- namic entropy. The discovery that black holes radiate like thermal bodies 24 Chapter 1. Introduction suggested that this analogy could be taken more seriously, and that black holes do have an entropy. This position seems to lead to some difficulties, however. Investigations into black holes in classical general relativity have led to the intuition that there is a unique black hole solution for a given set of asymptotically measurable charges (see, e.g. [47]). Statistical mechanics, however, dictates that entropy arises from counting the microstates of a particular macrostate. Black hole uniqueness would then seem to imply that such microstates should not exist. Understanding the microscopic origin of black hole entropy is therefore an important question. In string theory this origin was first understood in [48] for D-branes wrapping cycles in the space K3, and later for the D1-D5-P system [49]. We will focus our discussion on subsequent work that is similar in spirit to the original analyses, and which will segue into work done in this thesis. A similar study can be made of a gravitational singularity arising from a string wrapping a compact dimension. Lunin and Mathur [50] (see also the review [51]) considered IIB string theory on T 4 × S1. They considered a string wrapping a large number of times around the S1 and carrying a large number of units of momentum around the compact circle. There is a singular solution of type IIB supergravity that carries the appropriate charges that arise from such a wrapped string. To carry momentum in the compact direction, there must be travelling waves on the string, and since strings do not carry longitudinal waves, the waves must be transverse. The microstates that give this singular solution an entropy are taken to be different ways of distributing the momentum in the compact direction among the modes of oscillation. However, whereas the supergravity solution indicates that the singularity is a point, the string that gives rise to the singularity must be spread out by the transverse oscillations. The singular supergravity solution does not, then, correspond to any string state, which appears to give a contradiction. The singularity produced in this case was a naked singularity. Sen had argued previously [52] that such singularities should be given ‘stretched hori- zons’ at the location where the curvature becomes string scale. Supergravity should only be valid when the radius of curvature is large compared to the string scale, at which stringy effects become important. It is expected that these stringy effects would correct the geometry in regions with high curva- ture, and provide a horizon that cloaks the naked singularity–Sen’s stretched horizon. Lunin and Mathur [50] considered all of the different ways that the mo- mentum could be distributed in transverse waves. They observed that for a 25 Chapter 1. Introduction string carrying a typical distribution of transverse waves, the profile of the string would fill out to the distance of the stretched horizon. Outside the stretched horizon the geometry would look like the singular solution, how- ever inside the horizon the geometry would be quite different. This presents an interesting picture of the geometry inside the horizon. This geometry is frequently given the term ‘fuzzball’ in the literature because the string fills up the region inside the horizon [51]. In chapter 5 we will study an analogous phenomenon in the context of SU(2|4) symmetric supergravity solutions and the dual field theories. In this case the singularity is replaced by a region of space with complicated topology. 1.7 Plan In the remainder of this thesis we will apply and tie together the topics we have described above. The focus of chapters 2, 3 and 4 will be on taking information from gravity to learn about the dual SU(2|4) symmetric field theories. In the next two chapters we will find a Lagrangian definitions of Little String Theory as limits of the SU(2|4) supersymmetric field theories. The aim of these chapters is to make progress towards a better understand- ing of what this theory is, and to clarify physics that arises in the various limits that can be taken in the SU(2|4) supersymmetric field theories. In chapter 4 we will find general SU(2|4) supersymmetric solutions of type IIA supergravity by solving the problem formulated by Lin and Maldacena. The purpose of finding these solutions is to provide the information that will allow us to use duality to study the SU(2|4) gauge theories at strong coupling in detail. As an example of this, we use some of the solutions to calculate new instanton9 amplitudes in the Plane-Wave Matrix Model. In chapter 5 we will use information from the gauge theory to learn about the dual geometry in the gravitational theory. We use statistical mechanics to count the microstates of singular geometries that arise from taking a coarse- grained view of the regular supergravity solutions, and define an entropy function for coarse-grained geometries. Chapter 6 provides a summary of the main results and discusses future directions. Various technical details are collected in the appendices. 9See appendix A for a discussion of instantons. 26 Chapter 2 Little String Theory from the Double-Scaled Plane-Wave Matrix Model 2.1 Introduction As we have described in the introduction, type IIA Little String Theory [46, 53] describes the degrees of freedom of NS5-branes in type IIA string theory in the decoupling limit in which gs → 0 with α′ fixed (i.e. focusing on energies of order (α′)−1/2). It is believed to be a six-dimensional interacting non- gravitational theory with a Hagedorn density of states.10 In the infrared, the theory flows to the interacting (0,2) conformal field theory, but in general the theory does not have the properties of a local quantum field theory, as it exhibits T-duality. For a review of Little String Theory, see [45]. Since there is no direct Lagrangian description of the theory (though a DLCQ 11 formulation [54] and a description via deconstruction [55] have been proposed), the main tool for analyzing Little String Theory has been its gravity dual, the near-horizon NS5-brane solution of type IIA string theory. This is given for large r, where the IIA picture is valid, by ds2 = N5α ′(−dt2 + d ~x52 + dr2 + dΩ23) eφ = gse −r with N5 units of H flux through the S 3. However, even this description is problematic since the dilaton, which controls the string coupling, varies with the coordinate r, which sends the theory to strong coupling in the infrared part of the geometry. This, in turn, means that the description in that region must be in terms of M-theory. 10A Hagedorn density of states means that the density of states grows exponentially with energy. 11DLCQ, or discrete light-cone quantization involves formulating quantizing a theory on a space with a periodic light-like direction. The Hamiltonian operator evolves the theory in this periodic light-like direction. 27 Chapter 2. LST from the PWMM Lin and Maldacena [40], however, have found a related supergravity so- lution in which the flat five-dimensional part of the geometry corresponding to the spatial NS5-brane worldvolume directions has been replaced by an S5. We reproduce the (somewhat complicated) full supergravity solution in appendix B, but for large radius, the metric and dilaton become simply ds2 = N5α ′ [2r (−dt2 + dΩ25)+ dr2 + dΩ23] eΦ = gse −r , again with N5 units of H flux through the S 3. This new solution retains the linear dilaton behaviour and constant volume S3 permeated by H-flux, so it is natural to associate this solution with NS5-branes on S5. An advantage of this solution over its flat cousin is that, in this case, the full solution has a tunable maximum value for the dilaton at r = 0 and a tunable maximum curvature, so there is a regime in which a type IIA supergravity description is everywhere valid.12 13 The main goal of this chapter will be to explicitly describe a field theory dual for string theory on this solution, and thus a Lagrangian field theory definition of Little String Theory on S5. The broader context for our story is a D0-brane quantum mechanics ana- logue of the model of Polchinski and Strassler [58] for D3-branes. The field theory we consider is the Plane-Wave Matrix Model [30], a mass-deformation of the maximally supersymmetric D0-brane quantum mechanics. In our case, the mass deformation is maximally supersymmetric, preserving 32 su- percharges including an SU(2|4) symmetry [36, 59, 60]. The theory has a discrete spectrum, a dimensionless parameter that acts as a tunable cou- pling constant [32], and a large number of degenerate supersymmetric vacua preserving the SU(2|4) supersymmetry. Before the mass-deformation, the D0-brane quantum mechanics is dual to string theory on the near-horizon D0-brane solution of supergravity [17]. As with the near-horizon NS5-brane solution discussed above, this becomes strongly coupled in the infrared region of the geometry, and we must go to an eleven-dimensional description. However, the mass-deformation provides an infrared cutoff for the theory, so we might expect that solutions dual to the various vacua of the Plane-Wave Matrix Model (for large enough mass) would have a IIA description that is valid everywhere. 12Unfortunately, the solution contains Ramond-Ramond fields, so string theory is diffi- cult. 13A similar situation occurs in [56, 57], though in the present case, more of the R- symmetry is preserved. 28 Chapter 2. LST from the PWMM This picture was verified by Lin and Maldacena [40]. As we have de- scribed above, following [42], they searched for type IIA supergravity solu- tions with SU(4|2) symmetry, and showed that solutions of type IIA super- gravity corresponding to all vacua of the Plane-Wave Matrix Model could be constructed in terms of the solutions to a class of axially symmetric electrostatics problems involving charged conducting discs in 3 dimensions. While they did not solve the electrostatics problem, except in certain limit- ing cases, they showed that solutions of this type generally contain throats with non-contractible 3-spheres carrying H-flux. These regions of the geom- etry may be associated with fivebrane degrees of freedom described by the matrix model. In this chapter, we focus on the simplest class of vacua of the Plane-Wave Matrix Model, for which the dual geometry has only a single NS5-brane throat. We solve the appropriate electrostatics problem to find an exact supergravity solution, and determine the precise limit of this solution (which depends on three parameters) needed to decouple the throat region, yielding the explicit infinite-throat solution of Lin and Maldacena, corresponding to Little String Theory on S5. By understanding how the parameters of the supergravity solution match up with the parameters of the gauge theory, we then see what this limit corresponds to in the matrix model. We find that the corresponding limit is a limit of large N with the ’t Hooft coupling also taken to infinity in a particular way, roughly λ ∼ log4(N), while focusing on the excitations around a specific vacuum of the matrix model (the one corresponding to our original supergravity solution). Since we are taking a strict large-N limit in the field theory, we might naively expect that the corresponding gravity dual should be a free string theory. This is true asymptotically, due to the linear dilaton background, but not for finite values of r, so there should still be a genus expansion14 on the string theory side. We conjecture that this is reproduced in the gauge theory in a way very similar to the double-scaling limits used to describe low-dimensional string theories in terms of matrix models.15 In our case, the ’t Hooft coupling is scaled towards a critical coupling λc =∞ in such a way that the various terms in the matrix model genus expansion all contribute, despite N being infinite. Assuming this picture is correct, we are able to make predictions for the large λ behaviour of the full set of genus n diagrams in perturbation theory (section 2.6.1). 14Genus refers roughly to the number of handles in the string worldsheet. Interacting strings are described by worldsheets with progressively larger genus, or more handles, as the interaction becomes more complicated. 15This conclusion was predicted by Herman Verlinde. 29 Chapter 2. LST from the PWMM This chapter is organized as follows. In section 2.2, we review vari- ous aspects of the Plane-Wave Matrix Model, including decoupling limits that have been discussed in the past. In section 2.3, we review the Lin- Maldacena ansatz for gravity duals to the matrix model vacua and the elec- trostatics problems that need to be solved in order to find the solutions. We then provide an exact solution for the simplest such problem, which re- quires determining the potential due to parallel charged conducting discs in a specified background potential. In section 2.4, we discuss the matching of parameters between the gravity solutions and the matrix model. In section 2.5, we determine how to scale the parameters in our solution to obtain the Lin-Maldacena solution for Little String Theory on S5, and then use the correspondence with the gauge theory to determine the matrix model description of Little String Theory on S5. We also discuss the limit that gives a solution dual to the maximally supersymmetric theory of D2-branes on S2, and the gravity interpretation of the ’t Hooft limit of the matrix model. In section 2.6, we discuss the results. In 2.6.1, we explore the con- sequences of our result that the Little String Theory on S5 is obtained as a double scaling limit of the matrix model. In section 2.6.2, we describe an infinite-parameter family of supergravity solutions similar to (and with the same symmetries as) the Lin-Maldacena solution, and speculate on the matrix model description of these. Finally in section 2.6.3, we describe an application of our results to calculating energies of near-BPS states16 in the geometry. We collect some of the more technical details in the appendices. 2.2 Gauge Theory Let us recall briefly some properties of the Plane-Wave Matrix Model that we have discussed in chapter 1. We have given the action for this theory in equation (1.9). There are two sets of scalar fields in this theory, those we denote byXa, which transform as a vector under SO(6), and those we denote by Xi, which transform as a vector under SO(3). As we have described, the set of classical vacua for the model are given by Xa = 0, Xi = 13gJ i, where J i generate any N dimensional representation of the SU(2) algebra. These vacua are in one-to-one correspondence with partitions of N , since we may have in general nk copies of the k-dimensional irreducible representation such that ∑ k knk = N . This model has several interesting large N limits (distinguished by which 16BPS states are states whose energies do not receive quantum corrections as a result of supersymmetry. 30 Chapter 2. LST from the PWMM combination of g and N we hold fixed); we will describe these now, as promised in chapter 1. The M-theory limit According to the Matrix Theory conjecture [31], in the limit N →∞, g2/N3 fixed , this model should describe M-theory on the maximally supersymmetric plane-wave background of eleven-dimensional supergravity,17 with µp+l2p = N g 2 3 , p−/µ = H . Note that the quantities on the left are boost-invariant and dimensionless. States of M-theory on the plane-wave with zero light-cone energy are BPS configurations involving concentric spherical membranes and/or concentric spherical fivebranes. These correspond to vacua of the Plane-Wave Matrix Model [30, 39]. We will be particularly interested in vacua involving only one type of irreducible representation, say N2 copies of the N5-dimensional irreducible representation, Xi = 1 3g   J iN5 {N2 copies} . . . J iN5   . (2.1) In the M-theory large N limit with N2 fixed, this configuration describes the state with N2 coincident M2-branes. This is plausible, since classically, such a configuration corresponds to N2 coincident fuzzy spheres. On the other hand, if we keep N5 fixed in the limit, while taking N2 to infinity, the configuration gives us N5 coincident spherical M5-branes. 18 In [39] it was pointed out that there are other interesting large N limits that do not describe all of the degrees of freedom of M-theory, but rather focus in on degrees of freedom associated with the spherical branes. 17This background has ds2 = ds2flat + ( µ2 9 xixi + µ 2 36 xaxa)dx+dx+ and F123+ = µ. 18Evidence for this comes from the fact that the BPS excitations about this vacuum match the expected BPS excitations of coincident spherical M5-branes [39]. 31 Chapter 2. LST from the PWMM The D2-brane limit To understand the next limit, consider the excitations around the vacuum (2.1) at finite N . Classically, this configuration corresponds to N2 coinci- dent (two-dimensional) fuzzy spheres, on which the modes have maximum angular momentum N5. Fluctuations about this configuration are then de- scribed by noncommutative gauge theory on a fuzzy sphere with gauge group U(N2). If we compare this action arising from the matrix model with an action written as a noncommutative field theory with noncommutativity parameter θ and coupling g2 on a sphere of radius r, we find that the field theory parameters are related to the matrix model parameters via θ r2 = 1 N5 , g22r = g2 N5 , Er = H . Thus, by taking N →∞, g2/N fixed, N2 fixed , (2.2) we obtain a commutative field theory on a sphere. This field theory, written down in [39, 40] is essentially the low-energy theory of D2-branes, with mass terms for the scalars and fermions and a coupling of the radial scalar to the magnetic field, such that the whole theory preserves SU(2|4) supersymme- try. Note that we end up with a D2-brane theory instead of an M2-brane theory, since the limit (2.2) does not decompactify the M-theory circle. We can also try to find a similar limit to describe decoupled fivebrane degrees of freedom: The ’t Hooft limit For small values of g (with fixed N), we can study excitations about the various vacuum states perturbatively. In [32], it was found that the param- eter controlling perturbation theory is different for different vacua. For the vacuum with N2 copies of the N5-dimensional irreducible representation, perturbation theory is controlled by the combination g2N2, so for example, the vacuum with N2 = 1, N5 = N is more weakly coupled than the vacuum with N5 = 1, N2 = N . In particular, in the limit N →∞ g2N fixed , 32 Chapter 2. LST from the PWMM the coupling associated with the fivebrane vacua (with fixed N5 with N2 → ∞) remains finite, while the coupling associated with the membrane vacua (fixed N2 with N5 →∞), or any other generic vacuum, goes to zero. Again, this limit does not decompactify the M-theory circle, so it was suggested in [39] (also based on supergravity arguments) that this limit describes NS5-branes on a sphere. Below, we will see that it is a somewhat modified limit that is dual to the Lin-Maldacena gravity solution for NS5- branes on S5. 2.3 Gravity In the previous section, we have described the Plane-Wave Matrix Model, and various interesting large-N limits. At finite N , we can think of the matrix model as a massive deformation of the maximally supersymmetric quantum mechanics describing low-energy D0-branes in flat-space, similar in spirit to the deformation of N = 4 SYM considered by Polchinski and Strassler [58]. The gravity dual of the undeformed theory is string theory on the near-horizon D0-brane geometry, so we expect that the gravity dual for the Plane-Wave Matrix Model should be some infrared modification of this. As we have described in chapter 1, Lin and Maldacena [40] (follow- ing [42]) searched for type IIA supergravity solutions preserving the same SU(2|4) symmetry as the vacua of the Plane-Wave Matrix Model. Using an ansatz with this symmetry (reproduced in appendix B), they were able to reduce the problem of finding supergravity solutions to the aforementioned problem of finding axially-symmetric solutions to the three-dimensional La- place equation, with boundary conditions involving parallel charged con- ducting discs and a specified background potential. We will now describe the electrostatics problem in detail. The Electrostatics Problem Common to all vacua, we have in the electrostatics problem an infinite con- ducting plate at z = 0 (on which we may assume that the potential vanishes), and a background potential19 V∞ = V0(r2z − 2 3 z3) . (2.3) 19This potential results by taking point charges ∓Z4/3 at r = 0, z = ±Z in a constant electric field ~E = − 2 3 Z2ẑ in the limit Z →∞. 33 Chapter 2. LST from the PWMM In addition, corresponding to a matrix model vacuum with Qi copies of the di-dimensional irreducible representation, we have conducting discs with charge Qi parallel to the infinite plate and centred at r = 0, z = di. In order that the supergravity solution is non-singular, the radii Ri of the discs must be chosen so that the charge density at the edge vanishes.20 Thus, for each vacuum of the Plane-Wave Matrix Model, we have an electrostatics problem, whose solution (a potential V (r, z)) feeds into the equations (B.1) to give a supergravity solution. Properties of the Supergravity Solutions We briefly review some properties of the supergravity solutions [40]. It is straightforward to show that, as expected, all of these supergravity solutions approach asymptotically the near-horizon D0-brane solution. In the infrared region, the solutions have interesting topology, as we now recall. The coordinates r and z in the electrostatics problem form two of the nine spatial coordinates in the geometry. In addition, for each value of r and z, we have an S2 and an S5 with varying radii. The S5 shrinks to zero size on the r = 0 axis, while the S2 shrinks to zero size at the locations of the conducting plates, so we have various non-contractible S3s and S6s corresponding to paths that terminate on different plates or on different segments of the vertical axis, respectively. This is illustrated in figure 2.1. As shown in [40], through an S6, corresponding to a path surrounding plates with a total charge of Q, we have N2 = 8Q/π 2 units of flux from the dual of the Ramond-Ramond four-form, suggesting the presence of N2 D2-branes. Similarly, through an S3, corresponding to a path between plates separated by a distance d, we have N5 = 2d/π units of H-flux, suggesting that this part of the geometry between the plates is describing the degrees of freedom of N5 NS5-branes. If we take large plates at a fixed separation, the region between the plates corresponds to a long throat in the geometry with NS5-brane flux. Below, we will understand how to take a limit where such a throat becomes infinite so that we recover the Lin-Maldacena geometry. 20To see that this should be possible, note that without a background field, the charge density on a conducting disc diverges as an inverse square root near the edge. On the other hand, we have an inward electric field coming from the background potential that increases linearly with the radius of the disc. Thus, for a large enough disc, the tendency for the charge on the disc to bunch up at the edge should be balanced by the action of the inward electric field so that the charge density vanishes. 