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Chern-Simons Gauge Theory: 20 Years After %17-4 7XYHMIWMR Advanced 1EXLIQEXMGW S.-T. Yau, 7IVMIW Editor Chern-Simons Gauge Theory: 20 Years After Jørgen E. Andersen Hans U. Boden Atle Hahn Benjamin Himpel Editors %QIVMGER1EXLIQEXMGEP7SGMIX]-RXIVREXMSREP4VIWW Chern-Simons Gauge Theory: 20 Years After https://doi.org/10.1090/amsip/050 AMS/IP Studies in Advanced Mathematics Volume 50 Chern-Simons Gauge Theory: 20 Years After Jørgen E. Andersen Hans U. Boden Atle Hahn Benjamin Himpel Editors American Mathematical Society • International Press Shing-Tung Yau, General Editor 2000 Mathematics Subject Classification. Primary 11F23, 14E20, 16S10, 19L10, 20C20, 20F99, 20L05, 30F60, 32G15, 46E25, 53C50, 53D20, 53D99, 54C40, 55R80, 57M25, 57M27, 57M60, 57M50, 57N05, 57N10, 57R56, 58D27, 58D30, 58E09, 58J28, 58J30, 58J52, 58Z05, 70S15, 81T08, 81T13, 81T25, 81T30, 81T45, 83C80. Library of Congress Cataloging-in-Publication Data Chern-Simons gauge theory: 20 years after / Jørgen E. Andersen ...[et al.], editors. p. cm. (AMS/IP studies in advanced mathematics ; v. 50) Proceedings of a workshop held at the Max Planck Institute for Mathematics in Bonn, Aug. 3–7, 2009. Includes bibliographical references. ISBN 978-0-8218-5353-5 (alk. paper) 1. Number theory—Congresses. 2. Algebraic topology—Congresses. 3. Associative rings— Congresses. 4. K-theory—Congresses. 5. Group theory—Congresses. I. Andersen, Jørgen E. 1965– QA241.C6355 2010 514.74—dc22 2011012166 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2011 by the American Mathematical Society and International Press. All rights reserved. The American Mathematical Society and International Press retain all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ Visit the International Press home page at http://www.intlpress.com/ 1098765432 16151413 Contents Preface ix Remarks on Wilson Loops and Seifert Loops in Chern-Simons Theory 1 Chris Beasley Quantum Field Theory and the Volume Conjecture 19 Tudor Dimofte and Sergei Gukov Computational Aspects in Reidemeister Torsion and Chern–Simons Theories 43 Jer´ omeˆ Dubois Functional Integration and Abelian Link Invariants 65 Enore Guadagnini Chern-Simons Invariants, SO(3) Instantons, and Z/2 Homology Cobordism 83 Matthew Hedden and Paul Kirk Extending the SU(3) Casson Invariant to Rational Homology 3-Spheres 115 Christopher M. Herald Decomposition of Witten–Reshetikhin–Turaev Invariant: Linking Pairing and Modular Forms 131 Kazuhiro Hikami Representations and the Colored Jones Polynomial of a Torus Knot 153 Kazuhiro Hikami and Hitoshi Murakami Eta-Invariants and Anomalies in U(1) Chern-Simons Theory 173 Lisa Jeffrey and Brendan McLellan Delta-Groupoids and Ideal Triangulations 201 Rinat M. Kashaev Invariants of Knots and 3–Manifolds Derived from the Equivariant Linking Pairing 217 Christine Lescop Chern–Simons Theory, the 1/N Expansion, and String Theory 243 Marcos Marino˜ v vi Contents Global Lorentzian Geometry from Lightlike Geodesics: What Does an Observer in (2+1)-Gravity See? 261 Catherine Meusburger Spin Foam State Sums and Chern-Simons Theory 277 Aleksandar Mikovic´ and Joao Faria Martins Representations of the Ptolemy Groupoid, Johnson Homomorphisms, and Finite Type Invariants 285 Robert C. Penner Yang-Mills in Two Dimensions and Chern-Simons in Three 307 Ambar N. Sengupta Intersection Pairings on Spaces of Connections and Chern-Simons Theory on Seifert Manifolds 317 George Thompson Fermionization and Convergent Perturbation Expansions in Chern-Simons Gauge Theory 331 Jonathan Weitsman Analytic Continuation of Chern-Simons Theory 347 Edward Witten Preface The workshop Chern-Simons gauge theory: 20 years after washeldatthe Max-Planck-Institute for Mathematics in Bonn from August 3 until August 7, 2009. It brought together mathematicians and physicists from many countries, in- cluding Austria, Brazil, Canada, Denmark, France, Germany, Japan, Korea, Lux- embourg, Netherlands, Poland, Portugal, Romania, Spain, Sweden, Switzerland, United Kingdom and the United States of America, to discuss their research pro- grams in geometric topology, stochastic analysis and mathematical physics. What was shared by this diverse group was a strong interest in Chern-Simons theory, and this