Chern-Simons Gauge Theory: 20 Years After

Home , Chiara Nappi, Robert Penner, Sergei Gukov

%17-4 7XYHMIWMR Advanced 1EXLIQEXMGW S.-T. Yau, 7IVMIW Editor

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

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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, including 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 number 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 theory Dana Fine: A geometric alternative to gauge ﬁxing in Chern- Simons theory on S1 × Σ Stavros Garoufalidis: Chern-Simons theory and arithmetic Enore Guadagnini: Functional integration and abelian link invariants Sergei Gukov: Exact results for perturbative Chern-Simons theory with complex gauge group Christopher M. Herald: An SU(3) Casson invariant for rational homology spheres Kazuhiro Hikami: WRT invariants and modular forms Rinat M. Kashaev: On rings associated with ideal triangulations of knot complements Louis H. Kauffman: Khovanov homology and the Potts model Mikhail Khovanov: Categoriﬁcation of quantum groups Paul Kirk: Non-abelian representations, homology 3-spheres, and knot concordance Albrecht Klemm: Chern-Simons theory and topological string theory on non-compact Calabi-Yau manifolds Christine Lescop: On the cube of the equivariant linking pairing for closed 3-manifolds of rank one Marcos Marino:˜ Chern-Simons theory, the 1/N expansion, and string theory Gregor Masbaum: Integral structures in TQFT and the mapping class group Catherine Meusburger: Getting physics from 3d gravity: What does an observer in 3d gravity see? Aleksandar Mikovic:´ Invariants of spin networks embedded in 3- manifolds

xi xii List of Speakers

Hitoshi Murakami: SL(2, C)-representations and asymptotic behav- iors of the colored Jones polynomial of a knot Robert C. Penner: Fatgraphs and ﬁnite type invariants Albert Schwarz: Generalizations of Chern-Simons theory Ambar N. Sengupta: Functional integrals in low-dimensional gauge theories George Thompson: Chern-Simons theory on Seifert 3-manifolds Jonathan Weitsman: Fermionization and convergent perturbation expansions in Chern-Simons gauge theory Edward Witten: Branes and quantization List of Participants

Sergio Albeverio, University of Bonn Jørgen E. Andersen, University of Aarhus Dave Auckly, Kansas State University Dror Bar-Natan, University of Toronto Stefan Bauer, University of Bielefeld Chris Beasley, Stony Brook University Stefan Behrens, University of Bonn Adara Blaga, West University of Timisoara Hans U. Boden, McMaster University Francesco Costantino, IRMA, University of Strasbourg Tien Cuong Dinh, Professor at Paris 6 University Jérome Dubois, Université Paris 7, Denis Diderot Magnus Engenhorst, Institut für Mathematik, Universität Augsburg João Faria Martins, Centro de Matemática da Universidade do Porto Alexander Felshtyn, University of Szczecin and MPIM, Bonn Dana Fine, University of Massachusetts Stavros Garoufalidis, Georgia Institute of Technology Masha Gordina, University of Connecticut/University of Bielefeld Enore Guadagnini, University of Pisa Sergei Gukov, California Institute of Technology Atle Hahn, Universidade de Lisboa Luiz Hartmann, USP - Universidade de São Paulo Andriy Haydys, University of Bielefeld Chris Herald, University of Nevada Michael Heusener, Université Blaise Pascal Clermont II Kazuhiro Hikami, University of Tokyo Benjamin Himpel, University of Bonn Iulia Elena Hirica, University of Bucharest Fuji Hiroyuki, Nagoya University Lotte Hollands, University of Amsterdam Saeid Jafari, College of Vestsjaelland South (VUC) Franck Jedrzejewski, CEA (French Atomic Commission) Alexander Kahle, University of Göttingen Uwe Kaiser, Boise State University Rinat Kashaev, University of Geneva Louis H. Kauffman, Math Dept, University of Illinois at Chicago Gerald Kelnhofer, University of Wien Mikhail Khovanov, Columbia University Hoil Kim, Kyungpook National University Paul Kirk, Indiana University xiii xiv List of Participants