34 Chapter 2. LST from the PWMM 3Σ 6Σ r z 3 g 1 X   = Gravity Electrostatics Gauge Theory 3Σ 6Σ 1 k . . . dJ k . . . dJ 1 . . . dJ 1 J Q kQ kd 1d kR 1R kQ 1Q d Figure 2.1: Mapping between matrix model vacua, electrostatics configu- rations, and geometries. For illustrative purposes, we have replaced the S2 × S5s associated to each point (r, z) with S0 × S0. In the full geom- etry, the dotted segment maps to a submanifold Σ6 that is topologically S6 × S2 (simply connected) rather than the S1 × S0 shown here. Similarly, the dashed segment maps to a submanifold Σ3 that is topologically S 5×S3 rather than the S0 × S1 here. 35 Chapter 2. LST from the PWMM 2d −Q Q R E z r Figure 2.2: Electrostatics problem (cross section) corresponding to gravity solutions with single NS5 throat. The method of images has been used to replace the infinite conducting plate with an image disc. Solution to the Electrostatics Problem (Simplest Case) We will now solve the electrostatics problem above in the simplest case of a single disc above the infinite plate, with the space in between corresponding to a single NS5-brane throat. Thus, we would like to find the potential for a conducting disc of charge Q = π2N2/8 a distance d = πN5/2 above the infinite conducting plate, in the presence of the background field (2.3), as shown in figure 2.2. This will give the geometry dual to the matrix model vacuum with N2 copies of the N5 dimensional irreducible representation. We can simplify the problem somewhat using the the fact that electro- statics is linear and scale-invariant. Thus, if V = (r2z − 2 3 z3) + φκ(r, z) 36 Chapter 2. LST from the PWMM is the solution to the electrostatics problem above with V0 = 1, R = 1, and d = κ (with Q = q(κ) determined by the condition of vanishing charge density at the edge of the disc), then the solution to the general problem will be V = V0(r 2z − 2 3 z3) +R3V0 φ d R (r/R, z/R) , (2.4) and the charge on the disc will be Q = q(d/R)V0R 4 . (2.5) Note that φ is the part of the potential that vanishes at infinity, arising from the charges on the discs. We would now like to determine φκ(r, z). We may assume that the potential vanishes on the infinite conducting plate. Let us call the (constant) potential on the disc V = ∆. This will be determined in terms of κ by the condition that the charge density vanishes at the edge of the discs, but for now, we will take it to be arbitrary and solve for the potential in general. By the method of images, the potential will be the same as for a pair of conducting discs at z = ±κ with potentials V = ±∆, with the potential going like V → zr2− 23z3 at infinity. Without the background potential, this is a classic problem in electrostatics, considered by Maxwell, Claussius and Helmholtz, Kirchhoff, Polya and Szego, and eventually solved by Nicholson [61] and Love [62]. For these references and a nice summary, see [63]. For- tunately, the method of solution may easily be extended to our case with the background potential, as we describe in appendix C.1. To give the solution, it is convenient to define β = ∆+ 2 3 κ3 . (2.6) Then the potential is given by φκ(r, z) = β π ∫ 1 −1 Gκ(r, z, t)fκ(t)dt , (2.7) where Gκ(r, z, t) = − 1√ r2 + (z + κ+ it)2 + 1√ r2 + (z − κ+ it)2 and fκ(t) = f (0) κ (t)− 2 κ β f (2)κ (t) . (2.8) 37 Chapter 2. LST from the PWMM The functions f (n) κ (t) are special functions solving the integral equation f (n)κ (t)− ∫ 1 −1 Kκ(t, x)f (n) κ (x) = t n, (2.9) with kernel Kκ(t, x) = 1 π 2κ 4κ2 + (x− t)2 . This is a Fredholm integral equation of the second kind (see, e.g. [64]), and the solution may be written as a series f (n)κ (t) = ∞∑ m=0 Kmκ ◦ tn (2.10) where (K ◦ g)(t) ≡ ∫ 1 −1 K(x, t)g(x) . It may be shown that the series converges for any value of κ > 0 to define a bounded continuous function f (n) κ (t). The function f is related to the charge density on the disc as fκ(t) = 2π β ∫ 1 t rσκ(r)dr (r2 − t2) 12 , σκ(r) = β π2 [ fκ(1) (1− r2) 12 − ∫ 1 r f ′κ(t)dt (t2 − r2) 12 ] , (2.11) so that the total charge on the disc is q(κ) = β π ∫ 1 −1 fκ(t)dt . (2.12) Vanishing Charge Density Constraint The solution in the previous section was for arbitrary potential ∆, and will generally have a charge density that is nonvanishing at the tip of the disc. We will now determine a formula for ∆(κ) for which the charge density vanishes. From (2.11), it follows that σ(1) = 0 if and only if f(1) = 0 and f ′(1) is bounded. From the series solution, it is straightforward to prove that the latter condition is always satisfied for the functions fn(t), so our constraint 38 Chapter 2. LST from the PWMM comes from requiring f(1) = 0. Now, from (2.8) and the definition (2.6) of β, we see that the condition f(1) = 0 determines ∆ to be ∆(κ) = 2κ f (2) κ (1) f (0) κ (1) − 2 3 κ3 . (2.13) Finally, the charge on the disc is given in terms of κ by q(κ) = f (2) κ (1) f (0) κ (1) 2κ π ∫ 1 −1 f (0)κ (t)dt − 2κ π ∫ 1 −1 f (2)κ (t)dt . (2.14) Via equation (2.5) this function q determines the radius of the disc in the original problem in terms of the charge Q, the potential V0, and the separation d. The function q is plotted in figure 2.3. We show in appendix C.1.1 that its limiting behaviour for small and large κ is q(κ) → 18 , κ→ 0 , q(κ) → 83pi , κ κ→∞. (2.15) We will see in section 2.6.3 that the function q(κ) is physically important since it computes the energies of certain near-BPS states in the theory. Summary In summary, to generate the supergravity solution dual to the vacuum of the Plane-Wave Matrix Model corresponding to N2 copies of the N5 dimensional irreducible representation, we: • choose κ (ultimately related to a choice of coupling g), determine φκ(r, z) from (2.7) and q(κ) from equation (2.14), • take R = (πN5)/(2κ) so that d = Rκ = πN5/2, • choose V0 = (2κ4N2)/(q(κ)π2N45 ) so that (using (2.5))Q = q(κ)V0R4 = π2N2/8 . Then the gravity dual is given by equations (B.1), with V given by (2.4). 2.4 Matching Parameters with Gauge Theory In discussing the various scaling limits of the theory, we will need to un- derstand how the gauge theory parameters match with the parameters in 39 Chapter 2. LST from the PWMM 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.5 1 1.5 2 q κ Figure 2.3: Plot of q(κ). Dashed lines are the asymptotes 83piκ for large κ and 1/8 + 0.711κ for small κ. the supergravity solution, or equivalently, with the parameters d, R, Q, and V0 in the electrostatics problem. As we mentioned, the parameters d and Q are proportional to the number of units of NS5-brane and D2-brane flux through the noncontractible S3 and S6, respectively, in the geometry. As shown in [40], this allows us to associate 21 d = π 2 N5 and Q = π2 8 N2 . The remaining electrostatics parameter is V0 (or equivalently, R), which we interpret in the gauge theory as follows. The asymptotic form of the geometry is determined by the background field in combination with the leading dipole fields arising from the charges on the plates. These depend, respectively, only on V0 and the combination dQ, which is proportional to the dipole moment. These asymptotics should be the same for all vacua 21Note that we have taken all quantities in the electrostatics problem to be dimension- less. 40 Chapter 2. LST from the PWMM of a given theory, so the parameters V0 and dQ must depend only on g and N = N2N5, the parameters that determine which matrix model we are talking about. This is clearly true for dQ, but we must also have V0 = f1(g 2, N2N5) . (2.16) To further constrain V0 it is useful to note that in the gauge theory, the planar amplitudes depend only on N5 and the combination g 2N2. This suggests that the free string theory on the dual spacetime should be con- trolled by these two parameters, and in particular, that these parameters should control the metric. In terms of the electrostatics parameters, it is straightforward to see that the metric depends only on d and R (scaling V0 while holding these fixed scales the dilaton and Ramond-Ramond fields, but leaves the metric fixed), so d and R should each be some function of N5 and g2N2. This is clearly true for d, but we must also have R = f2(g 2N2, N5) . (2.17) Using the relation Q = V0R 4q(d/R), together with (2.16) and (2.17), we may conclude that V0 = 1 g2 h(g2N2N5) (2.18) for some function h. In section 2.5.1, we will see that the D2-brane limit of the gauge the- ory discussed in section 2.2 matches with the corresponding limit of the supergravity solution only if the function h approaches some constant h∞ at large values of its argument. Since we will mostly be interested in this regime (g2N2N5 is always large when supergravity is valid), we get the iden- tification V0 = h∞ g2 . (2.19) The electrostatics parameter R is a more complicated function of the gauge theory parameters, but follows from the other identifications via (2.5). 2.5 Scaling Limits In this section, we consider various scaling limits of the gravity theory in which one of the three parameters is scaled to infinity, with the others scaled such that we end up with something nontrivial. 41 Chapter 2. LST from the PWMM 2.5.1 Large d: the D2-brane Limit We begin by considering a limit of large d with fixed R. To understand how we should scale V0 to leave us with a nontrivial supergravity solution, note that for large d/R, the formula (2.5) determining the charge on the discs becomes Q = 8 3π V0dR 3 , (2.20) where we have used the large κ behaviour of q(κ). Also, the potential (2.3), taken near the position of the disc by replacing z = d+ η, becomes V = −2 3 V0d 3 − 2V0d2η + V0d(r2 − 2η2) + V0(ηr2 − 2 3 η3) . (2.21) The first two terms here have no effect on the supergravity solution, since the solution (B.1) depends only on ∂2zV and ∂rV . Thus, from (2.21) and (2.20), we see that in order to leave a finite nontrivial background potential and a finite non-zero charge on the disc, we must take V0 to scale like 1/d. Thus, our limit is d→∞, Q fixed, V0d =W0 fixed . (2.22) In this limit, we have a single charged conducting disc (with no infinite plate) in a background potential V =W0(r 2 − 2η2) . (2.23) In [40] Maldacena and Lin wrote down explicitly the geometry corre- sponding to this situation. In appendix C.2, we give an alternate derivation of the solution by explicitly solving the electrostatics problem, verifying that (2.20) correctly gives the charge necessary to ensure that the charge density vanishes at the edge of the plates. For large r, the solution approaches the solution for near-horizon D2-branes but with the flat directions along the D2-branes replaced by an S2 [40]. Thus, we expect that this limit should correspond to the limit of the matrix model giving rise to D2 branes on S2. Using the correspondence between matrix model parameters and electro- statics parameters (in particular, assuming that the function h in (2.18) is simply a constant at large argument), we find that the limit (2.22) becomes, N5 →∞, N2 fixed, g 2 N5 fixed, which is precisely the D2-brane limit discussed in section 2.2. 42 Chapter 2. LST from the PWMM 2.5.2 Large V0: the ’t Hooft Limit The next limit we consider is the ’t Hooft limit discussed in section 2.2, N2 →∞, N5 fixed, g2N2 fixed, which appeared to be a decoupling limit retaining interacting fivebrane de- grees of freedom. Using the correspondence between matrix model parame- ters and electrostatics parameters, we find that this is a limit with Q→∞, d fixed, R fixed V0 →∞ From the supergravity point of view, this is a limit in which the metric is held fixed with the maximum value of the dilaton going to zero. Thus, we have free string theory on the background corresponding to a single finite-sized disc above the infinite conducting plate. This geometry contains a finite throat region with NS5-brane flux, but also a noncontractible S3 with D2-brane flux. Thus, while we are describ- ing fivebrane degrees of freedom, this limit of the gauge theory does not correspond to the infinite-throat Lin-Maldacena solution for NS5-branes on S5. 2.5.3 Large R: Little String Theory on S5 Finally, we would like to understand precisely what limit of our solution is required to obtain the Lin-Maldacena infinite-throat solution for NS5- branes on S5. This corresponds to an electrostatics problem with two infinite conducting plates, with the potential between the plates equal to V = Ṽ0 sin (πz d ) I0 (πr d ) . (2.24) To obtain this from our solution, we certainly need to take a limit where R is going to infinity with d fixed. However, generically, we would simply end up with a constant vertical electric field between the plates. This does not give rise to any metric (the supergravity fields depend only on ∂2zV and ∂rV ) so we must take V0 large enough so that the leading corrections to this constant electric field remain nonzero in the limit. To understand the proper scaling, we start by considering the electro- statics solution for finite R. In the r < R region between the plates, we have an axially-symmetric solution to the Laplace equation that is regular at r = 0, so we can write V (r, z) = Vz=d z d + ∞∑ n=1 cn sin (nπz d ) I0 (nπr d ) . 43 Chapter 2. LST from the PWMM Here, we have separated off a constant electric field term that does not affect the metric such that the remaining piece vanishes at z = 0 and z = d for r < R. Now the cns are determined by the potential at r = R, cn = ( I0 (nπR d ))−1 V0R 3 2 d ∫ d 0 dz sin (nπz d ) ( 1 R3 (R2z − 2 3 z3) + φ d R (1, z R )−∆ d R z d ) (2.25) where we have used the expression (2.4) for V . Using our solution for φ, it is simple to show that the integral here has only a power law dependence on R, so the large R behaviour of cn is dominated by the exponential damping coming from the Bessel function at large argument, (I0(z)) −1 ∼ √ 2πze−z . To compensate for this damping, we must scale V0 exponentially in R, V0 ∼ e piR d , (2.26) which allows us to keep c1 finite in the limit. All the other coefficients cn>1 still vanish in the limit, so we indeed end up with the Lin-Maldacena solution. To be more precise, we can evaluate the integral in (2.25) to find the prefactor in (2.26). Using the results of appendix C.1.1, we find the behaviour c1 → V0 C (Rd) 3 2 e −piR d , (2.27) where we have numerically estimated C to be C ≈ 0.082. Thus, the precise limit we need to take to recover (2.24) is R→∞, d fixed, V0 → Ṽ0 1 C (Rd)− 3 2 e piR d , which also implies Q→∞. In supergravity language, the limit we are taking is designed to take the NS5-brane throat infinite while holding the dilaton at the bottom of the throat fixed. The fact that V is exponentially damped as we go towards the middle of the plates gives rise to the linear dilaton behaviour of the final supergravity solution. 44 Chapter 2. LST from the PWMM Field theory description Using the correspondence of parameters between field theory and the elec- trostatics we find that the limit of the Plane-Wave Matrix Model that defines the dual of the Lin-Maldacena solution, i.e. the field theory description of Little String Theory on S5, is N2 →∞, N5 fixed, 1 g2 λ 3 8 e−aλ 1 4 /N5 fixed . (2.28) where a is a numerical coefficient related to the constant in (2.19) by a = 2(π2/h∞) 1 4 . Thus, rather than holding the ’t Hooft coupling fixed, we scale it to infinity in a controlled way λ ∼ N45 ln4(N2) . (2.29) Unfortunately, because the coupling constant must be large, this implies that perturbation theory is not useful on the field theory side, though per- haps there are some near-BPS sectors of the theory where the expansion parameter is not the naive ’t Hooft coupling. 2.6 Discussion 2.6.1 The Double-Scaling Limit We have seen that to obtain the field theory dual of the Lin-Maldacena supergravity solution for Little String Theory on S5, we need to take a large N2 limit while scaling the ’t Hooft coupling to infinity in a controlled way. This double scaling limit is reminiscent of limits used to define low- dimensional string theories in old matrix models (see for example [65]). There, the ’t Hooft coupling is scaled to some critical value in a controlled way as N goes to infinity, such that all terms in the genus expansion continue to contribute even though N becomes infinite. We suspect that this is also the situation here, except that in our case, the “critical” value of the ’t Hooft coupling is infinity. The fact that N2 becomes infinite is consistent with the fact that the dilaton vanishes asymptotically in the supergravity solution. On the other hand, we still have a string genus expansion in the bulk of the supergravity solution, so we can understand the scaling of λ to infinity as necessary for the gauge theory to reproduce nontrivial string interactions in the bulk. 45 Chapter 2. LST from the PWMM To understand this in more detail, consider the matrix model genus expansion for some physical observable. It takes the form F = ∑ n fn(g 2N2, N5) Nn2 , (2.30) where fn(λ,N5) gives the sum of genus n diagrams. In the ’t Hooft limit with λ = g2N2 fixed and N2 taken to infinity, only the planar n = 0 term contributes. What we are suggesting is that the scaling (2.29) is such that all terms in the expansion (2.30) contribute. If this is true, it predicts that the large λ behaviour of fn is fn(λ)→ an [ λ 5 8 e aλ 1 4 N5 ]n . (2.31) The quantity in square brackets divided by N2, which we can call g̃, is the inverse of the quantity being held fixed in (2.28), so the genus expansion (2.30) becomes F = ∑ n ang̃ n . Thus, the constant g̃ serves as the effective string coupling. The behaviour (2.31) is a nontrivial prediction of our results and the assumption that the string theory genus expansion is still related to the gauge theory genus expansion. It should apply to the behaviour of any physical observable that survives the scaling limit, for example the energy of any state in the matrix model corresponding to some excitation in the NS5-brane throat. It would be interesting to understand more precisely from the gauge theory point of view which set of observables remain in the limit. Unfortunately, it seems difficult to check the behaviour (2.31) directly from the gauge theory, since it would involve summing infinite sets of dia- grams. However, it may be that this is possible for certain BPS or near-BPS observables, as for the circular Wilson loop in N = 4 supersymmetric Yang- Mills theory, where for example the full contribution at the planar level is given by an infinite set of ladder diagrams that can be summed explicitly [66]. Intriguingly, that result, 〈W 〉N=∞ = √ 2 π λ− 3 4 e √ λ, takes a rather similar form to our prediction here.22 A specific limit in which 22Note that the √ λ in this example and the λ 1 4 in our case both represent the squared radius of the respective S5s in string units. 46 Chapter 2. LST from the PWMM some matching similar to [30] might be possible is in a Penrose limit of the geometry [67], associated with geodesics around the S2 at r = 0. 2.6.2 An Infinite Parameter Family of NS5-brane Solutions Starting from our exact supergravity solution for the simplest class of vacua, we have found a specific limit that gives the Lin-Maldacena solution corre- sponding to the region between two infinite conducting plates. It is interest- ing to note that this solution is actually the simplest in an infinite-parameter family of solutions. In the electrostatics language, we can have any function V = ∞∑ n=1 cn sin (nπz d ) I0 (nπr d ) , with cn chosen to fall off fast enough so that the sum converges for all r. The supergravity solution corresponding to any such potential will have an infinite throat with noncontractible S3 carrying fivebrane flux. While it does not seem possible to obtain these more general solutions as limits of the solution we considered in this chapter, it is plausible that we could obtain them as limits of solutions corresponding to more general vacua of the Plane-Wave Matrix Model. Specifically, we could imagine starting with a solution containing some arbitrarily large number of discs, taking the size of the lowest disc to infinity as before, but now tuning all of the parameters V0, d2, Q2, . . . , dn, Qn in such a way that all of the upper discs have some non-trivial influence on the potential between the plates. It would be interesting to understand better the physical interpretation of these more general solutions. 2.6.3 Energies of Near-BPS States A useful result coming from our solution is the formula (2.5) that determines the radius of the disc in terms of the other parameters d, Q, and V0. From the supergravity solution (B.1), it is straightforward to show [40] that R determines the radius of the S5 at the point (r = R, z = d) corresponding to the edge of the disc as23 R2S5/α ′ = 4R. As described in [40] section 2.2, this in turn determines the energies of certain near-BPS states with large angular momentum on the S5 (see equation 23To see this, we use the Laplace equation to rewrite V ′′ and note that ∂rV vanishes on the discs. 47 Chapter 2. LST from the PWMM (2.42)).24 Thus, our function q(κ) in (2.5), defined in (2.14) and plotted in figure 3, determines the near-BPS energies for large R and d but arbitrary d/R, interpolating between the small d/R and large d/R results given in [40], equations (2.84) and (2.57) respectively. 2.6.4 Other Definitions of Little String Theory on S5 Finally, we note that while the limit we have defined may be the simplest description of the Little String Theory on S5, there should be many other field theoretic definitions. In the electrostatics picture, it should arise any time two nearby discs are scaled to infinite radius at fixed separation, with the background potential scaled so that the potential between the plates remains nontrivial. Thus, we could start with more general vacua of the Plane-Wave Matrix Model, vacua of maximally supersymmetric YM on R× S2, or vacua of N = 4 SYM on R × S3/Zk.25 In the next chapter we will show precisely how this occurs for for simple vacua of the latter two theories. 24These are the states of string theory on a plane-wave obtained by a Penrose limit associated with geodesics going around the S5 at fixed r and z. 25This latter possibility was noted in [40]. 