is reflected in the talks of the conference as well as in the papers in this volume. Witten introduced Quantum Chern-Simons theory to knot theory in 1989, when he described the Jones polynomial of a link in S3 as a certain (non-rigorous) Feyn- man path integral, more specifically as the quantum field theory whose action is a multiple of the Chern-Simons function. He interpreted these invariants using the axioms of topological quantum field theory (TQFT) as well as via an asymptotic expansion. Since then, there have been several advances in understanding both approaches to Chern-Simons theory, starting with the first rigorous construction of the TQFT version by Reshetikhin and Turaev. This gave rise to quantum topology, a vibrant field which is currently moving in a number of directions including: • categorification, • modular forms, making contact with number theory, • Gromov-Witten theory via Large N duality, and • asymptotic analysis making contact with finite type invariants and hy- perbolic geometry, for example via the volume conjecture and the AJ- conjecture. Many of these topics were touched upon in the talks delivered at the conference and are discussed further in the papers of this volume. One of the outstanding open problems in Chern-Simons theory is to provide a mathematically rigorous treatment for the Feynman path integral, and the difficulty has to do with the problem of constructing a suitable gauge-invariant measure on the infinite dimensional space of all connections. A new approach to this difficult problem is provided by stochastical analysis, and this approach was also discussed at the conference and is the focus for several papers in this volume. Another important aspect of Chern-Simons gauge theory that was featured at the conference has to do with Floer homology theory and related invariants for 3-manifolds. The starting point is Taubes’ result from 1989 which allows one to interpret the Casson invariant as the Euler characteristic of the instanton homology defined by Floer, and some new directions of research in Floer homology and Casson invariants are presented in the papers of this volume. ix xPreface In addition to its many scientific activities, the conference featured a num- ber of pleasant social activities, including a boat ride on the Rhine, a hike up the Drachenfels, a visit to the Biergarten in the Rheinau, and a fun wine and pretzel party at the Hausdorff Institute with musical performances in classical, jazz, and rock music by a number of talented individuals, including Dana Fine, Chris Her- ald, Ben Himpel, Diane and Louis Kauffman, Paul Kirk, Markus Land, Cristina Martinez, Justin Roberts, Roland van der Veen and Lucy Zhang. We are especially grateful to Matthias Kreck for making the Hausdorff Institute available to us for this memorable evening. There were many people who helped to make the conference a success, and we would like to take this opportunity to recognize their contributions. To begin, the conference received funding through the Hausdorff Center for Mathematics in Bonn and the ITGP network of the European Science Foundation, and we would like to thank all the staff at the Hausdorff Institute and the Max-Planck-Institute in Bonn for their friendly assistance during the conference. The editors would also like to thank Arthur Greenspoon for his sage advice and Debbie Iscoe for her expertise in preparing the final versions of the manuscripts. Finally, we would like to thank all the authors who contributed their work to this volume, and especially Edward Witten, who delivered an inspirational lecture at the conference and contributed another monumental paper that is sure to stimulate further activity in this and related fields. Jørgen E. Andersen, Hans U. Boden, Atle Hahn, and Benjamin Himpel November 2010 List of Speakers Dave Auckly: Gauge-string duality and the structure of large rank Chern-Simons invariants Dror Bar-Natan: Convolutions on Lie groups and Lie algebras and ribbon 2-knots Dave Auckly: Gauge-string duality and the structure of large rank Chern-Simons invariants Dror Bar-Natan: Convolutions on Lie groups and Lie algebras and ribbon 2-knots Chris Beasley: Localization for Wilson loops in Chern-Simons
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