Albrecht Klemm, University of Bonn Valentin Krasontovitsch, University of Bonn (Math student) Markus Land, University of Bonn Jonatan Lenells, Cambridge University Christine Lescop, University of Grenoble I Poon Chuan Adrian Lim, University of Luxembourg Christoph Luedeling, University of Bonn Marcos Mariño, University of Geneva Cristina Martinez, Autonoma University of Barcelona Gregor Masbaum, Université Paris 7 Gwenael Massuyeau, CNRS, University of Strasbourg Andrew McIntyre, Bennington College Catherine Meusburger, University of Hamburg Jouko Mickelsson, University of Helsinki and Royal Inst. of Technology Aleksandar Mikovic, Universidade Lusofona, Lisbon Hitoshi Murakami, Tokyo Institute of Technology Chiara Nappi, Institute for Advanced Study Robert Penner, University of California, Los Angeles Arturo Prat-Waldron, University of California, Berkeley and MPIM Bonn Justin Roberts, University of California, San Diego Nobuya Sato, Rikkyo University Albert Schwarz, UCDavis/MPIM Ambar Sengupta, Louisiana State University Jan Swoboda, MPIM, Bonn George Thompson, International Centre for Theoretical Physics ICTP Giorgio Trentinaglia, Georg-August-Universität Göttingen Alessandro Valentino, University of Göttingen Roland van der Veen, University of Amsterdam Jonathan Weitsman, Northeastern University Edward Witten, Institute for Advanced Study Don Zagier, Max-Planck-Institute for Mathematics José Miguel Zapata Rolón, University of Bonn Lucy Liuxuan Zhang, University of Toronto Titles in This Series

50 Jørgen E. Andersen, Hans U. Boden, Atle Hahn, and Benjamin Himpel, Chern-Simons Gauge Theory: 20 Years After, 2011 49 Matthias Plaue, Alan Rendall, and Mike Scherfner, Advances in Lorentzian Geometry, 2011 48 Lizhen Ji, Kefeng Liu, Lo Yang, and Shing-Tung Yau, Fourth International Congress of Chinese Mathematicians, 2010 47 Alexander Grigor’yan, Heat Kernel and Analysis on Manifolds, 2009 46.2 Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Editors, Lagrangian Intersection Floer Theory, 2009 46.1 Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian Intersection Floer Theory, 2009 45 Lydia Bieri and Nina Zipser, Extensions of the Stability Theorem of the Minkowski Space in General Relativity, 2009 44 Eric Sharpe and Arthur Greenspoon, Editors, Advances in String Theory, 2008 43 Lizhen Ji, Editor, Arithmetic Groups and Their Generalizations, 2008 42.1 Ka-Sing Lau, Zhou-Ping Xin, and Shing-Tung Yau, Editors, Third International Congress of Chinese Mathematicians, 2008 41 Wen-Ching Winnie Li, Editor, Recent Trends in Coding Theory and its Applications, 2007 40 Ovidiu Caliin, Der-Chen Chang, and Peter Greiner, Editors, Geometric Analysis on the Heisenberg Group and Its Generalizations, 2007 39 Zhijie Chen, Sheng-Li Tan, Jianpan Wang, and Stephen S.-T. Yau, Editors, Proceedings of the International Conference on Complex Geometry and Related Fields, 2007 38 Noriko Yui, Shing-Tung Yau, and James D. Lewis, Editors, Mirror Symmetry V, 2006 37 Lizhen Ji, Jian-Shu Li, H. W. Xu, and Shing-Tung Yau, Editors, Lie Groups and Automorphic Forms, 2006 36 Chuu-Lian Terng, Editor, Integrable Systems, Geometry, and Topology, 2006 35 Felix Finster, The Principle of the Fermionic Projector, 2006 34 Ren-Hong Wong, Editor, Computational Geometry, 2003 33 Eric D’Hoker, Duong Phong, and Shing-Tung Yau, Mirror Symmetry IV, 2002 32 Xi-Ping Zhu, Lectures on Mean Curvature Flows, 2002 31 Kiyoshi Igusa, Higher Franz-Reidemeister Torsion, 2002 30 Weiman Han and Mircea Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, 2002 29 S. T. Yau and Shuxing Chen, Editors, Geometry and Nonlinear Partial Diﬀerential Equations, 2002 28 Valentin Afraimovich and Sze-Bi Hsu, Lectures on Chaotic dynamical Systems, 2002 27 M. Ram Murty, Introduction to p-adic Analytic Number Theory, 2002 26 Raymond Chan, Yue-Kuen Kwok, David Yao, and Qiang Zhang, Editors, Applied Probability, 2002 25 Donggao Deng, Daren Huang, Rong-Qing Jia, Wei Lin, and Jian Zhong Wong, Editors, Wavelet Analysis and Applications, 2002 24 Jane Gilman, William W. Menasco, and Xiao-Song Lin, Editors, Knots, Braids, and Mapping Class Groups—Papers Dedicated to Joan S. Birman, 2001 23 Cumrun Vafa and S.-T. Yau, Editors, Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds, 2001 22 Carlos Berenstein, Der-Chen Chang, and Jingzhi Tie, Laguerre Calculus and Its Applications on the Heisenberg Group, 2001 21 J¨urgen Jost, Bosonic Strings: A Mathematical Treatment, 2001 20 Lo Yang and S.-T. Yau, Editors, First International Congress of Chinese Mathematicians, 2001 TITLES IN THIS SERIES