48 Chapter 3 Little String Theory from Double-Scaled Field Theories 3.1 Introduction In the preceding chapter we have shown that type IIA Little String Theory on S5arises as a double-scaling limit of the Plane-Wave Matrix Model. The argument we presented was based on understanding the scaling of the su- pergravity solutions dual to each of these theories. We found that by taking a region of the geometry containing a throat carrying NS5-brane flux, and taking a limit isolating this throat, we could start with a geometry dual to a vacuum of the Plane-Wave Matrix Model and end up with a vacuum dual to little string theory on S5. This led us to a corresponding scaling argument on the field theory side that gave us a Lagrangian definition of Little String Theory. We noted in the previous chapter, however, that in the electrostatics picture of the dual geometries, the limit isolating a throat with NS5-brane flux involved taking the size of adjacent charged conducting discs to infinity, and that, as the supergravity duals of maximally supersymmetric YM on R × S2 and N = 4 SYM on R × S3/Zk generically contain charged conducting discs, it is reasonable to think that a similar limiting procedure could be used to find Lagrangian definitions of Little String Theory based on these other theories. According to the proposal in [40], the supergravity solutions arising from configurations with a finite number of discs correspond to the (classically degenerate) vacua of maximally supersymmetric YM on R × S2. Config- urations with an infinite number of discs, arranged in a periodic fashion, correspond to the vacua of N = 4 SYM on R × S3/Zk. Finally, configu- rations with one infinitely large disc, and a finite number of discs above it, correspond to the vacua of the Plane-Wave Matrix Model (see figure 3.1). The relations among these field theories have been discussed in chapters 1, 2 and [39, 68, 69]. In this chapter, we extend the argument given in 2 that Little String 49 Chapter 3. LST from Field Theories (a) (b) (c) ... ... Figure 3.1: The three generic types of electrostatics configurations. The isolated set of discs in (a) is a configuration dual to a vacuum of SYM theory on R × S2 with sixteen supercharges. The periodic configuration in (b) is dual to a vacuum of N = 4 SYM theory on R × S3/Zk. The set of discs above an infinite conducting plane in (c) is dual to a vacuum of the Plane-Wave Matrix Model. 50 Chapter 3. LST from Field Theories Theory arises as a double-scaling limit of the Plane-Wave Matrix Model to find similar limits for maximally supersymmetric YM on R×S2 and N = 4 SYM on R × S3/Zk. We solve the electrostatics problems corresponding to specific simple vacua of these field theories and determine the scaling of parameters in the supergravity solutions that is required to obtain the Lin-Maldacena solution for NS5-branes on S5. By considering the matching between the parameters in the field theories and those in the correspond- ing electrostatics problems, we thereby determine the precise scaling of the gauge theory parameters that is required to obtain Little String Theory on S5. The proposed prescriptions are found to be double-scaling limits, similar to the one found in the case of the Plane-Wave Matrix Model [1]. Whereas in the Plane-Wave Matrix Model case it was found that the ’t Hooft coupling must be scaled like ln4N [1], we will show below that for the SYM theories on R × S2 and R × S3/Zk the ’t Hooft coupling must be scaled like ln3N and ln2N respectively. 3.2 The Gauge Theories and Their Dual Supergravity Solutions In this section we will recall some facts about the gauge theories in question and their supergravity duals, that we have described in chapter 1. In [40], Lin and Maldacena found a class of solutions of type IIA super- gravity with SU(2|4) symmetry depending on one single function V . This function V solves the three dimensional Laplace equation and satisfies the same boundary conditions as the electrostatic potential of an axisymmet- ric arrangement of charged conducting discs in a background electric field. By specifying the positions and sizes of the conducting discs, the charges on the discs, and the asymptotic form of V at infinity, V is determined uniquely. Each different specification of these parameters leads to a dif- ferent V , however not all such choices give rise to physically acceptable supergravity solutions. Flux quantization in the supergravity solution tells us that the charges on the discs and the spacing between discs are quantized. Positive-definiteness of various metric components in the supergravity solu- tions imposes constraints on the form of the asymptotic potential. Finally, the regularity of the supergravity solutions tells us that the surface charge density on the discs must vanish at the edge of the discs. This final condi- tion suggests that for a fixed asymptotic potential, the positions, charges, and sizes of the discs cannot be independently specified. For example, the sizes of the discs may be fixed once the other parameters are freely specified. 51 Chapter 3. LST from Field Theories For an extensive discussion of the general properties of these supergravity solutions see [40]. Here we are interested in the supergravity solutions dual to the vacua of the SYM theory on R × S2 and N = 4 SYM theory on R × S3/Zk. For all vacua of these two field theories, Lin and Maldacena determined the asymptotic form of V to be W0(r 2 − 2z2), where W0 > 0. The choice of vacuum is then given by specifying the charges, positions, and sizes of the discs. Let us review in some detail the connection between these parameters for the supergravity solutions and the parameters defining the field theory vacua. First consider N = 4 SYM on R × S3/Zk. The space S3/Zk can be described most directly by choosing coordinates on the unit S3 such that the metric takes the form dΩ23 = 1 4 [ (2dψ + cos θdφ)2 + dθ2 + sin2 θdφ2 ] , (3.1) where the ψ coordinate is 2π periodic, and θ, φ are the usual coordi- nates for S2. Then the orbifold S3/Zk is obtained by identifying ψ ∼ ψ + 2π/k. The vacua of this field theory are given by the space of flat connections on S3/Zk. Up to gauge transformations, these are of the form A = −diag(n1, n2, . . . , nN ) dψ, where e2pini/k are k-th roots of unity (clearly, to label the vacua uniquely, we should restrict the values of the integers ni to be in some fixed interval of length k). To understand intuitively how these vacua map to configurations of discs in the electrostatics problem, consider the field theory as a theory of D3-branes wrapped on an S3/Zk. Now apply a T-duality transformation in the isometry direction ψ. The T-dual coordinate ψ̃ is periodic ψ̃ ∼ ψ̃ + 2πk, and the background gauge field is mapped to an arrangement of D2-branes located at the positions ψ̃ = 2πn1, 2πn2, . . . , 2πnN (along with their images under translations by integer multiplies of 2πk). Naturally, this suggests that the dual supergrav- ity solution is obtained by considering a periodic configuration of discs with period proportional to k. The integers ni that specify the gauge theory vac- uum now determine the positions and charges of the discs within one period in the obvious manner. Presumably, the sizes of the discs are then fixed by demanding regularity of the supergravity solution. In rest of this chapter, we will be interested in the simplest vacuum state of the theory, given by the trivial gauge field ni = 0 for all i. In the normalization conventions of Lin and Maldacena [40], the dual supergravity solution is generated by the ax- isymmetric electrostatic potential V (r, z) for an arrangement of equal-sized discs at z = (π/2)km for all integers m, where the charge on each disc is 52 Chapter 3. LST from Field Theories Q = (π2/8)N . Now we consider the case of the SYM theory on R × S2. As discussed in [40], we can think of this theory as N = 4 SYM on R × S3/Zk, in the limit where k → ∞ and g2YM3 → 0 while keeping g2YM3k fixed. Up to a numerical constant, the limiting value of g2YM3k is the coupling g 2 YM2. If we start with a vacuum in the S3/Zk theory with background gauge field A = −diag(n1, n2, . . . , nN )dψ and take k →∞ with the integers ni fixed, then we obtain a vacuum of the S2 theory with a vacuum expectation value for one of the adjoint scalars Φ = −diag(n1, n2, . . . , nN ) and a background gauge field with associated flux F = dA = Φsin θdθdφ. All of the vacua of N = 4 SYM on R×S2 discussed in [40] can be obtained in this way. This limit has a clear interpretation in the T-dual picture. We start with a configuration of a finite number of D2-branes, repeated periodically by translating the whole arrangement by integer multiples of 2πk. In the limit k →∞, we are left with only one copy of the configuration of D2-branes, the images being pushed off to infinity. This naturally suggests that the dual supergravity solution is obtained by considering a configuration of a finite number of discs. It is clear that the integers ni determine the positions and charges of the discs in a manner analogous to the situation in the R × S3/Zk theory. Again, the sizes of the discs are presumably fixed by demanding regularity of the supergravity solutions. Note that the total sum of the charge on the discs must equal the rank of the gauge group N . In the rest of this chapter, we consider non-trivial vacua of the form Φ = (n, . . . , n,−n, · · · −n), where the integers n and −n each appear N/2 times. In this case the dual supergravity solution is generated by the potential V (r, z) corresponding to two equal- sized discs at z = ±(π/2)n with charge (π2/8)(N/2) on each disc. The final issue we need to discuss in this section is the normalization of the asymptotic potential at infinity. For the SYM theory on R×S2, we can relate W0 to g 2 YM2 by using the results in [1]. As discussed in [1, 39, 69] the SYM theory on R × S2 can be obtained as a limit of the Plane-Wave Matrix Model. This statement, together with the matching of parameters in the Plane-Wave Matrix Model discussed in [1], tell us that we must have W0 = h2 g2YM2 , (3.2) where the positive constant h2 does not depend on the parameters N , g 2 YM2, which define the gauge theory, and the eigenvalues of Φ, which label its vacua. For SYM theory on R×S3/Zk, the above mentioned relation between this theory and the theory on R×S2 suggests that we make the identification 53 Chapter 3. LST from Field Theories W0 = h3 g2YM3k , (3.3) where h3 is a positive constant that does not depend on N, k, g 2 Y M3 and the integers that label the vacua of the gauge theory. 3.3 Little String Theory from SYM on R × S2 In this section, we consider in detail the supergravity solution corresponding to the electrostatics problem for two identical discs of radius R located at z = ±d with charge Q on each disc and a background potentialW0(r2−2z2). We wish to solve the electrostatics problem explicitly and determine the required scaling to obtain the Lin-Maldacena NS5-brane solution. The Electrostatics Problem for the Case of two Identical Discs Following the approach of [1], we first solve the electrostatics problem for the specific case W0 = 1, R = 1, d = κ (the solution for the general case is then obtained by linear rescaling of the coordinates and an overall rescaling of the potential). In this case the solution must have the form V (r, z) = (r2 − 2z2) + φκ(r, z), (3.4) where φκ is an axisymmetric solution of the Laplace equation that vanishes at infinity. We can expand φκ in terms of Bessel functions, and in the region between z = −d and z = d, this expansion takes the form φκ(r, z) = ∫ ∞ 0 du u e−uκA(u) ( e−uz + euz ) J0(ru). (3.5) The potential on the two conducting discs, ∆, must be constant, and the electric field must be continuous at all points not on the discs. Imposing these boundary conditions leads to the following dual integral equations∫ ∞ 0 du u (1 + e−2κu)J0(ru)A(u) = ∆− r2 0 < r < 1∫ ∞ 0 duJ0(ru)A(u) = 0 r > 1. (3.6) Following [63] we find that the solution of these integral equations can be given in terms of the solution to a Fredholm integral equation of the second 54 Chapter 3. LST from Field Theories kind. The problem in this case is very similar to the one considered in [1]. We have A(u) = 2u π ∫ 1 0 dt cos(ut)f(t), (3.7) where f(t) satisfies the integral equation f(t) + ∫ 1 −1 dxK(t, x)f(x) = ∆− 2t2, (3.8) and K(t, x) = 1 π 2κ 4κ2 + (t− x)2 . (3.9) For each value of ∆, the integral equation for f can be solved numeri- cally. From the resulting electrostatics potential, we can compute the surface charge density on the discs σ(r) = 1 π2 [ f(1)√ 1− r2 − ∫ 1 r dt f ′(t)√ t2 − r2 ] . (3.10) We can adjust the constant ∆ until we find the value ∆κ for which the corresponding solution fκ satisfies fκ(1) = 0. Then the surface charge dis- tribution σκ(r) for this solution vanishes at the edge of the discs. This final condition ensures the regularity of the corresponding supergravity solutions. The total charge on each disc is given by qκ = 2 π ∫ 1 0 dtfκ(t). (3.11) Figure (3.3) shows a plot of qκ. For large κ the charge on each disc ap- proaches 8/3π, and for small κ the charge on each disc approaches 4/3π. Finally the solution for the general case is obtained by rescaling. The electrostatics potential is given by V (r, z) =W0(r 2 − 2z2) +W0R2φd/R(r/R, z/R), (3.12) and the total charge on each disc is given by Q =W0R 3qd/R. (3.13) 55 Chapter 3. LST from Field Theories z r 2d R E Figure 3.2: The electrostatics problem for two identical discs. The dotted lines show the background electric field configuration. 56 Chapter 3. LST from Field Theories 0.1 1 0.01 0.1 1 10 100 q κ κ 4 3pi 8 3pi Figure 3.3: The charge on each disc in the two-disc case. The solid and dashed lines show the asymptotes for small and large κ respectively. The Limit of the Lin-Maldacena Solution Now we can determine the limit of this solution that gives the Lin-Maldacena solution for NS5-branes on S5. In the region between the discs with 0 < r < R, our solution is an axisymmetric solution of the Laplace equation that is regular at r = 0, so we can expand the solution in terms of modified Bessel functions V (r, z) = Vz=d + ∞∑ n=1 cn cos ( (2n+1)piz 2d ) I0 ( (2n+1)pir 2d ) . (3.14) The coefficients cn may be determined by using the potential at r = R. This gives cn = ( I0 ( (2n+1)piR 2d ))−1 2W0R 2 ∫ 1 0 dz cos ( (2n+1)piz 2 ) ( 1− 2(κz)2 −∆κ + φκ(1, κz) ) . (3.15) 57 Chapter 3. LST from Field Theories Using our numerical solution for φκ, the above integral can be performed numerically. In the limit d≪ R, this gives c1 ≈ 1.56W0Rd ( I0 ( piR 2d ))−1 . (3.16) For large R/d this expression will be dominated by the Bessel function, which takes the asymptotic form (I0(z)) −1 ∼ √ 2πze−z To preserve some non-trivial geometry, we must then scaleW0 exponentially. Doing so keeps c1 finite in the limit, but sends all the other coefficients to zero so that we recover the Lin-Maldacena solution. More precisely, the Lin-Maldacena solution is obtained in the limit R→∞ d fixed W0 ∼ R−1(Rd)−1/2e piR 2d . (3.17) The Gauge Theory Interpretation Having understood the correct scaling on the gravity side, we can translate this into a condition on the gauge theory parameters. This amounts to N →∞ n fixed 1 g2YM2 λ1/2n1/2e−bλ 1/3/n fixed, (3.18) where the ’t Hooft coupling is λ = g2YM2N and b is a numerical coefficient related to the constant appearing in (3.2) by b = (π/4)(3/h2) 1/2. We see that this is a large N limit, where the ’t Hooft coupling is also scaled to infinity in a controlled way, and is very similar to the limit that was found in the case of the Plane-Wave Matrix Modelin [1]. Note that the number of NS5-branes is N5 = 2n. 3.4 Little String Theory from N = 4 SYM on R× S3/Zk Now we wish to perform a similar detailed analysis for the supergravity solution corresponding to a periodic array of discs of radius R, where the discs are located at z = (2m+1)d (m is any integer), the charge on each disc is Q, and the background electric field is given by the potentialW0(r 2−2z2). Again we first solve the electrostatics problem, then find the limit that recovers the NS5-brane solution of Lin and Maldacena. 58 Chapter 3. LST from Field Theories z r 2d R E ... ... Figure 3.4: The electrostatics problem in the case a periodic array of discs. The dotted lines show the background electric field configuration. The Electrostatics Problem for a Periodic Array of Discs As in the previous section, we solve the electrostatics problem for the special case W0 = 1, R = 1, d = κ, and obtain the solution for the general case by rescaling. In the absence of the background potential the charge distribution on each disc will be the same. Adding the background field will affect the charge distribution on each disc, but since the radial part of the electric field it creates is identical on each disc, the charge distribution will remain the same on each disc (see figure 3.4). We can separate the potential into the sum of the background field and the part due to the charge on the discs. V = r2 − 2z2 + φκ(r, z), (3.19) where φκ is periodic in z because the charge on each disc is identical. For- 59 Chapter 3. LST from Field Theories mally, we can expand φκ(r, z) in terms of Bessel functions as φκ(r, z) = ∫ ∞ 0 du u J0(ru)A(u) ∞∑ n=−∞ e−u|(2n+1)κ−z|, (3.20) and then try to determine the function A(u) by the imposing the boundary conditions. If we take the value of the potential V to be ∆−2κ2 on the disc at z = κ, then by imposing the boundary conditions we obtain the following dual integral equations∫ ∞ 0 du u ( 1 + 2e−2κu 1− e−2κu ) J0(ru)A(u) = ∆− r2 0 < r < 1∫ ∞ 0 duJ0(ru)A(u) = 0 r > 1. (3.21) However, direct attempts to solve these equations are met with divergences and various difficulties. The reason is that these equations hold only for- mally, because the sum in the expression for the potential (3.20) actually diverges. Physically, there is no divergence because the electric field remains finite. This is the same type of situation encountered for an infinite number of equally spaced point charges (or an infinite line of charge) on the z-axis, which occurs simply because we try to express the potential as a sum of the Coulomb potential for each charge. If we consider the potential difference between any two points, there is no divergence, so we can regularize (3.20) by subtracting the potential at any fixed reference point. In this case, it is more convenient to consider the first integral equation (3.21) as a condition on the electric field rather than the electric potential∫ ∞ 0 du ( 1 + 2e−2κu 1− e−2κu ) J1(ru)A(u) = 2r 0 < r < 1. (3.22) The dual integral equations can then be solved by introducing a function satisfying a Fredholm integral equation of the section kind, fκ(x) + ∫ 1 0 duK(x, u)fκ(u) = − 8x√ π , (3.23) where A(u) = − 1√ π ∫ 1 0 dξ sin(uξ)fκ(ξ). (3.24) The kernel is given by K(x, u) = 1 π ∫ ∞ 0 dtk(t)(− cos(u+ x)t+ cos |u− x|t), (3.25) 60 Chapter 3. LST from Field Theories where k(u) = 2e−2κu 1− e−2κu . (3.26) These integrals can be evaluated and the result is K(x, u) = 1 2πκ ( Ψ ( 1 + i(x+ u) 2κ ) +Ψ ( 1− i(x+ u) 2κ ) −Ψ ( 1 + i|x− u| 2κ ) −Ψ ( 1− i|x− u| 2κ )) , (3.27) where Ψ is the digamma function. We solved (3.23) numerically using the Nyström method (e.g. [70]). In contrast to the two disc case, since we considered the integral equation corresponding to a condition on the electric field, there is no ∆ to adjust to ensure that the surface charge density at the edge of the disc vanishes. In fact, for the form of the solution given in (3.24), this condition is automatically satisfied as long as fκ is bounded. In terms of fκ, the charge on each disc is qκ = − 1√ κ ∫ 1 0 dt tfκ(t). (3.28) Using our numerical solution for fκ we found that qκ approaches 8/3π for large κ and approximately 1.99κ for small κ (see figure 3.4). In principle, it is possible to determine the regularized potential com- pletely from this solution for fκ (however, the integrals involved are rather computationally expensive). Then the potential for the case of general W0, R and d is obtained by a linear rescaling of coordinates and an overall rescal- ing of the potential. Specifically, we note that the charge on each disc in the general case is Q =W0R 3qd/R. (3.29) The limit of the Lin-Maldacena Solution To determine how the Fourier coefficients of the potential scale with κ, we found it was most efficient to use the method of conformal mapping. Near the edge of the discs, when their radial size is much larger than their separation, the electrostatics problem becomes two-dimensional. By defining the complex coordinates ζ = (r − R) + iz, and w = 2∂ζV any holomorphic function w(ζ) will be a solution of the Laplace equation. As described in [40] the appropriate mapping in this case is ∂wζ = α tanh (πw β ) (3.30) 61 Chapter 3. LST from Field Theories 0.001 0.01 0.1 1 0.01 0.1 1 10 100 q κ κ 1.99κ 8 3pi Figure 3.5: The charge on a disc as a function of the spacing between discs. The numerical result is given by the dashed line. The solid line is the asymp- totic behaviour for small κ, q ∼ 1.99κ. For large κ the charge approaches 8 3pi . 62 Chapter 3. LST from Field Theories and so ζ = αβ π log cosh ( πw β ) , (3.31) where α, β are constants. Inverting this we find w = β π cosh−1 ( e piζ αβ ) . (3.32) If we fix the positions of the discs to be at ζ = i(2md), where m is an integer, we have d = αβ/2. The vertical electric field at any disc should be −4W0ℑ(ζ), so that β = 8W0d and α = 1/4W0. Expanding the potential in terms of modified Bessel functions, as in the two-disc case, we find that c1 ≈ 16W0d 2 π (I0( piR 2d )) −1(0.659). (3.33) Again, therefore, to preserve non-trivial geometry we must scale W0 expo- nentially. The precise scaling form to obtain the Lin-Maldacena solution is R→∞ d fixed W0 ∼ R−1/2d−3/2e piR 2d . (3.34) The Gauge Theory Interpretation In terms of the gauge theory parameters, we have N →∞ k fixed 1 g2YM3 λ1/4k1/2e−cλ 1/2/k fixed, (3.35) where the ’t Hooft coupling is λ = g2YM3N and c is a numerical coefficient related to the constant appearing in (3.3) by c = (2π/1.99h3) 1/2. This is again a double-scaling limit in which the ’t Hooft coupling is scaled to infinity in a controlled way. Note that the number of NS5-branes in this case is N5 = k. 3.5 Discussion We have given an explicit prescription for taking double-scaling limits of maximally supersymmetric YM on R × S2 and N = 4 SYM on R × S3/Zk to obtain Little String Theory on S5. These limits were obtained by using the family of supergravity solutions found by Lin and Maldacena [40]. With 63 Chapter 3. LST from Field Theories the similar result in 2, we have demonstrated that it is possible to take such a limit in each of the three generic examples of this family of solutions, and in each of the three field theories to which they are dual. In each case, the precise form of the double-scaling limit is similar. Whereas in the Plane-Wave Matrix Model it was found the correct limit was [1] N2 →∞ N5 fixed N2 ∼ λ5/8eaλ1/4/N5 , (3.36) we found above that for the SYM theory on R× S2 we have N →∞ n fixed N ∼ λ1/2n−1/2ebλ1/3/n , (3.37) and for N = 4 SYM theory on R× S3/Zk we have N →∞ k fixed N ∼ λ3/4k−1/2ecλ1/2/k . (3.38) As noted in [1], it is sensible that the correct limit to obtain Little String Theory from these field theories is a double-scaling limit as opposed to a strict ’t Hooft limit. If the correct limit was the ’t Hooft limit, then it would seem strange that the field theory could produce string loop interactions. That the ’t Hooft coupling should also be scaled to infinity in a controlled way allows the field theory to reproduce the string genus expansion. Suppose we consider the genus expansion for some physical observable in one of these theories F = ∑ g N2−2gfg(λ, α), (3.39) where α represents the other parameters. The double-scaling limit should be such that all terms in this expansion contribute. For this to occur, the terms in the expansion would have to take a particular form when λ is large. In the case of the Plane-Wave Matrix Model, this form was found to be [1] fg(λ)→ ag ( λ5/8eaλ 1/4/N5 )2g−2 , (3.40) where the bracketed expression divided byN2 serves as the effective coupling constant. Here we find for the SYM theory on R× S2 we must have fg(λ)→ ag ( λ1/2ebλ 1/3/n )2g−2 , (3.41) and for N = 4 SYM theory on R× S3/Zk fg(λ)→ ag ( λ3/4ecλ 1/2/k )2g−2 . (3.42) 64 Chapter 3. LST from Field Theories Interestingly, although these field theories live in different numbers of di- mensions, it is possible to recover Little String Theory from each of them by similar double-scaling limits. Obvious difficulties arise in checking these predictions. One might hope that there are some BPS observables for which such a check might be feasible. In the case of the circular Wilson loop in N = 4 SYM the full set of planar diagrams can be summed [66]. The result in that case took the form 〈 W 〉 N=∞ = √ 2 π λ−3/4e √ λ. (3.43) This result has been extended to all orders in [71], where it was shown that the asymptotic behaviour goes like e √ λ at each order. That behaviour also arises from modified Bessel functions. It would be interesting to calculate the circular Wilson loop in N = 4 SYM on R × S3/Zk, and to compare it with our results here. Other open questions remain. For example, as noted in chapter 1, the solution for Little String Theory on S5 given by Lin and Maldacena [40] is the simplest of an infinite family of solutions that have an infinite throat with H-flux. It would be interesting to understand if these solutions could arise from limits of more general disc configurations. It would also be interesting to understand more about the vacua of Little String Theory dual to these solutions. 