D-Brane Charge Groups and Fusion Rings in Wess-Zumino-Witten Models David Ridout Supervised by Prof. Peter Bouwknegt This thesis is presented for the degree of Doctor of Philosophy in the Department of Physics and Mathematical Physics at The University of Adelaide March, 2005 This work contains no material which has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previ- ously published or written by another person, except where due reference has been made in the text. I give consent to this copy of my thesis, when deposited in the Univer- sity Library, being available for loan and photocopying. Abstract This thesis presents the computation and investigation of the charges and the corre- sponding charge groups for untwisted symmetry-preserving D-branes in a Wess-Zumino- Witten model over a compact, connected, simply-connected, simple Lie group. First, some general ideas from conformal field theory are reviewed and applied to Wess-Zumino- Witten models. Boundary conformal field theory is then introduced with the aim of de- riving the Cardy constraint relating the consistent boundary conditions to fusion. This is used to justify certain dynamical processes for branes, called condensation, which lead to a conserved charge and constraints on the corresponding charge group (following Freden- hagen and Schomerus). These constraints are then used to determine the charge groups for untwisted symmetry-preserving branes over all compact, connected, simply-connected, simple Lie groups. Rigorous proofs are detailed for the Lie groups SU(r + 1) and Sp(2r) for all ranks r, and the relevance of these results to K-theory is discussed. These proofs rely on an explicit presentation of the corresponding fusion rings (over Z), which are also rigorously derived for the first time. This computation is followed by a careful treatment of the Wess-Zumino-Witten model actions; the point being that the consistent quantisation paradigm developed can also be applied to brane charges to determine the charge groups. The usual (string-theoretic) D-brane charges are introduced, and are proved to exactly reproduce the charges of Fre- denhagen and Schomerus when certain quantisation effects are brought into play. This is followed by a detailed investigation of the constraints induced on the corresponding charge groups by insisting that the string-theoretic charges be well-defined. These con- straints are demonstrated to imply those of Fredenhagen and Schomerus except when the Wess-Zumino-Witten model is over a symplectic Lie group, Sp(2r). In the symplectic case, numerical computation shows that these constraints can be strictly stronger than those of Fredenhagen and Schomerus. A possible resolution is offered indicating why this need not contradict the K-theoretic interpretation. iii Acknowledgements First and foremost, I would like to thank my supervisor Peter Bouwknegt for all the time, effort, encouragement, patience, and knowledge that he has invested in me over the last few years. He has motivated this study, guided my efforts, and above all, taken the time to understand my progress, despite my often incoherent explanations and frequent logical errors. I am especially grateful for the latter as anecdotal evidence from friends suggests that it is, alas, all too rare in this world. I also wish to extend a special thankyou to Danny Stevenson for the many hours he has spent discussing mathematics with me, from subtleties in the theory of Lie groups to algebraic topology, bundle gerbes, and K-theory. Likewise, I owe Nick Buchdahl a debt of gratitude for teaching me the basics of differential geometry and cohomology theory. Thanks also to Volker Braun, Alan Carey, Grant Cairns, Mike Eastwood, Ken Harrison, Marcel Jackson, Devin Kilminster, Michael Murray, Mathai Varghese, and Tony Williams, for helpful discussions through the years. Additional thanks must be extended to Peter Dawson and Peggy Kao, who had the dubious honour of having to share an office with me, and yet survived as friends and occasional collaborators. The same goes for Paul Martin, who also bears the load of being in an entirely different field, yet still cheerfully acknowledges my ramblings, and has never once complained about my monopoly on office whiteboard space. This project has been financially supported by an Australian Postgraduate Award and a National Research Scholarship from the University of Adelaide, and I would like to express my thanks to the University for this. I would also like to thank the Department of Mathematics at La Trobe University for their hospitality during the latter stages of my candidature. I am also extremely grateful to the Alumni Association of Adelaide Univer- sity, the Australian Federation of University Women, and my family, whose generosity enabled me to experience an international conference in my field. Last but not least, I have to thank my partner Penny. She’s had to put up with this before, but that’s hardly a reason not to record here the gratitude, to her, that I feel on a daily basis. v Contents Abstract ........................................... .............. iii Acknowledgements ................................... ............. v Chapter1. Introduction .............................. .............. 