19 So-Chin Chen and Mei-Chi Shaw, Partial Differential Equations in Several Complex Variables, 2001 18 Fangyang Zheng, Complex Differential Geometry, 2000 17 Lei Guo and Stephen S.-T. Yau, Editors, Lectures on Systems, Control, and Information, 2000 16 Rudi Weikard and Gilbert Weinstein, Editors, Differential Equations and Mathematical Physics, 2000 15 Ling Hsiao and Zhouping Xin, Editors, Some Current Topics on Nonlinear Conservation Laws, 2000 14 Jun-ichi Igusa, An Introduction to the Theory of Local Zeta Functions, 2000 13 Vasilios Alexiades and George Siopsis, Editors, Trends in Mathematical Physics, 1999 12 Sheng Gong, The Bieberbach Conjecture, 1999 11 Shinichi Mochizuki, Foundations of p-adic Teichmüller Theory, 1999 10 Duong H. Phong, Luc Vinet, and Shing-Tung Yau, Editors, Mirror Symmetry III, 1999 9 Shing-Tung Yau, Editor, Mirror Symmetry I, 1998 8 Jürgen Jost, Wilfrid Kendall, Umberto Mosco, Michael Röckner, and Karl-Theodor Sturm, New Directions in Dirichlet Forms, 1998 7 D. A. Buell and J. T. Teitelbaum, Editors, Computational Perspectives on Number Theory, 1998 6 Harold Levine, Partial Differential Equations, 1997 5 Qi-keng Lu, Stephen S.-T. Yau, and Anatoly Libgober, Editors, Singularities and Complex Geometry, 1997 4 Vyjayanthi Chari and Ivan B. Penkov, Editors, Modular Interfaces: Modular Lie Algebras, Quantum Groups, and Lie Superalgebras, 1997 3 Xia-Xi Ding and Tai-Ping Liu, Editors, Nonlinear Evolutionary Partial Differential Equations, 1997 2.2 William H. Kazez, Editor, Geometric Topology, 1997 2.1 William H. Kazez, Editor, Geometric Topology, 1997 1 B. Greene and S.-T. Yau, Editors, Mirror Symmetry II, 1997

In 1989, Edward Witten discovered a deep relationship between quantum field theory and knot theory, and this beautiful discovery created a new field of research called Chern-Simons theory. This field has the remarkable feature of intertwining a large number of diverse branches of research in mathematics and physics, among them low-dimen- American sional topology, differential geometry, quantum algebra, functional and Mathematical stochastic analysis, quantum gravity, and string theory. Society The 20-year anniversary of Witten’s discovery provided an opportu- www.ams.org nity to bring together researchers working in Chern-Simons theory for a meeting, and the resulting conference, which took place during the summer of 2009 at the Max Planck Institute for Mathematics in -RXIVREXMSREP Bonn, included many of the leading experts in the field. This volume 4VIWW documents the activities of the conference and presents several original www.intlpress.com research articles, including another monumental paper by Witten that is sure to stimulate further activity in this and related fields. This collection will provide an excellent overview of the current research directions and recent progress in Chern-Simons gauge theory.

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