65 Chapter 4 General Lin-Maldacena Solutions and Plane-Wave Matrix Model Instantons from Supergravity 4.1 Introduction In the preceding two chapters, we have been able to solve the electrostatics problem that arises in finding regular SU(2|4) symmetric solutions of type IIA supergravity that are dual to vacua of the Plane-Wave Matrix Model and maximally supersymmetric YM on R × S2. We have also found that these solutions can be used to find definitions of Little String Theory as double-scaling limits of the SU(2|4) symmetric field theories. Buoyed by our success, it is interesting to attempt to find more general solutions, and to determine what other aspects of the field theories at strong coupling might be elucidated using knowledge of the dual gravity side. The solutions that we have found so far, and the ones found by others [40] are all special cases. One particular solution given by Lin and Maldacena [40] corresponds to two infinitely large discs held at fixed separation. This is the solution dual to Little String Theory that has played a key role in chapters 2 and 3. The gravity dual was used in [67] to argue that Little String Theory on S5 has interesting features that differ from the theory in flat space. An explicit solution has also been given in the case of a single isolated disc, dual to a vacuum of the maximally supersymmetric Yang-Mills theory on R×S2 [1, 40]. Also, in the region very close to the tip of a disc, the problem becomes two dimensional, and it is possible to solve it by conformal mapping [40]. Solutions in the case of two discs were given above. More general solutions, however, are not known. In general this has prevented this set of dualities from being used to study the SU(2|4) symmetric field theories at strong coupling. It is interesting 66 Chapter 4. General LM Solutions and PWMM Instantons to consider what questions can be addressed from the information we do know on the gravity side. Recently, Lin [72] has made some progress in applying this correspondence to instanton calculations in the Plane-Wave Matrix Model and maximally supersymmetric Yang-Mills theory on R×S2. In the case of the matrix model, this question had been studied directly in the field theory [73]. Lin gave explicit results for weak coupling from the supergravity side, and found precise agreement with the gauge theory analysis. It would certainly be desirable to be able to perform other gauge theory calculations using the dual gravity description. It is, therefore, quite inter- esting to obtain more general supergravity solutions that would allow this to be done. In this chapter we will demonstrate that it is possible to reduce the generic electrostatics problem to a simple linear system that can be solved very simply using numerical methods. We will then use this technique to find some explicit results using Lin’s prescription for instanton calculations on the dual gravity side. For a simple example electrostatics configuration, dual to a vacuum of the Plane-Wave Matrix Model, we will give an explicit expression for the superpotential at strong coupling, and also the leading correction to Lin’s result at weak coupling. 4.2 Supergravity Solutions In this section we will review the Lin-Maldacena formulation of supergravity solutions in terms of electrostatics problems, full details can be found in [40]. Then in subsections 4.2.1 and 4.2.2 we will discuss the solution of these problems. To find the supergravity duals to field theories with SU(2|4) symmetry, Lin and Maldacena looked for similarly symmetric supergravity solutions. In particular, the bosonic part of this symmetry group is R×SO(3)×SO(6) so the supergravity solutions should contain an S2 and an S5. Interestingly, with this restriction all of the supergravity fields can be expressed in terms of a single function of the two remaining coordinates. For the supergravity equations to be satisfied, this function must be an axisymmetric solution to the Laplace equation in three dimensions. The full supergravity solution in terms of this function, in the string 67 Chapter 4. General LM Solutions and PWMM Instantons frame, is ds210 = ( V̈ − 2V̇ −V ′′ ) 1 2 { −4V̈ V̈ − 2V̇ dt 2 − 2V ′′ V̇ (dρ2 + dη2) + 4dΩ25 + 2 V ′′V̇ ∆ dΩ22 } e4Φ = 4(V̈ − 2V̇ )3 −V ′′V̇ 2∆2 C1 = − 2V̇ ′ V̇ V̈ − 2V̇ dt F4 = dC3, C3 = −4 V̇ 2V ′′ ∆ dt ∧ d2Ω, H3 = dB2, B2 = 2 ( V̇ V̇ ′ ∆ + η ) d2Ω, ∆ ≡ (V̈ − 2V̇ )V ′′ − (V̇ ′)2 . (4.1) where V is the electrostatics potential, and dots and primes indicate deriva- tives with respect to log r and z, respectively. To avoid conical singularities in (4.1), when the size of the S2 or S5 shrinks, requires that either V is reg- ular at r = 0 (S5 shrinks) or ∂rV = 0 (S 2 shrinks). Different supergravity solutions can, therefore, be specified by inserting some conducting discs of various radii Ri at positions zi. See figure 4.1. Inserting a disc will create separate two regions on the z-axis on which the S5 shrinks and will there- fore mean adding a non contractible 6-cycle, which will carry NS5-brane flux. Similarly, the region between two discs, on which the S2 shrinks, will be a non-contractible 3-cycle carrying D2-brane flux. Two additional constraints on the electrostatics solution come from en- suring that all of the metric components are positive definite and that the transformation to these coordinates is well defined. Positive definiteness re- quires that the electrostatics potential takes a definite asymptotic form, and the coordinate transformation requires that the charge density vanishes at the edge of each disc. We will now describe a method for solving the electrostatics problems for generic configurations. 4.2.1 General Solutions Dual to SYM on R× S2 One of the field theories for which Lin and Maldacena found the corre- sponding electrostatics configurations is maximally supersymmetric SU(N) 68 Chapter 4. General LM Solutions and PWMM Instantons Σ3 Σ6 S2 S5 r z Figure 4.1: An example electrostatics configuration. The conducting discs are the horizontal solid lines. Fibred above the rz-plane are an S2 and an S5. The size of the S2 shrinks on the discs, whereas the S5 shrinks on the z axis. The dashed lines indicate topological 3- and 6-cycles in the geometry. Yang-Mills theory on R × S2. This theory is related to N = 4 SYM on R × S3 by dimensionally reducing that theory on the Hopf fibre of S3 [40]. The field content is similar to N = 4, however the theory on R× S2 admits vacuum configurations with non-trivial Φ, the scalar field resulting from the dimensional reduction. The vacua of this theory are parametrized by a set of integers where Φ = diag(n1, n2, . . . , nN ) [39]. General discussion of the relations among the vacua in the SU(2|4) symmetric field theories can be found in [68, 69]. The set of electrostatics configurations in question are given by an arbi- trary set of positively charged conducting discs. Each disc will be associated with some D2 brane charge, so it is natural to think of the supergravity so- lutions as being dual to vacua of maximally supersymmetric Yang-Mills on R × S2 [40]. The integers in the vacuum configuration for Φ are related to positions of the discs by z = nπ/2, and the number of units of charge on the disc is related to the number of times that each integer appears by Q = π2N/8 [40]. The solutions to these electrostatics problems have been given in some specific cases [1, 2, 40]. For example the limit that the discs are very large, or only the geometry near the tip of a disc is of interest, the problem be- comes two dimensional and it is possible to treat it with conformal mapping [1, 2, 40, 72]. In the case that there is a single disc it is possible to find an 69 Chapter 4. General LM Solutions and PWMM Instantons exact solution [1, 40]. For two equally sized discs a formal solution can be found [1, 2]. We will show that the techniques of [1, 2, 63] to solve the elec- trostatics problem for two identical discs can be extended to more arbitrary disc configurations. We will discuss how these more general solutions may be found using these techniques. Consider the case of a collection of k charged conducting discs in the case of maximally supersymmetric Yang-Mills theory on R × S2. This problem is similar to the one for two discs considered in [2], however we will allow the discs here to sit at arbitrary positions, di and have arbitrary sizes, Ri. We can take the potential to be V =W0 ( r2 − 2z2 + ∑ i φi(r, z) ) , (4.2) where the first two terms ensure the correct asymptotic conditions, and the third is an asymptotically vanishing contribution that comes from the charges on the discs. It takes the form φi(r, z) = ∫ ∞ 0 du u J0(ru)Ai(u)e −u|z−di| . (4.3) Each function Ai will be shown to determine the charge density on the i th disc. To fix the form of these functions we impose the conducting boundary conditions on the discs. In particular, if the discs are held at fixed potentials ∆i, then we will find a set of dual integral equations similar in form to those in [1, 2, 63]. The conditions at the ith disc are that for r < Ri ∫ ∞ 0 du u J0(ur)  Ai(u) +∑ j 6=i Aj(u)e −u|dj−di|   = ∆i + 2d2i − r2 , (4.4) and for r > Ri ∫ ∞ 0 duJ0(ur)Ai(u) = 0 . (4.5) We can make the ansatz Ai(u) = 2u π ∫ Ri 0 dt cos(ut)fi(t) , (4.6) so that the conditions in (4.5) are automatically satisfied, and the conditions (4.4) become fi(r) + ∑ j 6=i ∫ Rj 0 dx K̄ij(x, r)fj(x) = gi(r) , (4.7) 70 Chapter 4. General LM Solutions and PWMM Instantons where K̄ij(x, r) = |di − dj | π [ 1 (x+ r)2 + |di − dj |2 + 1 (x− r)2 + |di − dj |2 ] , (4.8) gi(r) = βi−2r2, and βi = ∆i+2d2i . Since the gi are all symmetric functions, it is simpler to take the system as fi(r) + ∑ j 6=i ∫ Rj −Rj dxKij(x, r)fj(x) = gi(r) , (4.9) where Kij(x, r) = 1 π |di − dj | (x− r)2 + |di − dj |2 . (4.10) It is straightforward to show that the charge densities on the discs are given in terms of the fi as σi(r) = W0 π2   fi(Ri)√ R2i − r2 − ∫ Ri r du f ′i(u)√ u2 − r2   , (4.11) and that the total charges are Qi = W0 π ∫ Ri −Ri du fi(u) . (4.12) To find the fi we must solve the set of linear equations in (4.9), schematically this takes the form  1 K12 · · · K21 1 · · · ... ... . . .     f1 f2 ...   =   β1 − 2r2 β2 − 2r2 ...   . (4.13) Due to the complicated form of the kernels Kij this system is not easy to solve analytically, however, it is straightforward to solve it numerically using the Nyström method (see, e.g. [70]). This consists of discretizing the interval and solving the resulting linear system. An additional set of constraints comes from ensuring that the charge densities vanish at the edges of the discs. This amounts to enforcing that fi(Ri) = 0. We define f (j) i as the set of solutions to (4.9) with gi(r) = δ j i , where if there are N discs j = 1, . . . , N , 71 Chapter 4. General LM Solutions and PWMM Instantons and f (0) i as the set with gi(r) = 2r 2. The condition that the charge density vanishes at the edge of the disc is then that∑ j f (j) i (Ri)βj = f (0) i (Ri) . (4.14) There will be a unique solution for βi if det(f (j) i (Ri)) 6= 0. The full solution fi is then fi(r) = −f (0)i (r) + ∑ j βjf (j) i (r) , (4.15) with the potentials φi given by φi = ∫ Ri −Ri dtGi(r, z, t)fi(t) , (4.16) where Gi(r, z, t) = 1 π 1√ (|z − di|+ it)2 + r2 . (4.17) We have therefore reduced the electrostatics problem to a very simple linear system. In the case that there are only two discs, the problem is very straightforward and solution has been used to understand the relationship between SYM on R× S2 and Little String Theory [2]. 4.2.2 General Solutions Dual to the PWMM Another theory for which Lin and Maldacena found the corresponding elec- trostatics configurations is the Plane-Wave Matrix Model [30]. The Plane- Wave Matrix Model can be found by a consistent truncation of N = 4 SYM on R × S3 to the set of constant modes on the sphere [34]. The vacua of matrix model are given by the scalars that come from the former N = 4 gauge field taking values in a representation of SU(2) [30]. Lin and Malda- cena [40] associated these vacua with configurations of charged conducting discs above an infinite conducting plane. The method of solution is very similar to the case above. For the sake of brevity we will give the final solution. We will write the potential as V = V0 ( r2z − 2 3 z3 + ∑ i φi(r, z) ) , (4.18) where the first two terms are the background field and the φi arise from the charged discs as φi(r, z) = ∫ Ri −Ri dtGi(r, z, t)fi(t) . (4.19) 72 Chapter 4. General LM Solutions and PWMM Instantons The Green function is Gi(r, z, t) = 1 π ( 1√ (|z − di|+ it)2 + r2 − 1√ (|z + di|+ it)2 + r2 ) , (4.20) and fi is a solution of the integral equation fi(r) + ∑ j ∫ Rj −Rj dxKij(r, x)fj(x) = gi(r) , (4.21) in which the kernel is given by Kij(x, r) = 1 π [ |di − dj | (x− r)2 + |di − dj |2 − |di + dj | (x− r)2 + |di + dj |2 ] , (4.22) and gi(r) = βi − 2dir2, where βi = ∆i + 23d3i . The differences between this solution and the one presented in section 4.2.1 arise from the presence of the infinite conducting plane that, via the method of images, implies the presence of oppositely charged conducting discs below the image plane. However, the conditions on the charges on the discs (4.11),(4.12), still hold, so the requirement that the charge density vanishes at the edge of the disc is that fi(Ri) = 0. We may again consider solutions to (4.21) in which gi(r) = δ j i , which we will call f (j) i , and f (0) i for which gi(r) = 2dir 2. The condition that the charge density vanishes at the edge of each disc is again (4.14), and fi will be given by (4.15). As in the case of SYM on R × S2, the electrostatics problem has been reduced to a very simple linear system. We will now show how we can solve this system to study instantons at strong coupling on the field theory side using this method. 4.3 Instanton Calculations Recently, Lin [72] has considered tunnelling between vacua in the Plane- Wave Matrix Modeland in maximally supersymmetric YM on R× S2. It is possible to study this on both the gauge theory and gravity sides. In the gauge theory case, this can be approached by directly studying the instanton solutions [73]. Lin [72] has also shown that it is possible to introduce a superpotential that gives a bound for the instanton action according to Sinst = − 1 g2 ∆W . (4.23) 73 Chapter 4. General LM Solutions and PWMM Instantons N5N5 N2 N2 − 1 Figure 4.2: The Young diagrams associated with the initial and final vacua in the Plane-Wave Matrix Model, and the electrostatics problems for the dual supergravity solutions. Lin [72] further studied this on the gravity side. Explicit answers were given in the case that the discs in the corresponding electrostatics problem were small and could be approximated by point charges. Moreover, Lin [72] gave a prescription for how the instanton action could be expressed in terms of the electrostatics potential in more general cases, but was not able to give explicit expressions. For completeness, we will briefly review Lin’s prescription for finding the instanton action from the gravity side [72], and then we will demonstrate the use of the techniques developed above to calculate the instanton action for some non-trivial electrostatics configurations. 4.3.1 Instantons on the Gravity Side Since Lin and Maldacena [40] have found electrostatics configurations cor- responding to SU(2|4) symmetric field theory vacua, it is interesting to un- derstand how instantons in the field theories can be described in the gravity picture. Lin [72] has studied this question by first considering vacua in the field theories for which the electrostatics configurations do not differ dras- tically. These instantons can be addressed by calculating the action for a Euclidean D2-brane wrapping a non-contractible Σ3 in the geometry. As discussed in [40], since the brane will be wrapping a cycle carrying some N5 units of flux, it should have N5 D0-branes ending on it, and therefore describe the creation of N5 D0-branes in the throat, see figure 4.2. Using the mapping to an electrostatics configuration, the action for the 74 Chapter 4. General LM Solutions and PWMM Instantons Euclidean D2-brane can be expressed in terms of the solution to the electro- statics problem. Consider the case of a charged conducting disc at a position z0 above an infinite conducting plane, as shown in figure 4.2. Lin [72] has shown that the action for such a configuration takes the form SE = − 2 π [V (z0)− V (0)− z0V ′(0)] , (4.24) where V is the electrostatics potential evaluated along r = 0, and prime denotes differentiation with respect to z. This expression, however, is pro- portional to the change in energy of the electrostatics configuration, SE = 8∆U/π3. This led Lin [72] to identify the superpotential at strong coupling as W ≡ −8g 2 π3 U = −16g 2 π3 ∑ i QiVi , (4.25) where U is the energy of the electrostatics configuration. In the case that the discs are small relative to their separation, which is at weak coupling in the gauge theory, the superpotential is given by the energy of a system of point charges. Using the prescription (4.25), Lin found [72] W = 1 3 ∑ i N (i) 2 N (i)3 5 , (4.26) in perfect agreement with the weak coupling gauge theory results [72, 73]. In the case that the coupling is not weak, the charges arrange to form extended discs. We will find the superpotential at strong coupling by solving the electrostatics problem for a set of extended discs. 4.3.2 Instantons in the PWMM In this section we will consider instantons in the simplest non trivial electro- statics configuration, that of a single conducting disc carrying π2N2/8 units of charge at a distance πN5/2 above a conducting plane. This corresponds to a field theory vacuum with N2 copies of the N5 dimensional representa- tion. We will determine the superpotential for arbitrary N2 and N5, and calculate the action for Euclidean D2-brane wrapping the non-contractible Σ3. In the case of small changes to the background mediated by the Euclidean D2-brane, this will compute the action between a vacuum of the Plane-Wave Matrix Model with N2 copies of the N5 dimensional representation, and a vacuum with N2 − 1 copies of the N5 dimensional representation and N5 copies of the trivial representation. See figure 4.2. 75 Chapter 4. General LM Solutions and PWMM Instantons The electrostatics problem in this case can be approached using the technique outlined in section 4.2.2 applied to the case of a single disc above the conducting plane. We consider a disc of radius R, at a distance d from the plane, which is a generalization of the approach in [1]. The solution to the electrostatics problem will be V = V0 ( r2z − 2 3 z3 + φ ) , (4.27) where φ is φ(r, z) = ∫ R −R dtG(r, z, t)f(t) , (4.28) with Green function G(r, z, t) = 1 π ( 1√ (|z − d|2 + it)2 + r2) − 1√ (|z + d|2 + it)2 + r2) ) . (4.29) Here f satisfies the integral equation f(r) + ∫ R −R dxK(r, x)f(x) = g(r) , (4.30) with g(r) = β − 2dr2, β = ∆+ 23d3, and kernel K(r, x) = − 1 π 2d (x− r)2 + 4d2 . (4.31) We will solve the problem by finding a numerical solution to the integral equation (4.30). Solving this integral equation is straightforward. As a check on our nu- merical results, we ensured that the asymptotic form for the superpotential in the limit that the number of units of charge on the discs was small is given by (4.26). Indeed, we found that for N5 ≫ λ 13 ≫ 1 26 W ≈ 1 3 N2N 3 5 + a √ λN2N 3 2 5 , (4.32) where the numerical constant a ≈ 1.4. The solution of the electrostatics problem when the discs are large gives the superpotential at strong coupling. A plot of the superpotential for N5 = 26Here λ ≡ g2N2, where g is the Yang-Mills coupling of the matrix model. 76 Chapter 4. General LM Solutions and PWMM Instantons 105 106 107 108 100 101 102 103 104 105 106 107 108 109 W /N 2 λ √ 2λN 3 2 5 (1−N − 1 2 5 ) 1 3N 3 5 Figure 4.3: The superpotential for N5 = 100. The dashed lines indicate the asymptotic values for small and large λ compared to N5. 100 is given in figure 4.3. It is possible to extract the asymptotic form for the superpotential in the limit that λ 1 4 ≫ N5 ≫ 1. We find the result W ≈ b √ λN2N 3 2 5 (1−N − 1 2 5 ) + cλ 1 4N2N 2 5 (1−N − 3 4 5 ) , (4.33) where the numerical constants b ≈ 1.4142 ≈ √2, and c ≈ 0.7. In defining the superpotential, there was the freedom to choose an overall constant factor. Here we have defined the superpotential to be zero for the vacua given by N2N5 copies of the trivial representation. We can also use the electrostatics solution to determine the action ac- cording to (4.24) for the instanton shown in figure 4.2. A plot of the result for N5 = 100 is shown in figure 4.4. When λ is small compared to N5, SE ≈ 1 3 N2N 3 5 λ , (4.34) and it falls off faster than any power of λ when λ is large. When λ is small compared to N5, the potential on the disc in the electrostatics problem is negative, which is due to the form of the background potential. As the size of the disc increases the potential on the disc increases. The instanton action 77 Chapter 4. General LM Solutions and PWMM Instantons 10−10 10−8 10−6 10−4 10−2 100 102 105 106 107 108 109 1010 1011 S E /N 2 λ N35 /(3λ) Figure 4.4: The action for a euclidean D2 brane for the situation shown in figure 4.2. Here N5 = 100. The dashed line shows the asymptotic behaviour for λ small compared to N5. 