1 1.1. BackgroundandMotivation........................ ............ 1 1.2. Overview ....................................... .......... 4 Chapter2. ConformalFieldTheory ...................... ............. 9 2.1. ConformalInvariance ............................ ............ 9 2.2. Fusion......................................... ........... 19 Chapter 3. Wess-Zumino-Witten Branes I: Algebraic Considerations .......... 25 3.1. Wess-Zumino-WittenModels ....................... ........... 25 3.2. BoundaryConformal FieldTheoryandBranes . ........... 38 Chapter4. BraneChargeGroups ......................... ............ 53 4.1. BraneDynamicsandConservedCharges . ........... 53 4.2. ChargeGroupComputations ........................ ........... 58 4.3. Addenda: SymmetriesandK-Theory.................. ........... 67 Chapter5. FusionRings............................... ............. 77 5.1. FusionRingsandAlgebras ......................... ........... 77 5.2. FusionPotentials............................... ............. 81 5.3. Proofs......................................... ........... 84 5.4. Generalisations ................................ ............. 94 Chapter 6. Wess-Zumino-Witten Branes II: Geometric Considerations ......... 103 6.1. SomeAlgebraicPreliminaries ..................... ............. 103 6.2. Closed String Wess-Zumino-WittenModels. ............. 106 6.3. OpenString Wess-Zumino-WittenModels . ............ 112 Chapter7. BraneChargeGroupsRevisited.. .. .. .. .. .. .. .............. 125 7.1. GeometricChargeDefinitions ...................... ............ 125 7.2. ChargeComputations ............................. ........... 131 7.3. ChargeGroupConstraints ......................... ............ 136 Conclusions ........................................ .............. 143 vii viii CONTENTS Appendix A. Finite-Dimensional Simple Lie Algebras . ............... 147 A.1. Basics......................................... ........... 147 A.2. Automorphisms of g .........................................152 A.3. RepresentationsandCharacters. .................. .............. 155 AppendixB. UntwistedAffineLieAlgebras.. .. .. .. .. .. .. ............. 159 B.1. Basics......................................... ........... 159 B.2. Automorphisms of g .........................................162 B.3. Representations, Characters, and Modularity . ................ 167 b AppendixC. CompactLieGroups ......................... ........... 171 C.1. Basics......................................... ........... 171 C.2. ALittleTopology................................ ........... 174 C.3. TheTopologyofLieGroups......................... .......... 177 Bibliography....................................... ............... 183 CHAPTER 1 Introduction 1.1. Background and Motivation Wess-Zumino-Witten models are examples of string theories, although they need not necessarily satisfy all the consistency conditions that one expects [90, 131, 132]. The string field is given by a map g from the string worldsheet Σ to the target space. This is illustrated in Figure 1.1. What distinguishes Wess-Zumino-Witten models, and makes them particularly tractable, is that the target space is the underlying manifold of a Lie group G. This tractability allows for a more detailed study of various aspects of string theory than is usually possible on a topologically non-trivial target space. These models also provide convenient examples for testing new ideas in string theory. The action gov- erning Wess-Zumino-Witten models will be constructed in Chapter 6 (where the reasons for their appellation will also become evident). As string theories, Wess-Zumino-Witten models should be conformally invariant on Σ. That is, these models should define a conformal field theory on the string worldsheet, and this is also referred to as a Wess-Zumino-Witten model. These theories have a special place in the study of (two-dimensional) conformal field theories. They provide relatively accessible examples, with an extended symmetry algebra that is very well understood. Moreover, it is widely believed (though no proof has been offered) that Wess-Zumino- Witten models are the fundamental building blocks out of which all