Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics Number 111 Topology, C*-Algebras, and String Duality Jonathan Rosenberg American Mathematical Society with support from the National Science Foundation Topology, C*-Algebras, and String Duality http://dx.doi.org/10.1090/cbms/111 Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics Number 111 Topology, C*-Algebras, and String Duality Jonathan Rosenberg Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society Providence, Rhode Island with support from the National Science Foundation NSF-CBMS-Regional Conference in the Mathematical Sciences on Topology, C∗-Algebras, and String Duality held at Texas Christian University, Forth Worth, Texas May 18–22, 2009 Partially supported by the National Science Foundation. The author acknowledges support from the Conference Board of Mathematical Sciences and NSF Grant #0735233 2000 Mathematics Subject Classification. Primary 81T30; Secondary 81T75, 19K99, 46L80, 58B34, 55R10, 55P65, 55R50, 14J32, 53Z05. For additional information and updates on this book, visit www.ams.org/bookpages/cbms-111 Library of Congress Cataloging-in-Publication Data Rosenberg, J. (Jonathan), 1951– Topology, C∗-algebras, and string duality / Jonathan Rosenberg. p. cm. — (Regional conference series in mathematics ; no. 111) Includes bibliographical references and index. ISBN 978-0-8218-4922-4 (alk. paper) 1. Algebraic topology. 2. C∗-algebras. 3. Duality theory (Mathematics) I. Conference Board of the Mathematical Sciences. II. Title. QA612.R58 2009 514.2—dc22 2009032465 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. 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Visit the AMS home page at http://www.ams.org/ 10987654321 141312111009 Contents The symbol ✰ denotes either a more advanced section, or else a bit of a digression, which can be skipped without interrupting the flow of the rest of the book. Preface vii Chapter 1. Introduction and Motivation 1 1.1. Structure of Physical Theories 1 1.2. Some Basics of String Theory 3 1.3. Dualities Related to String Theory 7 1.4. ✰ More on S-Duality and AdS/CFT Duality 10 Chapter 2. K-Theory and its Relevance to Physics 13 2.1. A Quick Review of Topological K-Theory 13 2.2. K-Theory and D-Brane Charges 21 2.3. K-Homology and D-Brane Charges 22 Chapter 3. A Few Basics of C∗-Algebras and Crossed Products 25 3.1. Basics of C∗-Algebras 25 3.2. K-Theory of C∗-Algebras 29 3.3. Crossed Products 33 Chapter 4. Continuous-Trace Algebras and Twisted K-Theory 37 4.1. Continuous-Trace Algebras and the Brauer Group 37 4.2. Twisted K-Theory 40 4.3. ✰ The Theory of Gerbes 42 Chapter 5. More on Crossed Products and Their K-Theory 47 5.1. A Categorical Framework 47 5.2. Connes’ Thom Isomorphism 51 5.3. The Pimsner-Voiculescu Sequence 52 Chapter 6. The Topology of T-Duality and the Bunke-Schick Construction 55 6.1. Topological T-Duality 55 6.2. The Bunke-Schick Construction 57 Chapter 7. T-Duality via Crossed Products 63 7.1. Group Actions on Continuous Trace-Algebras 63 7.2. The Raeburn-Rosenberg Theorem 68 Chapter 8. Higher-Dimensional T-Duality via Topological Methods 71 8.1. Higher-Dimensional T-Duality 71 8.2. A Higher-Dimensional Bunke-Schick Theorem 72 v vi CONTENTS Chapter 9. Higher-Dimensional T-Duality via C∗-Algebraic Methods 77 9.1. Methodology and a Key Example 77 9.2. Uniqueness of Group Actions 78 9.3. The General Case 81 Chapter 10. Advanced Topics and Open Problems 85 10.1. Mirror Symmetry 85 10.2. Other Solutions to the Missing T-Dual Problem 87 10.3. ✰ Fourier-Mukai Duality 88 10.4. ✰ Refinements of K-Theory 93 10.5. Open Problems 94 Bibliography 95 Notation and Symbols 103 Index 107 Preface This book is the outgrowth of an NSF/CBMS Regional Conference in the Math- ematical Sciences, May 18–22, 2009, organized by Robert Doran and Greg Friedman at Texas Christian University. I am highly indebted to Bob and Greg for their tire- less work in getting together funding for the conference, making all the logistical arrangements, recruiting participants and other speakers, and for keeping me on track during this entire process. The subject of this book, identical to the subject of the conference, is inter- disciplinary. Thus it involves a more-or-less equal mixture of topology, operator algebras, and physics. There is also a bit of algebraic geometry, especially in the last chapter. While I assume most readers of this book are probably somewhat fa- miliar with at least one of these subjects, they may not necessarily be knowledgeable about all or even most of them. So I have tried to include some basics on each one. The expert reader can skip over these sections. Roughly speaking, each chapter of this book corresponds to a single lecture from the conference, but “fleshed out” a little more. I have also included some sections, marked with a star, that I didn’t go into in detail in the lectures. These are more advanced, and someone just wanting an overview of the subject can skip these, though they might interest the more advanced reader. Since the book covers a lot of ground, it differs from most con- ventional mathematics books which follow the methodical “theorem-proof” style. There are of course plenty of theorems and proofs, but my main objective has been to show how several seemingly disparate subjects are closely linked with one an- other, and to give readers an overview of some areas of current research. In some places this happens at the expense of trying to cover everything systematically. I would also like to the thank the Conference Board of the Mathematical Sci- ences and the National Science Foundation for their support. NSF Grant DMS- 0735233 supported the conference, NSF Grant DMS-0602750 supported the entire Regional Conference program, and NSF Grants DMS-0504212 and DMS-0805003 supported the research that went into the preparation of the lectures and the writing of this book. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. Finally, I would also like to thank the Ruth M. Davis Professorship at the University of Maryland for partial support during the writing of this book, and I would like to thank the many participants at the CBMS conference for their lively participation and penetrating questions at the meeting, and for their valuable vii viii PREFACE feedback on the first draft of this manuscript. They caught many misprints, errors, and omissions, and also made many useful suggestions for improvements, and for this I am very grateful. Jonathan Rosenberg College Park, Maryland Bibliography 1. Ofer Aharony, Steven S. Gubser, Juan Maldacena, Hirosi Ooguri, and Yaron Oz, Large N field theories, string theory and gravity,Phys.Rep.323 (2000), no. 3-4, 183–386, arxiv.org: hep-th/9905111. MR1743597 (2001c:81134) 2. Clara Luc´ıa Aldana Dom´ınguez, Representation of a gauge group as motions of a Hilbert space, Ann. Math. Blaise Pascal 11 (2004), no. 2, 131–153, available at http://ambp.cedram.org/item?id=AMBP 2004 11 2 131 0. MR2109604 (2005h:58018) 3. E. Alvarez, L. Alvarez-Gaum´e, J. L. F. Barb´on, and Y. Lozano, Some global aspects of duality in string theory,NuclearPhys.B415 (1994), no. 1, 71–100, arxiv.org: hep-th/9309039. MR1265457 (95a:81197) 4. M. Ando, M. J. Hopkins, and N. P. Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Invent. 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