78 Chapter 4. General LM Solutions and PWMM Instantons according to (4.24) begins to fall off from the behaviour at weak coupling near where the potential on the disc crosses zero. It is sensible that it should vanish when the coupling is infinite, since in that case we would expect the electric field to become constant between the discs near the origin, and so the potential difference and dipole contributions should cancel out. Let us briefly mention when these results should be valid. The euclidean brane approximation will be valid when the number of units of charge on the disc is large, the potential from the dipole at the origin is small at the surface of the disc, and the curvature is small. Therefore λ, N2 and N5 must all be large. It would be helpful to better understand the scaling behaviour that was found in (4.32) and (4.33). We have taken the normalization of the super- potential thus far to allow for direct comparison of vacua with N2 copies of the N5 dimensional representation to ones with N2N5 copies of the trivial representation. The asymptotic behaviours we found in (4.32) and (4.33) using this prescription have some interesting features. In particular, at a fixed order in λ, we found that the coefficients for the subleading terms in N5 are one (i.e. the factors of (1 − N−α5 ), where α is some positive num- ber). The reason for this is as follows. If we took the superpotential to be normalized to zero for the empty background instead, it would have been advantageous to take out a further scaling factor of N35 ∼ d3 in (4.27). In that case, the potential would be of the form V = V0N 3 5 V̄ (R/d). Likewise, the charge on the disc would then have the form Q = V0N 4 5 q̄(R/d). Since the total charge is proportional to N2, we must have that Q ∼ N2 ∼ N 4 5 g2 q̄(R/d) , (4.35) or q̄(R/d) ∼ g 2N2 N45 = λ N45 . (4.36) We see, then, that functions of R/d in the scaled electrostatics problem can only depend on the combination of gauge theory parameters λ/N45 . Therefore the superpotential with this alternative normalization must have the form W̄ (λ,N2, N5) = N2N 3 5 w̄ ( λ N45 ) . (4.37) Our numerical results confirm this. We find the following asymptotic be- 79 Chapter 4. General LM Solutions and PWMM Instantons haviour for w̄: w̄(x) ≈ 1 3 ( 1− 10 3 x 2 3 ) , x≪ 1 , (4.38) w̄(x) ≈ − √ 2x 1 2 + 0.7x 1 4 , x≫ 1 . The superpotential with the original normalization is given in terms of W̄ by W (λ,N2, N5) = W̄ (λ,N2, N5)− W̄ (λN5, N2N5, 1) . (4.39) If we combine the contributions from the asymptotic behaviour of each of the terms in this expression that come from the behaviour found in (4.38), then we will recover the asymptotic behaviour that was found in (4.32) and (4.33). The factors of (1−N−α5 ) occur as a result of those combinations. 4.4 Discussion In this chapter we have given a prescription for finding the supergravity solu- tions dual to general vacua of the Plane-Wave Matrix Model and maximally supersymmetric YM on R×S2 by using the mapping of Lin and Maldacena [40] on to axisymmetric electrostatics problems. The prescription extends the technique developed in the previous two chapters and [1, 2] to arbitrary electrostatics configurations. The electro- statics problems are reduced to a set of integral equations that can be solved quite straightforwardly using the Nyström method. We have shown that an application of the prescription to a specific case can be used to study instantons at strong coupling in the Plane-Wave Ma- trix Model. In particular we found that the instanton action for a transition between a vacuum described by N2 copies of the N5 dimensional represen- tation and one by N2 − 1 copies of the N5 dimensional representation and N5 copies of the trivial representation falls off faster than any power of λ at strong coupling (see figure 4.4). We also found that at strong coupling the superpotential for a vacuum with N2 copies of the N5 dimensional rep- resentation behaves like √ 2λN2N 3/2 5 , when N5 is large (see figure 4.3). This demonstrates that the techniques developed above are useful for obtaining strong coupling results in the field theory. One question that would be interesting to address using this method is to calculate the superpotential explicitly for more general electrostatics configurations. For example, studying the vacuum of maximally supersym- metric Yang-Mills theory on R×S2 in which Φ = diag(n, . . . , n,−n, . . . ,−n), 80 Chapter 4. General LM Solutions and PWMM Instantons a similar scaling could be applied as was done for the Plane-Wave Matrix Model. In that case we would expect that the corrections to the weak cou- pling results given by Lin [72] would depend on the parameter λ/n3. It would certainly be interesting to study that case in detail, as well as other more general vacua. One open question is to prove that requiring the charge density vanish at the edge of each disc implies that there is a unique solution to the electro- statics problem. The condition for a unique solution to exist is given above by requiring det(f (j) i (Ri)) 6= 0. We have not been able to prove that this is true in general. It would be interesting to do so. Finally, it would be very interesting to use this method for finding the dual geometry to study other strong coupling phenomena on the gauge the- ory side. 81 Chapter 5 Coarse-Graining the Lin-Maldacena Solutions 5.1 Introduction In the preceding chapters we have seen a class of examples of gauge/gravity duality in which the field theories have discrete, degenerate bases of vacuum states, each of which corresponds to a non-singular geometry. The field theories are those we have met in the the foregoing chapters: the Plane-Wave Matrix Model [30], maximally supersymmetric YM on R × S2 [39], and N = 4 SYM on R × S3/Zk [40]. The theories have degenerate vacua and their SU(2|4) supersymmetry may be used to argue that the classical vacua are degenerate in the quantum theory and are present at strong coupling [36]. Lin and Maldacena [40] looked for dual supergravity solutions and found solutions in one-to-one correspondence with the natural bases of vacuum states in the field theories. 27 Most of the discussion in this chapter will concentrate on the Plane-Wave Matrix Model, however in 5.6 we will generalize our analysis to the other theories. We have seen so far that although the geometries corresponding the ba- sis of vacuum states all share the same asymptotics. In the Plane-Wave Matrix Model, for example, they are asymptotically the near-horizon D0 brane solutions. In the infrared region, however, they differ greatly, even in their topology: they contain different sets of 3- and 6-cycles carrying dif- ferent amounts of flux. Since the generic vacuum state in the field theory is a linear superposition of basis elements, such a state cannot be dual to a single non-singular supergravity solution with fixed topology (assuming there are observables that can detect topology), but must simply be dual to a quantum superposition of the topologically different geometries. Simi- 27The construction is completely analogous to the construction of gravity duals to half BPS states of N = 4 SUSY Yang-Mills theory [42]. As in that case, the smooth supergrav- ity solutions corresponding to particular states can have large curvatures, and thus are only approximations to the true dual geometries which should minimize the α′-corrected low-energy effective action. 82 Chapter 5. Coarse-Graining the LM Solutions larly, generic mixed states in the field theory, such as the zero-temperature limit of the thermal state, involve microstates corresponding to many dif- ferent topologies so we might expect that a gravitational dual description in terms of a single geometry is impossible. On the other hand, there are many examples of geometries believed to be dual to thermal states of field theories, and these thermal states involve enormous numbers of microstates that can be very different in the infrared. Mathur and collaborators have advocated (see [74] for a review) that we should interpret the thermal state geometry as a coarse-grained description of the underlying microstates, just as the homogeneous magnetization that we use to describe the thermal state of a spin system is a coarse-grained description of the true microstates of the spins. Specifically, the macroscopic description of almost any state in the underlying ensemble of microstates is extremely close to one partic- ular coarse-grained configuration, the thermal equilibrium state. We will see that in our case also, there is a natural way to coarse-grain (i.e. give a macroscopic description of) geometries corresponding to typical microstates, and that most of the microstates have a coarse-grained description that is very close to a particular geometry, which we propose is the correct dual to the zero temperature limit of the thermal state. In this geometry, the complicated topological features that distinguish the individual microstate geometries are replaced by a singularity.28 The details of the coarse graining procedure are described in section 5.3 below, but we give the essential idea here. The supergravity fields in the Lin-Maldacena geometries are determined in terms of the potential for an ax- ially symmetric electrostatics problem involving a certain number of parallel coaxial charged conducting discs in a background electric field. The number, locations and charges of the discs are determined by the data specifying the field theory vacuum.29 We will find that typical field theory vacua corre- spond to electrostatics configurations with a large number of closely spaced discs whose radii are very small compared with the separation between the discs. At large N , such a configuration has a natural coarse-grained de- scription as a smoothly varying charge distribution on the axis. Inserting the potential arising from this coarse-grained configuration into the Lin- Maldacena supergravity solution, one finds a singular geometry. Since all of the nontrivial topological features are associated with the regions between the discs in the electrostatics configurations (these regions map to topo- 28For a general discussion of conditions under which field theory states can be associated with semiclassical geometries, see [75] in the LLM context and [76] in the D1-D5 context. 29The radii of the discs are determined by the other information via a constraint. 83 Chapter 5. Coarse-Graining the LM Solutions logically non-trivial throats in the supergravity solutions), we see that the complicated topologies that characterize individual microstates are replaced by a singularity in the coarse-grained description.30 A completely analogous coarse-graining has been discussed [77–80] for the half-BPS sector of N = 4 SUSY Yang-Mills theory. There, the mi- crostate geometries are the type IIB LLM geometries [42], constructed in terms of droplets of a two-dimensional incompressible fluid, and the coarse- grained description allows for configurations with arbitrary density of the fluid between zero and the maximal density. One significant difference is that all of the states we consider are ground states for the field theory, whereas the LLM discussion relates to a special class of excited states with energy equal to an R-charge. As emphasized in [77], a given coarse-grained configuration provides an approximation to a very large number of microstates, just as in the ther- modynamic description of ordinary physical systems. Further, there is one preferred coarse-grained configuration, analogous to the thermal equilibrium state, which is very close to the coarse-grained description of almost any ran- domly chosen microstate. For the type IIB LLM geometries, the geometry corresponding to this preferred state was determined in [77] and dubbed the “hyperstar” geometry. In section 5.3 of this chapter, we determine the corresponding geometry for the Plane-Wave Matrix Model. In our case, the ensemble of microstates we consider is just the set of vacuum states, or al- ternately the set of states that contribute (each with equal weight) to the T → 0 thermal state density matrix. Thus, we propose that our preferred geometry is the T → 0 limit of the geometry dual to the thermal state of the field theory. In section 5.5, we also derive geometries corresponding to the preferred states in other restricted ensembles, analogous to the type IIB superstar [81], and discuss thermal geometries for the remaining SU(2|4) symmetric theories in section 5.6. As for an ordinary thermodynamic system, the thermal states we de- rive should maximize an entropy functional that measures the number of microstates nearby an arbitrary coarse-grained configuration. In section 5.4, we derive such an entropy functional, and find that it may be written simply in terms of the data that specify the geometry. We find that this functional is indeed maximized by the thermal state geometry of section 5.4. Further, we note that for all the coarse-grained configurations, those 30We should note however, that for the case of closely spaced discs, the supergravity approximation is not valid for the region between the discs, so the classical topological features that we are discussing should be understood to be replaced by some stringy analogue. 84 Chapter 5. Coarse-Graining the LM Solutions for which the entropy functional vanishes are the ones that coincide the orig- inal non-singular microstate geometries. On the other hand, configurations with non-zero entropy are necessarily singular. In the general proposal by Mathur and collaborators, black hole geome- tries with horizons are to be understood as coarse-grained descriptions of underlying horizon-free microstate geometries. In the present setup, the coarse graining leads to geometries with naked singularities uncloaked by horizons, but this is to be expected since the number of microstates in our case is not large enough to give a classical finite-area horizon in the su- pergravity limit. It may be that a horizon develops as we move from the supergravity approximation to solutions minimizing the full low-energy ef- fective action, but, as we will see, realizing this would necessarily involve understanding both α′ and string loop corrections. 5.2 The SU(2|4) Symmetric Matrix Quantum Mechanics and the Dual Lin-Maldacena Geometries In this section, we review the Plane-Wave Matrix Model, its vacua, and the dual geometries constructed by Lin and Maldacena. The other SU(2|4) symmetric field theories are discussed in section 5.6. We will see that each of these theories has a large degeneracy of vacuum states at the classical level. This degeneracy remains at the quantum level, since the represen- tation theory of SU(2|4) does not allow for states with arbitrarily small non-zero energies, and therefore does not allow the zero-energy states in the classical limit of the theory to receive corrections to their energy [36, 59]. 5.2.1 The Plane-Wave Matrix Model The Plane-Wave Matrix Model [30] is a massive deformation of the su- persymmetric matrix quantum mechanics describing decoupled low-energy D0-branes in flat space.31 It is described by a dimensionless Hamiltonian H = Tr ( 1 2 P 2A + 1 2 (Xi/3) 2 + 1 2 (Xa/6) 2 + i 8 Ψ⊤γ123Ψ + i 3 gǫijkXiXjXk − g 2 Ψ⊤γA[XA,Ψ]− g 2 4 [XA,XB ] 2 ) ,(5.1) 31This is similar to the Polchinski-Strassler deformation of N = 4 SUSY Yang-Mills theory [58], but in this case, we preserve all 32 supersymmetries. 85 Chapter 5. Coarse-Graining the LM Solutions where A = 1, . . . , 9, i = 1, . . . , 3, and a = 4, . . . , 9. Here, the scalars XA and 16-component fermions Ψ are hermitian N × N matrices, and PA is the matrix of canonically conjugate momenta. Apart from N , the size of the matrices, the theory has one dimensionless parameter g, such that the theory is weakly coupled for small enough g.32 For this theory, the classical vacua, each with zero energy, are described by Xa = 0 a = 4, . . . , 9 Xi = 1 3g J i i = 1, 2, 3, where J i give any reducible representation of the SU(2) algebra. These vacua are in one-to-one correspondence with partitions of N , since we may have in general nk copies of the k-dimensional irreducible representation such that ∑ k knk = N . Below, it will be convenient to represent such a partition by a Young diagram with N boxes, containing nk columns of length k. In the D0-brane picture, a block-diagonal configuration with nk copies of the k-dimensional irreducible representation is associated classically with concentric D2-brane fuzzy spheres, with nk spheres at radius proportional to k. On the other hand, it was argued in [39] that at sufficiently strong coupling, such a configuration is better described as a collection of concentric fivebranes, with multiplicities and radii given in terms of the numbers and lengths of columns in the dual Young diagram.33 For general values of parameters, we can interpret the solution as a fuzzy configuration with both D2-brane and NS5-brane characteristics. This will be apparent from the dual gravitational solutions, which include throats carrying D2-brane flux and throats carrying NS5-brane flux in the infrared part of the geometry. 5.2.2 Electrostatics The vacua of the matrix model each preserve SU(2|4) symmetry. In [40], Lin and Maldacena searched for type IIA supergravity solutions preserving the same SU(2|4) symmetry (more precisely, with isometries given by the bosonic subgroup SO(6)×SO(3)×U(1) of SU(2|4)). Using an ansatz with this symmetry (reproduced in B.1), they were able to reduce the problem of 32The model was introduced originally as a matrix model for M-theory on the maximally supersymmetric eleven-dimensional plane-wave. For this we are required to take a limit N →∞ with g2N ∼ N4. In the present work, we will mainly be concerned with the usual ’t Hooft large N limit with λ fixed. 33In [39], the matrix model was discussed in the context of its conjectured description of M-theory on a plane-wave background. There, the fivebranes were M5-branes, while here we are considering a limit with fixed λ, dual to a IIA background, so the fivebranes are NS5 branes. 86 Chapter 5. Coarse-Graining the LM Solutions finding supergravity solutions to the problem of finding axially-symmetric solutions to the three-dimensional Laplace equation, with boundary condi- tions involving parallel charged conducting discs and a specified background potential. Corresponding to each classical vacuum and choice of parame- ters, we have a specific electrostatics problem, whose solution (a potential V (r, z)) feeds into the equations (B.1) to give the dual supergravity solu- tion. Further, the smooth supergravity solutions for which fluxes through non-contractible cycles are quantized appropriately are in one-to-one corre- spondence with the vacua. For the other SU(2|4) symmetric theories described in section 5.6, the construction differs only by a choice of boundary conditions (background potential or the presence/absence of infinite-sized conducting plates). The solution to these electrostatics problems has been discussed in [3]. We now describe the electrostatics problem in detail and then review some general features of the dual supergravity solutions. Common to all vacua, we have in the electrostatics problem an infinite conducting plate at z = 0 (on which we may assume that the potential vanishes), and a background potential V∞ = V0(r2z − 2 3 z3) . (5.2) In addition, corresponding to a matrix model vacuum with Qi copies of the di-dimensional irreducible representation, we have conducting discs with charge Qi parallel to the infinite plate and centred at r = 0, z = di. 34 In order that the supergravity solution is non-singular, the radii Ri of the discs must be chosen so that the charge density at the edge vanishes. The parameters of the matrix model are related to the parameters in the electrostatics problem as N = ∑ Qidi and g 2 ∝ 1/V0. 5.2.3 Gravity Duals The coordinates r and z in the electrostatics problem form two of the nine spatial coordinates in the geometry. In addition, for each value of r and z, we have an S2 and an S5 with radii that depend on (r, z). The S5 shrinks to zero size on the r = 0 axis, while the S2 shrinks to zero size at the locations of the conducting plates, so we have various non-contractible S3s and S6s corresponding to paths that terminate on different plates or on different segments of the vertical axis respectively. This is illustrated in figure 2.1. As shown in [40], through an S6 corresponding to a path surrounding plates with a total charge of Q, we have N2 = Q units of flux from the dual of 34Our conventions here are slightly different from the ones in [40] 87 Chapter 5. Coarse-Graining the LM Solutions the Ramond-Ramond four-form, suggesting the presence of N2 D2-branes. Similarly, through an S3 corresponding to a path between plates separated by a distance d, we have N5 = d units of H-flux, suggesting that this part of the geometry between the plates is describing the degrees of freedom of N5 NS5-branes. Since the matrix model is a massive deformation of the maximally su- persymmetric quantum mechanics describing low-energy D0-branes in flat- space, we should expect that the dual supergravity solutions correspond to infrared modifications of the near-horizon D0-brane geometry. Indeed, the solutions are asymptotically the same as the near-horizon D0-brane solu- tion, with the strong-coupling region in the infrared replaced by smooth topological features that depend on the choice of vacuum. 5.3 Coarse-Graining the Lin-Maldacena Geometries For large N , the Plane-Wave Matrix Model has of order exp( √ 6N/π) inde- pendent vacua labelled by reducible dimension N representations of SU(2). In this section, we will argue that as for standard thermodynamic systems (e.g. particles in a box), if we use coarse-grained, macroscopic variables to describe the states, then despite the large number of possible microscopic states, the description of a randomly chosen microstate will, with very high probability, be extremely close to the average or “thermal equilibrium” state. We will see explicitly what the coarse-grained description of this average state is in our case, and see that there is a natural way to associate a ge- ometry to this (and more general) coarse-grained configurations. We will interpret the resulting geometry as the zero-temperature limit of the ther- mal state, since this state has a density matrix with equal contributions from each basis vacuum state. Much of the discussion in this section follows ideas in [77] for the LLM geometries. 5.3.1 Macroscopic Variables We begin by understanding the macroscopic variables appropriate in our case. As we will see, typical gauge theory states for large N will correspond to electrostatics configurations with large numbers of charged discs at unit separation. The microstate configurations are specified by giving the (inte- ger) charge at each discrete location on the vertical axis. Since the extent of the disc configurations on this axis will be much larger than the disc 88 Chapter 5. Coarse-Graining the LM Solutions separations (typically by a factor of √ N as we will see), it is sensible to characterize configurations by a macroscopic charge density Q(z). This, we can define by averaging the microscopic charge over a distance much larger than the disc separations, but much smaller than the vertical extent of the disc configuration. Thus, in the coarse-grained description of states, Q(z) should be a smooth function. We still need to understand how the charge Q(z) should be arranged in the directions perpendicular to z (recall that for the microstates it spreads out dynamically on the charged conducting discs), but first it will be helpful to see what Q(z) looks like for typical states. 5.3.2 Typical States In the microscopic description, the charges Qn at position z = n label how many times the irreducible representation of dimension n appears, and are subject to the constraint ∞∑ n=1 nQn = N . (5.3) We would now like to ask what a typical randomly chosen representation looks like. To do this, we first note that the independent vacuum states of the matrix model are in one-to-one correspondence with the quantum states of a free massless boson on an interval (a.k.a. a quantum guitar string) with energy E−E0 = ~ωN , where ω is the frequency of the lowest mode. In this analogy, Qn give the number of particles of frequency nω. For largeN , where the energy and number of particles are large, we know that a thermodynamic description is appropriate, and that any macroscopic quantities evaluated for a randomly chosen microstate are extremely likely to be extremely close to the average values. For our discussion, we will be interested in the average coarse-grained charge distribution defined above, so we start by computing the expected value of Qn for each n. This is equivalent to calculating the expected particle numbers for our gas of free bosons in the microcanonical ensemble at energy E = N (setting ~ = ω = 1). For large N , this should agree up to tiny corrections with the result as computed in the canonical ensemble, so long as we choose the temperature such that the expected value of the energy is N . The calculation is much simpler in the canonical ensemble, since now we can sum over all states without a constraint. 89 Chapter 5. Coarse-Graining the LM Solutions To study the canonical ensemble, we write a partition function [82] Z = ∑ Qn e−β P nQn = ∏ n ∑ Qn e−βnQn = ∏ n 1 1− e−βn . (5.4) From this, the expectation value of Qn is found (for example by changing the β in front of Qn to α, differentiating ln(Z) with respect to −αn, and setting α = β) to be 〈Qn〉 = 1 eβn − 1 . (5.5) The expected value of energy is 〈N〉 = −∂β ln(Z) = ∑ n n eβn − 1 ≈ π 2 6β2 , where the last line assumes that the sum can be approximated by an integral (valid for large N). Solving for β in terms of N and plugging in to (5.5), we find 〈Qn〉 = 1 e pin√ 6N − 1 . (5.6) Thus, the coarse-grained approximation to a typical microstate will have a linear charge density very close to 〈Q(z)〉 = 1 e piz√ 6N − 1 . (5.7) Or, defining x = z/ √ N and √ Nq(x) to be the charge density in terms of x, we have 〈q(x)〉 = 1 e pix√ 6 − 1 . (5.8) 5.3.3 Supergravity Solution for the Average State We would now like to understand the supergravity solution corresponding to the average coarse-grained configuration we have found. To do this, we first need to understand precisely how the charge Q(z) should be distributed 90 Chapter 5. Coarse-Graining the LM Solutions in the horizontal directions. For the microstates, the actual distribution of charge is determined dynamically, since the charges are free to move on conducting discs whose radii are determined by the constraint that the charge density at the edge vanishes. However, we will now see that the typical configurations for large N with fixed λ have discs whose radii are much smaller than the separation between the discs. Thus, in the coarse- grained picture for typical states, we can take the charge distribution to sit on the vertical axis. To understand how large the discs should be, we note that for the mi- crostates, having conducting discs with the correct radii is necessary in order to avoid singularities in the supergravity solution. If we simply place all the charge on the axis, singularities should appear (wherever ∂rV = 0). These cannot be at radii much larger than the original radii of the discs, since at these large radii, the electrostatics potential should be modified only slightly when we move all the charge to the axis. Thus, the distance scale defined by the sizes of the discs should be the same as the typical coordinate distance from the axis where singularities appear in the modified configuration. We will now use this to estimate the radii of the discs for the typical configura- tions. For a charge distribution Q(z) on the vertical axis, the corresponding potential will be given by [82] V (r, z) =V0 ( r2z − 2 3 z3 ) + ∫ ∞ 0 dz′Q(z′) { 1√ r2 + (z − z′)2 − 1√ r2 + (z + z′)2 } , (5.9) where the second term arises from the image charges below the infinite conducting plate. It is straightforward to check that such a potential for smooth Q(z) always gives rise to a singular supergravity solution [82]. The singularity appears at the locus of points where the radial component of the electric field vanishes [40]. To estimate this radius, we note that for slowly varying Q(z), the radial electric field near the axis is given by Er(r) = −2rzV0 + 2Q(z) r , so the singularity is located at35 r = √ Q(z) zV0 . (5.10) 35This should be a good approximation so long as r is small compared with Q/Q′. 91 Chapter 5. Coarse-Graining the LM Solutions From (5.7), we see that for z of order √ N , the typical value of the charge on each disc is of order one, while for z of order one, the typical charge is of order √ N . Recalling that V0 ∼ 1/g2, we estimate that the typical radii of the discs will be r ∼ √ λ N 3 2 z = O( √ N) , r ∼ √ λ N 1 2 z = O(1) . In either case, for large N and fixed λ the typical radii go to zero. Thus, in the coarse-grained description of typical states in the ’t Hooft limit, we can take all the charge to be located on the z-axis. This leads us to the following conclusion: the geometry dual to the T = 0 thermal state of the Plane-Wave Matrix Model at large N is given by the Lin-Maldacena solution (B.1), with potential (5.9) determined in terms of the charge distribution (5.7). It may be that for some coordinate choice, the solution takes a simpler, more explicit form, but we have not investigated this. 5.3.4 Coarse-Grained Solutions for Large Discs For large N and fixed λ, we have seen that the typical states have electro- statics configurations for which the discs are small relative to their separa- tions, so that the charge can simply be taken to lie on the vertical axis in the coarse-grained description. However, it is also useful to have a coarse- grained description of states in cases where the radii of the discs is larger than their separations. This is relevant, for example, if we allow λ to scale as a power of N , or for fixed λ in restricted ensembles for which we restrict the number of fivebranes (as in section 5.5). In such cases, the coarse-grained picture will have the closely spaced discs replaced by a uniform material that conducts only in the directions perpendicular to the z-axis. This material will have some smooth profile described by a radius function R(z) and carry charges such that total charge on the conductor between heights z and z + dz is Q(z). Just as the radii of the discs in the original setup are determined by the charges, we should expect that R(z) in the coarse-grained situation will be determined by Q(z). Specifically, it turns out that the shapeR(z) of the conductor must be chosen such that the surface charge density vanishes. This R(z) gives the coordinate location of the singularity in the supergravity solution corresponding to a given coarse-grained Q(z). The details of this coarse-graining procedure 92 Chapter 5. Coarse-Graining the LM Solutions r rz z Figure 5.1: Coarse-graining for large discs. The shaded region represents a solid conductor that conducts only in the horizontal directions. 93 Chapter 5. Coarse-Graining the LM Solutions and the mathematical procedure that determines R(z) in terms of Q(z) are described in appendix D.1. 5.4 An Entropy Functional In thermodynamic systems, we can often associate an entropy with coarse- grained configurations that are more general than the state of thermal equi- librium for the whole system. In this section, we give a functional that associates an entropy to a general coarse-grained Lin-Maldacena geometry and discuss its properties. A similar entropy functional has been derived recently for the LLM geometries in [75, 83]. 5.4.1 A Familiar Example To motivate our definition of an entropy functional for the Lin-Maldacena geometries, we will first concentrate on a more familiar system: a collection of spins on a line. We will work this example out in detail so that the our procedure for the Lin-Maldacena geometries is clear. We will consider a line of independent spins interacting with a magnetic field with a strength scaled so that the partition function can be written Z = ∑ {si} e−β P i si , (5.11) where β is the inverse temperature, and we will take si = ±1. We will break up our system of N spins into macroscopic subsystems of ℓ spins according to where the size of the subsystems are such that 1≪ ℓ≪ N. (5.12) Since the size of each subsystem is large, we may use the canonical ensemble to study them. We find the entropy of each subsystem is given by S = ℓβ eβ − e−β eβ + e−β ℓ log(eβ + e−β), (5.13) and the magnetization (or energy) will be given by M = ℓ eβ − e−β eβ + e−β . (5.14) 94 Chapter 5. Coarse-Graining the LM Solutions We can then write an entropy density for the subsystem s = S/ℓ in terms of the average magnetization m =M/ℓ as s = 1 +m 2 log (1 +m 2 ) + 1−m 2 log (1−m 2 ) (5.15) This gives the total entropy associated with our coarse-grained spin system as S[m] = ∫ dx [ 1 +m 2 log (1 +m 2 ) + 1−m 2 log (1−m 2 )] , (5.16) subject to the constraint that the total magnetization is M = ∫ dxm. (5.17) Maximizing the entropy functional with respect to m, subject to the con- straint, recovers constant m, which is just equivalent to stating that the thermal state has maximal entropy. Our prescription for finding an entropy function is therefore the fol- lowing: Break up the system into macroscopic parts (the sets of spins). Determine the entropy for each part and express it as a function of the coarse-grained variables (the average magnetization or energy) of the part. Add up the entropies of each of the parts to find the total entropy, subject to the constraint that the coarse-grained variables are consistent with globally defined quantities (the total magnetization or energy). 5.4.2 Entropy for Coarse-Grained Matrix Model Vacua Now we move on to the Plane-Wave Matrix Model vacua. In this case, the variable that we use to describe our coarse-grained configurations is the charge density q(x) (recall that we defined x = z/ √ N . Let us now consider the interval [x, x+ dx) as a subsystem of our analogue thermodynamic sys- tem. The charge in this interval, q(x)dx is given as a sum of independent microscopic variables q(x)dx = Qn + · · ·+Qn+l , which are also independent of the variables that determine Q outside the interval. Here n = x √ N and l = dx √ N . We assume that the coarse- graining is over macroscopic distances, in other words that the number l of individual degrees of freedom contributing to Q(x)dx is large. Thus, we 95 Chapter 5. Coarse-Graining the LM Solutions should have 1 ≪ l ≪ n. Now, for the subsystem, we have the partition function Z = n+l∏ k=n 1 1− e−βk . This gives free energy F ≈ lT ln(1− e−nβ) , and energy Ē = 〈nQn + · · · + (n+ l)Qn+l〉 ≈ nl enβ − 1 . The entropy is then S = (E − F )/T = l [ nβ enβ − 1 − ln(1− e −nβ) ] . Note that this is proportional to the size of the interval, so it makes sense to define an entropy density s(z) = S/l or equivalently s(x) = √ NS/l. We would like to express this in terms of the average charge density Q(x) in the interval, given by Q = 〈Qn + · · · +Qn+l〉/dx ≈ √ NĒ/(nl) = √ N 1 enβ − 1 . Solving for β in terms of Q, and substituting into the formula for s, we find s(x) = √ N((q + 1) ln(q + 1)− q ln(q)) , where we have defined q = Q/ √ N . Thus, we can associate to a coarse-grained configuration described by a charge density q(x) an entropy S[q(x)] = √ N ∫ dx[(q + 1) ln(q + 1)− q ln(q)] . (5.18) Allowed vacua of the matrix model are subject to the constraint∫ dxxq(x) = 1 . (5.19) 96 Chapter 5. Coarse-Graining the LM Solutions We can now check that maximizing (5.18) subject to the constraint (5.19) gives the correct result for the charge density. Introducing a Lagrange mul- tiplier for the constraint and varying with respect to q, we find ln(q + 1)− ln(q) + Λx = 0 . This gives q(x) = 1 eΛx − 1 , and enforcing the constraint yields Λ = π/ √ 6 . Thus, we reproduce (5.8). For more general coarse-grained configurations, it is clear from (5.18) that the entropy will be nonzero if there is any interval (x1, x2) for which q(x) is continuous and nonzero. Thus, the only way to have a vanishing entropy functional with a nonzero net charge is to have the charge located at discrete points on the axis such that q(x) is a sum of delta functions, as we have in the microstate configurations.36 In this case, the entropy vanishes since for large q, we have (q + 1) ln(q + 1)− q ln(q) ∼ ln(q) (large q) and ∫ ln(δ(x − a))dx = 0 . Recalling that the D2- and NS5-brane fluxes are quantized properly in the supergravity solutions if and only if the charges are quantized and located at integer values of z, we conclude that the entropy function is zero if and only if q(x) corresponds to a microstate geometry.37 Consequently, all coarse- grained configurations with non-zero entropy correspond to singular super- gravity solutions. Our formula (5.18) gives the entropy as a simple expression in terms of q(x), which in turn directly determines the geometry. In this sense, it is a 36Technically, such a q(x) can only appear as a coarse-grained configuration in the limit where we take the coarse-graining scale to zero. Thus, for any non-zero coarse-graining scale, the entropy will be non-zero for all configurations. 37This is analogous to the statement in the spin system we have considered above that the entropy vanishes if we know the precise microstate of the spin system, m = ±1 everywhere. If we think of the spins as binary digits, this, of course, has to do with the usual notions of information entropy. 97 Chapter 5. Coarse-Graining the LM Solutions geometrical formula for the entropy. We might also ask whether there is any direct relation to a horizon area (or Wald’s generalization [84]) in this case. However, as is typical in examples with a large amount of supersymmetry, the singular coarse-grained geometries that we obtain have no horizons.38 On the other hand, both the curvature and the dilaton diverge at the singu- larities, so the supergravity solution should receive both α′ and string loop corrections. It is possible that the fully corrected solutions have horizons. Following [52], we might hope that an appropriate definition of a stretched horizon around the singularity39 would have area that reproduces the en- tropy (perhaps up to numerical factors). In fact, our setup should provide a very stringent test of any proposed definition of a stretched horizon, if we demand that it correctly reproduces the functional dependence of the entropy on q(x). Unfortunately, as we show in appendix D.2, the necessary location of a stretched horizon whose area would reproduce our entropy is parametrically closer to the singularity than either the radius where the cur- vature becomes large or the radius where the dilaton becomes large. At this scale, it is probably naive to expect that a simple area would reproduce the entropy. 5.5 Other Ensembles The T = 0 thermal solution we have found is analogous to the ‘hyperstar’ geometry of [77], dual to the coarse-grained typical state of N = 4 SUSY Yang-Mills theory on S3 with a U(1) ∈ SO(6) R-charge equal to energy. For that theory, there is a related geometry known as the ‘superstar’ that has been understood as the geometry dual to the equilibrium state in a more restricted ensemble for which the number of D-branes in the spacetime is fixed. There are similar restricted ensembles that are natural to consider in our case. To understand these, we recall that the microstate geometries contain various non-contractible S3 cycles carrying NS5-brane flux and non-contractible S6 cycles carrying D2-brane flux. For a given microstate, there will be some 38It was shown in [40] that the metric components in a general LM geometry B.1 will be continuous and nonzero (except for points on the conducting discs) for all potentials V satisfying the three dimensional Laplace equation. From this it is straightforward to see that the region outside of the coarse-grained conducting discs is causally connected. 39Possible definitions considered in the literature include the locus of points where the curvature becomes strong, where the dilaton becomes strong, where the local temperature equals the Hagedorn temperature, or where microstates begin to differ significantly from each other. 98 Chapter 5. Coarse-Graining the LM Solutions S3 S6 r z Figure 5.2: Example electrostatics configuration showing the non- contractible cycles S3 and S6 carrying the largest amount of NS5-brane and D2-brane flux respectively. 3-cycle in the geometry carrying a maximal number of units N5 of NS5-brane flux and some 6-cycle carrying a maximal number of units N2 of D2-brane flux, as shown in figure 5.2. We loosely refer to N5 and N2 as the number of NS5-branes and D2-branes in the geometry. Just as we understood the typical states in general, we can also ask about the form of the typical states in ensembles where either N2 or N5 or both are fixed. To do this, we note that the total number of units of NS5-brane flux is given by the largest j for which Qj 6= 0, while the number of units of D2-brane flux is given by the total charge ∑ j Qj. If we consider a Young diagram with Qj rows of length j, then N2 and N5 are the total number of rows and columns in the Young diagram respectively. The problem of study- ing typical Young diagrams with a fixed number of rows (or equivalently a fixed number of columns) is precisely the one studied in [77] to understand typical states in the hyperstar ensemble of LLM geometries, while the prob- lem of studying typical Young diagrams with a fixed number of rows and columns is precisely the one studied in [77] to determine the typical configu- rations in the (generalized) superstar ensemble. Thus, we can directly carry over those results to find the q(x). 5.5.1 Fixed N5 For fixed N5, we simply restrict the partition function (5.4) to n ≤ N5. The expected value of Qn is given by the same formula, 〈Qn〉 = 1 eβn − 1 , (5.20) 99 Chapter 5. Coarse-Graining the LM Solutions but now the expected value of N is 〈N〉 = N5∑ n=1 n eβn − 1 ≈ N25 f(βN5) , f(x) ≡ 1 x2 Li2(1− e−x) . Thus, we obtain a charge density Q(z) = 1 e z N5 f−1(N/N2 5 ) − 1 , z ≤ N5 . Note that in the unrestricted ensemble, the typical extent of the charge dis- tribution was of order √ N , so we only have a significant difference from the unrestricted ensemble when N5 is of order √ N or smaller. One interesting case is that where we fix N5 to some large but finite value in the large N limit. In this case, we find β = N5 N , and Q(z) ≈ N N5z . In this case, our estimate (5.10) for the size of the discs gives r ∼ √ λ/N35 , so the discs are large compared to their separations for λ ≫ N35 . In this case, we need to use the methods of appendix D.1 to determine the appropriate coarse-grained geometry. 5.5.2 Fixed N2 or Fixed N2 and N5 For fixed N2 (with either fixed or unrestricted N5), it is simplest to work in a grand canonical ensemble where we introduce a chemical potential for N2 and tune it to get the correct value. We will therefore consider the partition function Z(β, µ) = ∑ Qj e− P (βj+µ)Qj . (5.21) From this, we obtain a charge distribution 〈Q(z)〉 = 1 eβz+µ − 1 , (5.22) where β and µ are fixed by demanding N = 〈∑ j jQj 〉 = N5∑ j=1 j eβj+µ − 1 , (5.23) 100 Chapter 5. Coarse-Graining the LM Solutions as before, and 〈∑ j Qj 〉 = N5∑ j=1 1 eβj+µ − 1 . (5.24) In general, β and µ are complicated functions of N2 and N5, but as pointed out in [77], there is a simple special case where we take β → 0 with fixed µ. This gives the solution in the case where we restrict N2N5 = 2N . In this case, the charge density is constant Q(z) = N2 N5 , 0 ≤ z ≤ N5, and the supergravity solution may be written explicitly in terms of ordinary functions. This case corresponds to a triangular Young diagram, which in the LLM case gives rise to the original superstar geometry. We also get a simple expression for the charge distribution in the case where N2 is large but fixed in the large N limit with N5 unrestricted. In this case, a straightforward calculation gives Q(z) = N22 N5 e −N2 N5 z . 5.6 Higher Dimensional SU(2|4) Symmetric Theories In this chapter thus far, we have focused on the Plane-Wave Matrix Model. However, as we have described in the preceding chapters, Lin and Malda- cena [40] also identified supergravity duals to the vacua of other, higher dimensional, field theories with SU(2|4) supersymmetry. These are the aforementioned maximally supersymmetric YM on R × S2, N = 4 SYM on R×S3/Zk, and type IIA Little String Theory on S5. [1, 39, 40]. Aspects of the relations among these theories and the Plane-Wave Matrix Model have been discussed in [68, 69], and in chapter 1. In this section we will analyze these theories in the same way as we have the Plane-Wave Matrix Model. For the higher-dimensional theories, the construction of dual supergravity solutions differs only in the boundary conditions for the electrostatics problem. The individual microstates are still distinguished by the locations and charges of finite-sized conducting discs, so the coarse-graining procedure and the entropy functional are exactly the same as in the Plane-Wave Matrix Model. 101 Chapter 5. Coarse-Graining the LM Solutions 5.6.1 Maximally Supersymmetric Yang-Mills Theory on R× S2 Field theory We will first consider maximally supersymmetric YM on R × S2. We will briefly recall the relevant aspects of the theory, we have described above. This theory can be derived as a limit of the Plane-Wave Matrix Model [39], or of N = 4 SYM on R× S3/Zk in the limit k →∞ [40]. The field content of this theory is the same as the usual low-energy D2- brane gauge theory, with an SU(N) gauge field together with fermions and seven scalar fields. Six of the scalar fields are associated with the SO(6) R-symmetry of the theory. The remaining one comes from the dimensional reduction when the k → ∞ limit is taken in N = 4 SYM on S3/Zk. We will refer to this scalar as Φ. The vacua of this field theory are given by Φ = −diag(n1, n2, . . . , nN ), and F = dA = Φsin θdθdφ, where the ni are integers, and θ and φ are the usual coordinates on S2. The different vacua of the theory are labelled by the multiplicities of the integers in the vacuum configurations of Φ and F . Supergravity We will now recall the features of the dual gravity solutions, which we have described in the preceding chapters. The supergravity duals to vacua of the maximally supersymmetric YM on R × S2 have isolated sets of finite sized charged conducting discs and the background potential V∞ =W0(r2 − 2z2) . (5.25) A diagram of these solutions was given in figure 3.1. As before, non-singular solutions will have discs with radii Ri chosen so that the charge density vanishes at the edge of each disc. Corresponding to a vacuum with Ni copies of the integer ni will be an electrostatics configuration with discs at positions di = πni/2 carrying charge Qi = π 2Ni/8. The gauge theory parameters are related to the elec- trostatics ones as g2YM ∝ 1/W0, and N = ∑ Ni. In similar fashion to the Plane-Wave Matrix Model case, we can find the potential for the system with coarse-grained charge density Q to be V (r, z) =W0(r 2 − 2z2) + ∫ ∞ −∞ dz′ Q(z′)√ r2 + (z − z′)2 . (5.26) 102 Chapter 5. Coarse-Graining the LM Solutions Typical states As we have described above, the vacua of this theory are labelled by a set of integers and their multiplicities. Since the integers specifying the vacuum can be arbitrarily large (the only restriction is that the sum of multiplicities is N), we have an infinite number of vacua in this case. In the electrostatics picture, this corresponds to the fact that the discs are allowed to sit anywhere on the z-axis, with the only restriction that the total charge isN . As a result, quantities such as the charge at any location will average to zero, and we cannot see any natural way to define a typical configuration in this case for the unrestricted ensemble. On the other hand, we do get a well defined thermal configuration in an ensemble where we fix the number of NS5-branes, as in section 5.5. This corresponds to fixing the separation between the highest and lowest disc. For the SU(N) theory, we should demand also that the sum of integers times their multiplicities is zero, so we end up with a finite set of vacuum states. For coarse-grained typical states, the total charge N will be evenly distributed between the N5 plates, so the coarse-grained charge density will be Q(z) = N N5 , −N5 2 ≤ z ≤ N5 2 . Another way to obtain a non-trivial electrostatics configuration is to recall the definition of this theory as a k → ∞ limit of N = 4 SYM on S3/Zk. If we instead take a limit in which N →∞ and k →∞ with N/k = ξ fixed then the resulting theory will have a T = 0 thermal state arising from the electrostatics potential V (r, z) = W0(r 2 − 2z2) − (πξ)/(2) ln(r). The corresponding geometry will have a string like singularity with entropy density s = (1 + ξ) ln (1 + ξ)− ξ ln ξ. (5.27) 5.6.2 N = 4 Yang-Mills Theory on R× S3/Zk Field theory Let us now turn to N = 4 SYM on R×S3/Zk, and recall some of its relevant properties. This theory and its vacua can be obtained from N = 4 SYM on S3 in the following manner, as outlined in [2]. We can coordinatize the S3 using the metric ds2S3 = 1 4 [(2dψ + cos θdφ)2 + dθ2 + sin2 θdφ2] (5.28) 103 Chapter 5. Coarse-Graining the LM Solutions where θ and φ are the usual coordinates on S2, and ψ is an angular variable with period 2π. The orbifold is obtained by identifying ψ ∼ ψ + 2π/k. The vacua of the field theory are given by the space of flat connections, modulo gauge transformations, on S3/Zk. The orbifold allows for vacua of the form A = −diag(n1, n2, . . . , nN )dψ, so that e2pini/k are kth roots of unity. This ensures that A has unit holonomy around the full angular direction ψ, which is topologically trivial. To label the vacua uniquely, we will restrict the integers ni to be on the interval [0, k). Supergravity In the supergravity picture, the background potential for N = 4 SYM on R × S3/Zk is the same as in (5.25), but the electrostatics configuration is required to be periodic in z with period πk/2. Even though the background potential is not periodic in z, the part of the potential that determines the charge densities on the discs is. So the electrostatics solution will have a periodic part that arises from the charged discs in addition to the background piece. The periodic arrays of conducting discs are, in turn, related to the vacua of the field theory. For a vacuum that has Ni repetitions of the integer ni, the corresponding electrostatics configuration will have a set of charged conducting discs at positions z = πni/2, π(ni ± k)/2, π(ni ± 2k)/2, . . ., each carrying charge π2Ni/8. The gauge theory parameters are given in terms of the electrostatics parameters by g2YMk ∝ 1/W0 and N = ∑ Ni. Here the potential for the system with coarse-grained charge density Q is V (r, z) =W0(r 2 − 2z2) + ∫ ∞ −∞ dz′ Q(z′)√ r2 + (z − z′)2 , (5.29) where Q has a of period πk/2. Typical states Having described the field theory vacua and the corresponding auxiliary electrostatics configurations, we would like to consider the typical state. To find the typical configuration in this case, we can use the partition function (5.21) with β = 0. We can fix µ by imposing N = k 1 eµ − 1 , (5.30) which means e−µ = N N + k , (5.31) 104 Chapter 5. Coarse-Graining the LM Solutions and the typical vacuum will have q = N/k. Up to an overall constant, the electrostatic potential can be found outside the charge distribution to be V (r, z) = W0(r 2 − 2z2) − (πN)/(2k) ln(r). It is singular, and has an entropy of S = k (( 1 + N k ) ln ( 1 + N k ) − N k ln ( N k )) . (5.32) 5.6.3 Type IIA Little String Theory on S5 Field theory Type IIA Little String Theory on S5, has figured prominently in chapters 2 and 3 of this thesis. We recall that it was defined originally by its su- pergravity dual, found in [40] and described below. Using this supergravity dual, it has been argued in this thesis can be defined by particular double- scaling limits of either the Plane-Wave Matrix Model [1], the maximally supersymmetric YM on R× S2 or N = 4 SYM on R× S3/Zk [2]. Supergravity Let us recall briefly its supergravity solution. In this case, for the theory associated with k fivebranes we have two infinite conducting plates separated by a distance k. As shown by Lin and Maldacena [40], we can have a non- trivial potential V (r, z) = 1 g0 I0 ( r k ) sin ( z k ) (5.33) between the plates for which the corresponding geometry has an infinitely long throat carrying NS5-brane flux. The parameter g0 is related to the size of the sphere on which the NS5-branes sit, as measured in units of α′ (the dimensionful coupling of the Little String Theory). We can consider adding additional charged conducting discs to this sys- tem while keeping the number of units of NS5-brane flux fixed. In the elec- trostatics picture, this corresponds to adding some number of finite charged conducting discs in the region between the two infinite discs. The discs can sit at positions di = πni/2, where the integers ni are in the interval [1, k), and carry finite charges Ni. Typical states As for the 2+1 dimensional case, the number of vacua here is infinite if we allow arbitrary configurations finite discs in between the infinite conducting 105 Chapter 5. Coarse-Graining the LM Solutions plates. However, it is interesting to consider some restricted ensembles. First, we add some fixed number N of units of D0-brane flux. This requires that ∑ i iNi = N . In this case, the counting problem is identical to that is section 5.5.1, so we obtain the same typical charge distribution. Of course, the supergravity solution will be different here, since the background potential is now (5.33). Alternatively, we could consider an ensemble of geometries in which the number of units of D2-brane charge is fixed. In that case it is again conve- nient to use (5.21) with β = 0. Fixing the asymptotic charge we find that N2 = (N5 − 1) 1 eµ − 1 , (5.34) which can be inverted to give e−µ = N2 N2 +N5 − 1 . (5.35) The typical state will have 〈Qj〉 = 1 eµ − 1 = N2 N5 − 1 , (5.36) and the entropy of this configuration is, for N5 ≫ 1, S = N5 (( 1 + N2 N5 ) ln ( 1 + N2 N5 ) − N2 N5 ln ( N2 N5 )) . (5.37) 106 Chapter 6 Conclusion and Future Directions The present thesis has examined a variety of topics in the context of gauge/ gravity duality for field theories with SU(2|4) supersymmetry. We have argued that Little String Theory arises as a double-scaling limit of each of the SU(2|4) symmetric theories. We found that in each case the theory arises as a large N limit in which the coupling constant is scaled to infinity in a controlled way, thereby providing a Lagrangian definition of Little String Theory on S5. The double scaling limits we found in each case were such that λ/N → 0, in contrast to the M-theory limit in which λ/N →∞. This contrast is interesting because Little String Theory is a five-dimensional theory without gravity, and M-theory is an eleven-dimensional gravitational theory. We have also given a solution to the supergravity equations for any general configuration that is dual to a vacuum of the Plane-Wave Matrix Model or maximally supersymmetric YM on R×S2. Having these solutions will be very useful for strong coupling calculations in these gauge theories that require studying embeddings of strings or branes in the background geometry. There is also the possibility that these solutions may be of use in fluid mechanics, where problem of the radiation of waves from a single submerged disc can be reduced to the electrostatics problem for a pair of charged conducting discs. Our more general solutions may have applications to more general “dock problems” in fluid mechanics. An application of these supergravity solutions to the calculation of new instanton amplitudes was also given. Finally, we studied coarse-graining of the supergravity solutions, and proposed an entropy functional for singular geometries. We showed that this entropy functional vanishes for any microstate geometry, but is non-zero for any singular geometry. The results in this thesis concerning double-scaling limits and Little String Theory clearly indicate that the strong-coupling dynamics of the SU(2|4) symmetric field theories is very rich. One question that would be interesting to understand is the way in which the S5 on which the Lit- tle String Theory lives, or indeed the full background geometry, emerges at 107 Chapter 6. Conclusion and Future Directions strong coupling. It would seem that the application of the ideas set out by Berenstein and collaborators [85, 86], may provide some insight into this. In that work, in the context of N = 4 SYM, a simplified matrix model has been proposed to study supersymmetric sectors of the theory at strong coupling. In the vacuum state, the eigenvalues of the matrices spread out to form an S5 which is taken to be the S5 in the dual geometry, and the spectrum of excitations about the vacuum configuration were shown to correspond to the spectrum of near BPS states with large U(1) R-charge. In the present context, for certain vacua of the Plane-Wave Matrix Model, the same matrix model used by Berenstein and collaborators should describe the strong cou- pling dynamics. The S5 of eigenvalues ought to correspond to the S5 in the geometry that sits at the tip of a conducting disc, and the spectrum of exci- tations should correspond to near BPS states on a background that comes from taking a plane-wave limit near the tip of the disc. We noted in chapter 2 that information from the supergravity solution for a single disc above an infinite conducting plane can be used to extract the spectrum of near BPS excitations on the plane-wave geometry near the tip of the conducting disc. This is true more generally, however, and our general solutions to the electrostatics problems dual to general vacua of to the Plane-Wave Matrix Model and maximally supersymmetric YM on R×S2 can be used to extract near BPS spectra for the field theories in an arbitrary vacuum. It would be an interesting and non-trivial check of the gauge/gravity correspondence to determine if the spectrum agrees with a perturbative calculation in the field theory. This setting might also provide other laboratories for studying the integrability issues that are currently of great interest in N = 4 SYM. Another interesting and related issue stems from the various large N limits that can be taken in these theories. As we found in chapters 2 and 3, the double-scaling limits we have giving Little String Theory have λ/N → 0, whereas the M-theory limit of the Plane-Wave Matrix Model has λ/N →∞. The vastly different nature of the emergent geometries and theories in these limits suggests that coupling constant dependence is very important when discussing emergent geometries. It would be interesting to understand the nature of an intermediate limit. How fast must λ scale with N to produce a gravitational theory? How does the dimensionality of the theory change as we vary the ratio λ/N? To further allow the application of this duality to study the SU(2|4) theories at strong coupling, more investigation is needed to fill out the dic- tionary relating gauge theory observables to geometric calculations. A basic gauge theory observable is the Wilson loop. Preliminary investigations of supersymmetric versions of this operator suggest that the relative coupling 108 Chapter 6. Conclusion and Future Directions to the adjoint scalar fields in the field theories has a nice geometric inter- pretation as an angle in the (r, z)-plane in the electrostatics picture. This angle also seems to determine whether the Wilson loop operator is able to distinguish between the different gauge theory vacua. A possible future direction stemming from the work on coarse-graining is to further the understanding of the role of representation theory in AdS/CFT. One of the reasons why Lin and Maldacena were able to identify supergrav- ity solutions with vacua of the Plane-Wave Matrix Model is that these vacua are protected by supersymmetry, and so would be expected to correspond to some supergravity solutions. The representation theory of SU(2|4), in turn, determines that these states are protected by supersymmetry. It seems, therefore, that one could look for examples of bubbling geometries, starting by considering the representation theory of Lie superalgebras. This is not the path that is usually taken. One future direction would be to better understand the role that representation theory plays in the other known examples of bubbling geometries, for example the case of bubbling Wilson loops [87, 88]. A speculative benefit of such a study is that the extension of the coarse-graining ideas developed in this thesis are more challenging in that context. However, they might be facilitated by a better understand- ing of the representation theory underlying that case. Further study of the representation theory of other Lie superalgebras might also lead to the iden- tification of other gauge/gravity duality candidates. See also [89] for related discussion. In addtion to this represntation theoretic question, another future di- rection is to get a better understanding of the Plane-Wave Matrix Model at strong coupling, and there now exist numerical techniques [90, 91] that should enable this to be done. The gauge/gravity correspondence is most well understood when comparisons are made of quantities that are super- symmetric. However, it would be extremely useful to be able to push this understanding to non supersymmetric quantities as well. Numerical simula- tion provides an avenue for doing this, and we are currently engaged in such an investigation. 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They are particularly important in situations where a system has minima that are classically degenerate, and determine the rate of tunnelling between the minima. To be concrete, we will consider a non-relativistic particle in one dimen- sion with a potential of the form V = 1 a2 (1− cos ax) = 2 a2 sin2 ax 2 (A.1) where a is some parameter with dimensions of length−1. This is a periodic potential with minima at x = 2nπ/a. The action for the system has the form S = ∫ dt ( 1 2 ẋ2 − 2 a2 sin2 ax 2 ) . (A.2) The corresponding euclidean action is SE = ∫ dt ( 1 2 ẋ2 + 2 a2 sin2 ax 2 ) . (A.3) To maintain consistency with the discussion in chapter 4, we will define a “superpotential” W according to 1 2 ( ∂W ∂x )2 = V (x) = 2 a2 sin2 ax 2 , (A.4) which means that W = − ( 2 a )2 cos ax 2 + const. (A.5) 119 Appendix A. Quantum Mechanical Instantons Having introduced this superpotential, we can write the action as SE = 1 2 ∫ dt ( ẋ2 + ( ∂W ∂x )2) . (A.6) We will complete the square by adding and subtracting 2ẋ∂W∂x under the integrand, giving SE = 1 2 ∫ dt ( ẋ− ∂W ∂x )2 + ∫ dtx ∂W ∂x = 1 2 ∫ dt ( ẋ− ∂W ∂x )2 + W |xfxi , (A.7) where in the second line we have used the chain rule to evaluate the second integral, and xi and xf are the initial and final points of the motion of the particle. Note that the first term in this expression is a complete square, and is therefore positive. The minimum action will, therefore, be given by a configuration for which this term is identically zero, and the corresponding minimum value of the action will be W |xfxi . In the language of chapter 4 the superpotential provides a bound for the Euclidean action, and instanton configurations that satisfy ẋ − ∂W∂x = 0 saturate the bound, or are “BPS”. Explicit solutions for a particle that tunnels between the minima at 0 and 2π/a are given by x(t) = 4 a tan−1 ( et−t0 ) , (A.8) where t0 is a constant that determines the time the particle is at the point π/a. Let us verify that the instanton configurations ẋ − ∂W∂x = 0 solve the Euclidean equation of motion, which is ẍ = ∂V ∂x = ∂W ∂x ∂ ∂x ( ∂W ∂x ) . (A.9) Considering the left hand side we have LHS = ẍ = d dt ẋ = dx dt ∂ ∂x ẋ = ∂W ∂x ∂ ∂x ( ∂W ∂x ) = RHS. (A.10) 120 Appendix A. Quantum Mechanical Instantons Therefore these configurations solve the Euclidean equation of motion. To make the interpretation of this clearer, we can compare this with the transmission coefficient for a potential barrier that can be calculated using the WKB approximation in quantummechanics (e.g. [92]). The transmission coefficient is given by T = e2 R xf xi dx √ 2(E−V ). (A.11) Consider a particle in with the Hamiltonian H = 1 2 p2 + 2 a2 sin2 ax 2 . (A.12) A particle with zero energy will have a transmission coefficient given by T = e−2 R dx √ 2V (A.13) with the potential we have defined in (A.1). Using the superpotential we gave in (A.5), this becomes T = e−2W | xf xi , (A.14) so we see that the superpotential determines the rate of tunnelling between vacua in this system. 121 Appendix B Lin-Maldacena Supergravity Solutions The general Lin-Maldacena SU(2|4)-symmetric supergravity ansatz (sup- pressing an overall factor of α′ in the metric) is given by [40] ds210 = ( V̈ − 2V̇ −V ′′ ) 1 2 { −4V̈ V̈ − 2V̇ dt 2 − 2V ′′ V̇ (dρ2 + dη2) + 4dΩ25 + 2 V ′′V̇ ∆ dΩ22 } e4Φ = 4(V̈ − 2V̇ )3 −V ′′V̇ 2∆2 C1 = − 2V̇ ′ V̇ V̈ − 2V̇ dt F4 = dC3, C3 = −4 V̇ 2V ′′ ∆ dt ∧ d2Ω, H3 = dB2, B2 = 2 ( V̇ V̇ ′ ∆ + η ) d2Ω, ∆ ≡ (V̈ − 2V̇ )V ′′ − (V̇ ′)2 . (B.1) 122 Appendix B. Lin-Maldacena Supergravity Solutions Their explicit solution corresponding to Little String Theory on S5 is ds210 = N ( −2r √ I0 I2 dt2 + 2r √ I2 I0 dΩ25 + √ I2 I0 I0 I1 (dr2 + dθ2) + √ I2 I0 I0I1s 2 I0I2s2 + I 2 1c 2 dΩ22 ) B2 = N ( −I21cs I0I2s2 + I21c 2 + θ ) d2Ω eΦ = g0N 3/22−1 ( I2 I0 )3 4 ( I0 I1 ) 1 2 ( I0I2s 2 + I21c 2 )− 1 2 C1 = −g−10 1 N 4r I1 I2 (I20s 2 + I21c 2)dt C3 = −g−10 4I0I 2 1s 3 I0I2s2 + I21c 2 dt ∧ d2Ω (B.2) where In(r) are the usual Bessel functions, s ≡ sin(θ), and c ≡ cos(θ). 123 Appendix C More Details of the Electrostatics Solutions in Simple Cases C.1 Solution of the Electrostatics Problem with Two Discs In this appendix, we describe the solution of the electrostatics problem de- scribed in section 2.3, namely to find a solution of Laplace’s equation ∇2V = 0 such that V = 0 at z = 0, V = ∆ for {z = κ, 0 < r < 1} and V behaves as zr2 − 23z3 for large z2 + r2. To start, we write V = zr2 − 2 3 z3 + φ so that φ vanishes at infinity and at z = 0 and is given by φ(r) = ∆+ 2 3 κ3 − κr2 ≡ β(1− αr2) on the upper plate. Axially symmetric solutions to the Laplace equation that are regular at r = 0 and vanish as r → ∞ are linear combinations of functions e±zuJ0(ru), where u is a continuous parameter. If we denote by V+ and V0 the function φ in the regions z ≥ d and 0 ≤ z ≤ d respectively, then we must have V0 = ∫ ∞ 0 B(u) sinh(zu)J0(u) V+ = ∫ ∞ 0 C(u)e−zuJ0(u) (C.1) 124 Appendix C. Simple Electrostatics Solutions taking into account the boundary conditions at z = 0 and z =∞. We now take into account the boundary conditions at z = d. First, we must have V0 = V+ at z = d, so it must be that B(u) = 2βu−1e−duA(u) C(u) = 2βu−1 sinh(du)A(u) for some function A(u) (the prefactors have been chosen for later conve- nience). Also, we have ∂zV0 = ∂zV+ at z = κ for r > 1 and V0 = V+ = β(1 − αr2) for 0 < r < 1. These will be satisfied if∫ ∞ 0 A(u)J0(ru)du = 0 r > 1 (C.2)∫ ∞ 0 duu−1(1− e−2κu)A(u)J0(ru) = 1− αr2 0 < r < 1.(C.3) This set of equations is of the type considered in [63], chapter 4.6, “Dual integral equations with Hankel kernel and arbitrary weight function”. For completeness, we review the solution of this type of equation in appendix C.3. For our specific case, the solution is given by A(u) = 2u π ∫ 1 0 f(t) cos(ut)dt , where f(t) is an even function of t satisfying the integral equation f(x)− ∫ 1 −1 K(x, t)f(t)dt = g(x) , (C.4) where g(x) = 1− 2αx2 (C.5) and K(x, t) = 1 π 2κ 4κ2 + (x− t)2 . The solution f of the integral equation may be given as a series (2.10). Substituting our solution for A in (C.1) gives the simpler result (2.7), valid for all z. C.1.1 Limiting Forms of the Solution In our discussion of scaling limits of the theory, it will be useful to have some explicit results for the behaviour of the electrostatics solution for large and small values of κ. 125 Appendix C. Simple Electrostatics Solutions Large κ For large κ, the kernel K has a small norm, so the series solution (2.10) is well approximated by its leading term. Thus, f(x)→ g(x) = 1− 2αx2 . In order that the charge density vanishes at the edge of the plate, equation (2.11) implies that f(1) must vanish. Thus, we must have α = 1 2 , which (for large κ) implies ∆→ −2 3 κ3 + 2κ . This implies that β = 2κ, so that using (2.12), we have q → 8 3π κ . As a check, we can match these results with those in appendix C.2, where we have solved the electrostatics problem after first taking the limit of large d with fixed R. Small κ For small κ, the problem is more difficult to study, since the norm of the kernel K approaches one (the kernel approaches a delta function), so the series solution (2.10) converges very slowly. The work [94] studied in detail the case of charged conducting discs at small separation without a back- ground potential, but some of the general discussion there is helpful in our case also. Up to corrections of (fractional) order √ κ, the leading small κ behaviour of the solution of the integral equation (C.4) is [94] f(t)→ κ−1 ∫ 1 −1 k(s, t)g(s)ds , where k(s, t) = 1 2π log { 1− st+ (1− s2) 12 (1− t2) 12 1− st− (1− s2) 12 (1− t2) 12 } . 126 Appendix C. Simple Electrostatics Solutions This corresponds to the approximation that σ(r) varies as κ−1φ(r). In our case, g is given by (C.5), so we find that f(t)→ 1 2κ { (1− t2) 12 − α(1− t2) 12 + 2 3 α(1− t2) 32 } . (C.6) Now, in order that σ vanishes at the edge of the disc, (2.11) implies that f(1) vanishes and that the f ′(1) is bounded. We see that the latter condition is satisfied only if α→ 1 in this limit.40 Thus, in the limit of small κ, we have fκ(t)→ 1 3κ (1− t2) 32 . (C.7) From α→ 1, we infer that ∆→ κ (C.8) in the limit, and using (2.12) we have q(κ)→ 1 8 . We are particularly interested in the small κ behaviour of the integral (2.25), which after a change of variables becomes I = ∫ 1 0 dy sin(πy) [ φκ(1, κy) − { ∆κy − κy + 2 3 κ3y3 }] . (C.9) The largest terms are the first two terms in curly brackets, each of which give a contribution of order κ, but (C.8) implies that these cancel. Using numerical methods described in appendix E, we have estimated the remain- ing contributions. We find that there is a further cancellation between the φ term and the term in curly brackets, both of which give contributions of order κ2 ln(κ). The net result is that the integral behaves for small κ as41 I ≈ 0.04κ2. 40For finite values of κ, the series solution (2.10) may be used to show that f ′(1) is actually bounded for all t. On the other hand, corrections to the formula (C.6) are generally non-zero at t = ±1. Thus, the vanishing charge density constraint for the exact solution at finite κ comes from the condition f(1) = 0. We have verified numerically in appendix E that this condition gives the same result α → 1 in the limit κ → 0 as the condition that f ′(1) is bounded applied to the leading approximation. 41Note that here we cannot simply use the leading approximation (C.7) for f . The reason is that the integral over t receives most of its contribution close to the boundaries of the interval [−1, 1] where the leading approximation goes to zero like a 3/2 power. Corrections to (C.7), which are fractionally small in the bulk of the interval, become more important in the region near the boundaries, and result in a modified behaviour for the integral. 127 Appendix C. Simple Electrostatics Solutions While this result is numerical, we have performed an analytic check on the method. If we normalize the function φ̃(x, y) = C [ φκ(1 + x, κy)− { ∆κy − κy + 2 3 κ3y3 }] by choosing C such that (for example) φ̃(0, 0.5) = 1, then φ̃ should have a well defined limit as κ → 0. This must be a nontrivial solution to the Laplace equation in the case where we have one infinite conducting plate (at y = 0), and one semi-infinite conducting plate (at y = 1, x < 0), with vanishing potential on both plates (recall that the term involving ∆ ensures that φ̃ vanishes on both plates). This reduces to a two-dimensional problem that may be solved using conformal mapping techniques [40]. As explained in [40], if we define a complex coordinate z = x + iy and another complex variable w = 2∂zV = ∂xV − i∂yV . Then the Laplace equation ensures that the mapping between z and w is an- alytic. Further, ∂xV is everywhere non-negative and vanishes at the plates, so the region outside the plates must map to the right half plane, with the plates mapping to the imaginary axis. The explicit transformation that achieves this (unique up to transformations generated by translations, scal- ings, and inversions that fix the imaginary axis) is [40] z = iw + 1 π ln(w) + i 2 + 1 π + 1 π ln(π) . (C.10) This implies V = Re( ∫ z w(z′)dz′) = u( 1 π − v) , where w = u + iv. From (C.10), we find that u and v are determined in terms of x and y by x = −v + 1 2π ln(u2 + v2) + 1 π + 1 π ln(π) y = u 1 π tan−1 (v u ) + 1 2 , so we have a relatively explicit result for the potential in the limit. Af- ter normalizing the result in the same way that we normalized φ̃, we find excellent agreement with our numerical results for φ̃ in the limit of small κ. 128 Appendix C. Simple Electrostatics Solutions C.2 Single disc solution In this appendix, we solve the electrostatics problem obtained in the limit of large d with V0d =W0 fixed and R fixed, to give a more direct derivation of the asymptotically near-horizon D2-brane solution derived by Lin and Maldacena. To define this solution, we want to find the potential for a charged con- ducting disc of radius R and charge Q in a background potential V =W0(r 2 − 2η2) . (C.11) In this case, the potential W0 will eventually be related to the charge, Q, and the radius, R, by the condition that the charge density vanishes at the edge of the disc. By the superposition and scale-invariance properties of electrostatics, the solution must take the form V =W0R 2φ(r/R, η/R), where φ is the solution to the problem with W0 = 1 and R = 1. Further if q is the total charge on the disc required in this simplified problem so that the charge density vanishes at the edge of the disc, we must have Q =W0R 3q. To start, we would like to find a solution to Laplace’s equation with boundary conditions that the potential is fixed to ∆ on a disc of radius 1 at z = 0 and becomes r2 − 2z2 at infinity. The potential, ∆, and the total charge on the disc, q, will be fixed by demanding that the charge density vanishes at the edges of the disc. We begin by writing V = r2 − 2z2 + φ , such that φ vanishes at infinity and φ(r, z = 0) = ∆− r2; . If we denote by V+ the potential φ for z > 0, then separating variables gives V+(r, z) = ∫ ∞ 0 u−1A(u)e−uzJ0(ur)du . (C.12) 129 Appendix C. Simple Electrostatics Solutions By symmetry, the potential V− for z < 0 must be V−(r, z) = V+(r,−z) . Finally, we require that V+ = V− = ∆− r2 0 ≤ r ≤ 1 , ∂zV+ − ∂zV− = 0 r > 1. These imply the dual integral equations∫ ∞ 0 A(u)J0(ru)du = 0 r > 1 (C.13)∫ ∞ 0 duu−1A(u)J0(ρu) = ∆− 2r2 0 < r < 1, (C.14) which are of the type considered in appendix C.3, with k(u) = 0 and h(r) = ∆− r2. In this case, the integral equation is trivial, so the result is A(u) = 2u π ∫ 1 0 f(t) cos(ut)dt with f(t) = g(t) = ∆− 2r2 . Substituting for A in (C.12) gives V+ = 1 π ∫ 1 −1 f(t)√ r2 + (z + it)2 The charge density on the disc is given by σ(r) = 1 4π (∂zV−(r,−ǫ)− ∂zV+(r, ǫ)) = 1 2π ∫ ∞ 0 A(u)J0(ur) = 1 2π [ f(1) (1− r2) 12 − ∫ 1 r ds f ′(s) (s2 − r2) 12 ] = ∆− 2 π2 1√ 1− r2 + 4 π2 √ 1− r2 . In order that the charge density vanishes on the tip, we need ∆ = 2 , 130 Appendix C. Simple Electrostatics Solutions so finally σ(r) = 4 π2 √ 1− r2 and the total charge is q = 8 3π . If we like, we can write an explicit solution for the potential using oblate spherical coordinates (see [63], section 3.3), but we will not need it here. C.3 Dual integral equations In this appendix, we review the solution of dual integral equations of the form ∫ ∞ 0 A(u)J0(ru)du = 0 r > 1 (C.15)∫ ∞ 0 duu−1(1 + k(u))A(u)J0(ρu) = h(r) 0 < r < 1 ,(C.16) following chapter 4.6 of [63]. In general, the solution is given as A(u) = 2u π ∫ 1 0 f(t) cos(ut)dt , where f(t) is the solution to a Fredholm integral equation of the second kind, f(x) + ∫ 1 0 K̃(x, t)f(t) = g(x) , with g(x) = d dx ∫ x 0 uh(u)du√ x2 − u2 . The kernel K is given in terms of k(u) by K̃(x, u) = x u 1√ 2π {Kc(|x− u|)−Kc(x+ u)} , with Kc(ξ) = √ 2 π ∫ ∞ 0 k(t) cos(ξt)dt . 131 Appendix D More Details on Coarse-Graining D.1 Coarse-Graining for Large Discs For certain parameter values, or in restricted ensembles, the typical states are such that the radii of the discs are large compared to their separations. As we noted above, in this case, the macroscopic description will replace the closely spaced discs with a solid material that conducts only in the horizontal directions. Such a conductor has the following properties. Since the charges are free to rearrange themselves in the directions perpendicular to z, they will do so in such a way that the final potential inside the conductor is a function only of z, ensuring that the electric field in the r and θ directions is zero. There will generally be some charge distribution inside the conductor, given by ρ(z) = − 1 4π V ′′(z) , (D.1) so ρ is also a function only of z. The remaining charge will build up at the surface of the conductor. In general, the shape R(z) for the conductor, and the linear charge distribution Q(z) on the conductor, together with some fixed background potential will determine the charge density ρ(z) inside the conductor and the surface charge density σ(z), determined from ρ(z) via Q(z) = πR2(z)ρ(z) + 2πR(z)σ(z) √ 1 + (R′(z))2 . (D.2) On the other hand, for some special choice of R(z), the surface charge density will vanish. This is the coarse-grained analogue of the constraint that the charge density should vanish at the tip of the discs. D.1.1 The Variational Problem We will now set up the mathematical problem that determines R(z) and ρ(z) from Q(z). We start by assuming some fixed R(z) and Q(z). 132 Appendix D. More Coarse-Graining Outside the conductor, the potential will be given by V+(r, z) = V0(r, z) + Ṽ (r, z) , where Ṽ is the potential due to the charges in the conductor, which should vanish at large r and z. Since Ṽ is an axially-symmetric solution of Laplace’s equation, we can expand it in terms of Bessel functions, Ṽ (z) = ∫ ∞ 0 du u A(u)e−zuJ0(ru) . Inside the conductor, the potential will be some function V−(z). The un- known functions A(u) and V−(z), together with the charge density ρ(z) inside the conductor and the charge density σ(z) on the surface of the con- ductor will be determined by the two equations (D.1) and (D.2), and the boundary condition ~E+(R(z), z) − ~E−(z) = 4πσ(z)n̂ . (D.3) In our case, we wish to fix R(z) by the constraint that the surface charge density vanishes. Then the electric field must be continuous across the boundary of the conductor, and since the electric field is vertical inside, we must have ∂rV (R(z), z) = 0. Explicitly, we have ∂rV0(R(z), z) − ∫ ∞ 0 due−zuA(u)J1(R(z)u) = 0 . (D.4) This determines R(z) in terms of A(u). Given this, the potential inside the conductor is determined by the z component of the boundary condition (D.3), or simply by continuity of the potential across the boundary, so V−(z) = V0(R(z), z) + ∫ ∞ 0 du u A(u)e−zuJ0(R(z)u) . Finally, we can use (D.1) and (D.2) to write an equation relating A(u) and Q(z), Q(z) = −1 4 R2(z)(∂2zV0(z) + ∫ ∞ 0 duuA(u)e−zuJ0(R(z)u)) . (D.5) To summarize, A(u) is determined by the integral equation (D.5) where R(z) is determined in terms of A via (D.4). In practise, it is far simpler to determine R(z) and Q(z) given some A(u), or more generally some solution to the Laplace equation that arises from any 133 Appendix D. More Coarse-Graining set of axially symmetric localized charges. We could also parametrize our solution to the Laplace equation via the multipole data rather than the function A(u). As an example of this approach, we can come up with an explicit coarse-grained supergravity solution starting with the simplest non- trivial solution Ṽ , namely the potential from a dipole localized at the origin (the infinite conducting plane at z = 0 forces the potential to be an odd function of z.). In this case, we have Ṽ (r, z) = p z (r2 + z2) 3 2 . The radial electric field for the full potential is then Er(r, z) = −∂rV+(r, z) = −2V0rz + 3p rz (r2 + z2) 5 2 . Requiring that this is zero gives r = 0 or z = 0 or z2 + r2 = x2 , where we define x = ( 3p 2V0 ) 1 5 . Thus, in this case, the profile of the conductor is spherical. From (D.5), we can now determine the corresponding charge density Q(z). We find Q(z) = 5 2 V0z(x 2 − z2) . As a check, we find that the total dipole moment for this configuration is∫ ∞ 0 dz2zQ(z) = p . So we have at least one example where we know both the geometry and the Young diagram explicitly. Note that for this case, the typical height for the plates and the typical size are the same, of order x. In terms of the field theory parameters, we have V0 ∼ 1/g2 and p = 2N , so x ∼ λ 15 . Thus, our coarse-grained description should be valid as long as λ is large. The typical charge on one of the plates in the corresponding microstate geometries is Q ∼ V0x3 ∼ N/λ 25 . In section 5.3, we saw that this charge is of order one for typical distributions, so it is only for λ ∼ N 52 that the geometry we have constructed has an entropy of the same order of magnitude as the thermal state. (It is important to note that for a fixed configuration of discs (i.e. fixed p/V0 ∼ λ), the corresponding charge distribution changes as a function of N .) 134 Appendix D. More Coarse-Graining D.2 Stretched Horizons In this appendix we investigate the possibility that the area (or some gener- alization of area) of a suitably defined stretched horizon might reproduce the entropy formula (5.18).42 We focus on a particularly simple specific example of a coarse-grained geometry, and find that a stretched horizon whose area would reproduce the entropy would necessarily be parametrically closer to the singularity than both the scale xs where the string coupling becomes of order one, and the scale xc where the curvature becomes string scale. The geometry we focus on is the thermal state geometry of N = 4 SYM on R× S3/Zk. In this case the potential is simply −Nπ 2k log ρ+ V0(ρ 2 − 2η2) (D.6) where V0 ∼ 1g2YMk as identified in chapter 3. The potential is singular at ρ = 0, which violates the regularity condition on the LM geometry. The boundary of the coarse-grained conducting discs is at ρ = r0 √ piN 4kV0 , and the supergravity solution is ds210 = ( N 4V0kπ ) 1 2 { −44V0kρ 2 Nπ dt2 + 16kV0 4kV0ρ2 −Nπ (dρ 2 + dη2) + 4dΩ25 + 4kV0ρ 2 −Nπ Nπ dΩ22 } , e4Φ = Nπk 8V 30 (4kV0ρ 2 −Nπ)2 , C3 =− (4kV0ρ 2 −Nπ)2 Nπk dt ∧ d2Ω, B2 =2ηd 2Ω. (D.7) We see explicitly that the geometry is singular at ρ = r0 ∼ √ g2N , which is exactly the edge of the discs, but there is no horizon in this geometry. This solution has been considered in [42], where it was pointed out that the singularity is related to the Zk orbifold singularity in the IIB language. We will assume the stretched horizon to be a constant ρ surface respect- ing the translational symmetry along the η direction. Using ρ = r0 + x, we 42These ideas have been explored in the context of coarse-grained LLM microstates [83, 95, 96], though a prescription for defining a stretched horizon that generally reproduces the entropy of coarse-grained states has not emerged. 135 Appendix D. More Coarse-Graining find the string coupling becomes of order one at xs = 1 8 √ πV 20 ∼ (g2Y Mk)2. (D.8) The Ricci scalar can be calculated noticing the fibred structure of the metric, Rstring = 3 √ V0k Nπ 8V0kρ 2 −Nπ 4V0kρ2 −Nπ . (D.9) We see that it diverges at exactly the boundary of the coarse-grained con- ducting discs. The curvature becomes of string scale at xc ∼ 1 . (D.10) In the above we have assumed g2YMN ≫ 1 in order for the supergravity approximation to be valid. As a result, we will be interested in the scale where g2YMN ≫ 1 ≫ g2YMk, and in particular N/k ≫ 1. The area of an 8-surface at constant t and ρ = R(η) can be calculated to be (in the Einstein frame) A = 211/2ω2ω5 √ 1 +R′2(z) √ (V̈ − 2V̇ )V̇ 3/2, (D.11) where ω2, ω5 are the volume elements of the two-sphere and five-sphere re- spectively. Specializing to R(z) = r0 + x and to the metric (D.7), we get A = 16ω2ω5 √ Nπ (4V0kρ 2 −Nπ)3/2 k2 . (D.12) We note that it is a monotonically increasing function with the distance from r0. Using this and evaluating at xc, xs we find Ac ∼ N 5/4 g 3/2 YMk 2 , As ∼ (g2Y Mk)3Ac . (D.13) The Bekenstein-Hawking entropy formula S = AGN givesGN = g 2 s = (g 2 Y Mk) 2) Sc ∼ 1 g 11/2 YM k 11/4 ( N k )5/4 , Ss = (g 2 YMk) 3Sc. (D.14) As expected Sc ≫ Ss. According to the entropy functional (5.18), the entropy associated with the geometry (D.7) is S = −k ln(N/k) + (N + k) ln(N/k + 1). (D.15) 136 Appendix D. More Coarse-Graining In the large N/k limit it becomes S ∼ k(ln(N/k) + 1), (D.16) which is much smaller than both Ss, and Sc. Here, both α ′ and string loop corrections are very important. Further, if a horizon (or some stringy analogue) does exist in the fully corrected solution, we may require a highly stringy generalization of area to compare with the entropy. While we have studied only one particular example, we expect that the qualitative features will apply in the general case. 137 Appendix E Numerical Methods In this appendix we will provide more detail on the numerical methods used in this thesis. An important problem that has occurred throughout this thesis is to find SU(2|4) symmetric solutions of type IIA supergravity. Because the of the symmetries in this ansatz, as Lin and Maldacena showed, the super- gravity equations are reduced to a Laplace equation. Due to the boundary conditions for regular solutions, however, solving the supergravity equations is non-trivial, but in many cases possible, as we have discussed above. Al- though we have found a solution of the supergravity equations under general circumstances, it turns out that to extract detailed information about the solution it is necessary to apply some approximations, and for the types of information we have required, numerical methods are the most appropriate. We have been able to cast the supergravity equations into a form so that the solution is given by the solution of a Fredholm integral equation of the second kind. Equations of this type are encountered frequently in physics, for example the Lippman-Schwinger equation in quantum mechanics is of this form. In some cases, when the kernel is of a special form, the equations can be solved exactly. Unfortunately, analytic solutions are not known for the present case. One approach to solving integral equations of this type is to expand the kernel in terms of some basis of known functions, for example the Legendre polynomials would be suitable in the present case, and then to truncate the expansion after a suitable number of terms to give a solution of the desired accuracy (see, e.g. [64]). This technique is not particularly helpful in the present case because an analytic form for integral of the kernel against standard sets of complete functions is not known. These integrations must then be performed numerically, and to obtain reasonably accurate solutions it is computationally expensive. Another numerical approach, called the Nyström method, turns out to be more appropriate for the present problem. This method is described many places and a nice discussion can be found in [70]. We will summarize the main details here. The crux of this method is to notice that for integral 138 Appendix E. Numerical Methods equations of the type ψ(x) + ∫ β α dtK(x, t)ψ(t) = f(x) (E.1) the operator K = ∫ β α dtK(x, t) is a linear operator, and so we can write the integral equation as (1 + K)ψ = f . (E.2) The difficulty then arises in inverting the operator (1 + K). To obtain a tractable problem, one can restrict the interval (α, β) to a finite number of points, and thereby reduce the problem to solving a finite sized linear system, which is straightforward to do numerically. The remaining challenge is then to ensure that the interval is discretized in a suitable way so that there are a sufficient number of points, suitably distributed in the interval to provide a sufficiently accurate numerical solution. The number and distribution depend on the form of the kernel. Fortunately for us we are provided with a starting point, because the integral equation that we have to solve is a special case of a similar integral equation that is found in the context of fluid mechanics [97]. In that case, the problem under consideration was the radiation of water waves from an oscillating submerged disc. The kernel in question is K(x, t) = 1 π 2κ 4κ2 + (x− t)2 (E.3) where κ is a parameter that gives the separation between the discs. In [97] the Nyström method was applied to an integral equation with this kernel, and it was found that for κ ∼ 10−1 a discretization using 60 point Gauss- Legendre quadrature gave good numerical results. In our application, to ensure the constraint that the charge density vanished at the edge of the conducting disc, it was necessary to compare solutions of the integral equa- tion for different right hand sides. The behaviour near the edge of the interval was of particular importance for ensuring the vanishing charge den- sity condition. We found that using the midpoint quadrature rule, instead of Gauss-Legendre, gave better numerical results for the region near the edge of the interval. To solve the numerical system we used the GNU Scientific Library LU-decomposition routine, implemented in the C programming lan- guage. Numerical calculations were run on a Linux based workstation with dual 2.80 GHz Intel Pentium 4 CPUs. We solved the integral equation over a range of values for the parameter κ, and checked that the number of points in the discretization was suffi- cient so that the results did not depend on the number of points. We also 139 Appendix E. Numerical Methods compared the solution of the integral equation in the bulk of the interval to ensure that it gave the same behaviour with both midpoint and Gauss- Legendre quadrature. We checked that the eigenvalues of the matrix in the linear system were sufficiently different from zero to allow the system to have a unique solution. A further check came from considering the solutions of the integral equation for large κ, in which case an analytic form of the solution can be found. We ensured that the numerical results were consis- tent with this form when we took κ ≫ 1, and that the relevant physical parameters changed continuously as we scaled κ away from this limit. A final check on the numerical results was done for the form of the potential in chapter 2, and was described above in C.1. That was to compare the numerical result for the electrostatics potential with a solution that can be found by conformal mapping in the case where κ≪ 1. Although it was not possible to fix the overall normalization of the potential from the conformal mapping, excellent agreement was found between the form of the potential obtained by conformal mapping and the result from the Nyström method. We were therefore able to see that the numerical methods appeared to inter- polate smoothly between the two extremes in the parameter space in which we had some analytic control over the form of the potential. 140 Appendix F Note on Publication Chapter 2 is based on the paper [1] describing work done in collaboration with H. Ling, A.R. Mohazab, H.-H. Shieh and M. Van Raamsdonk, and contains sections from that paper. I contributed to this project at all lev- els including the formulation of the method we used, the calculations we performed, and preparing the manuscript. In particular, our main results on the form of the double scaling limits come from a numerical code that I wrote. Chapter 3 is based on the paper [2] describing work done in col- laboration with H. Ling and H.-H. Shieh, and contains sections from that paper. The main results of that paper come from results of numerical codes that I wrote, and I wrote the majority of the original manuscript. Chapter 4 is based on the paper [3] describing work I carried out independently, and contains sections from that paper. Chapter 5 is based on [4] describing work done in collaboration with H.-H. Shieh and M. Van Raamsdonk, and con- tains sections from that paper. I was involved at all stages of the project. Particular contributions include the generalizations to ensembles with dif- ferent combinations of fixed charges, and a significant amount of writing to the original manuscript. 141

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