Proceedings of Symposia in PURE MATHEMATICS

Volume 98

String-Math 2016

String-Math 2016 June 27–July 2, 2016 College` de France, Paris, France

Amir-Kian Kashani-Poor Ruben Minasian Nikita Nekrasov Boris Pioline Editors Proceedings of Symposia in PURE MATHEMATICS

Volume 98

String-Math 2016

String-Math 2016 June 27–July 2, 2016 College` de France, Paris, France

Amir-Kian Kashani-Poor Ruben Minasian Nikita Nekrasov Boris Pioline Editors 2010 Mathematics Subject Classification. Primary 14D24, 14H60, 14D21, 14J33, 58E20, 81T60, 81T30.

Library of Congress Cataloging-in-Publication Data Names: String-Math (Conference) (2016: Paris, France) | Kashani-Poor, Amir-Kian, 1974– editor. | Minasian, Ruben, 1967– editor. | Nekrasov, Nikita, 1973– editor. | Pioline, Boris, 1972– editor. Title: String-Math 2016: June 27–July 2, 2016, Coll`ege de France, Paris, France / Amir-Kian Kashani-Poor, Ruben Minasian, Nikita Nekrasov, Boris Pioline, editors. Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Proceed- ings of symposia in pure mathematics; volume 98 | Includes bibliographical references. Identifiers: LCCN 2017052181 | ISBN 9781470435158 (alk. paper) Subjects: LCSH: Geometry, Algebraic–Congresses. | Quantum theory–Mathematics–Congresses. | AMS: Algebraic geometry – Families, fibrations – Geometric Langlands program: algebro- geometric aspects. msc | Algebraic geometry – Curves – Vector bundles on curves and their moduli. msc | Algebraic geometry – Families, fibrations – Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory). msc | Algebraic geometry – Surfaces and higher-dimensional varieties – Mirror symmetry. msc | Global analysis, analysis on manifolds – Variational problems in infinite-dimensional spaces – Harmonic maps. msc | Quantum theory – Quantum field theory; related classical field theories – Supersymmetric field theories. msc | Quantum theory – Quantum field theory; related classical field theories – String and superstring theories; other extended objects (e.g., branes). msc Classification: LCC QA564 .S77 2016 | DDC 516.3/5–dc23 LC record available at https://lccn.loc.gov/2017052181

DOI: http://dx.doi.org/10.1090/pspum/098

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Preface v List of participants xi Three-dimensional N = 4 gauge theories in omega background Mathew Bullimore 1 3d supersymmetric gauge theories and Hilbert series Stefano Cremonesi 21 Quantized Coulomb branches of Jordan quiver gauge theories and cyclotomic rational Cherednik algebras Ryosuke Kodera and Hiraku Nakajima 49 Supersymmetric field theories and geometric Langlands: The other side of the coin Aswin Balasubramanian and Jorg¨ Teschner 79 A journey from the Hitchin section to the oper moduli Olivia Dumitrescu 107 S-duality of boundary conditions and the Geometric Langlands program Davide Gaiotto 139 Pure SU(2) gauge theory partition function and generalized Bessel kernel P. Gavrylenko and O. Lisovyy 181 Reduction for SL(3) pre-buildings Ludmil Katzarkov, Pranav Pandit, and Carlos Simpson 207 Conformal nets are factorization algebras Andre´ Henriques 229 Contracting the Weierstrass locus to a point Alexander Polishchuk 241 Spectral theory and mirror symmetry Marcos Marino˜ 259

iii

Preface

String-Math is the annual conference that was founded to reflect the most signif- icant progress at the interface of string theory and mathematics. It is a young offspring of the annual Strings conferences, which have been gathering world ex- perts and young researchers working in the field of string theory since 1989. The first String-Math conference was held in 2011 at the University of Pennsylvania in Philadelphia, followed by the conferences in Bonn (2012), Stony Brook (2013), Edmonton (2014), and Hainan (2015). The String-Math 2016 conference took place at Coll`ege de France in Paris from June 27th till July 2nd, and this volume records the contributions which were presented there.

The fruitful exchange of ideas between physicists and mathematicians has a long history. The new era that began about 30 years ago with the advent of string theory and topological field theories is remarkable not only due to the increased intensity of mutually beneficial interactions between mathematicians and physicists, but also because each side is encountering the other on her own turf. Indeed, string theory has been contributing to areas deep in the heart of mathematics, far (or so we thought) from the math–physics frontier: from the classification of four manifolds using Seiberg-Witten theory, to holomorphic curve counting and other questions in enumerative geometry via topological string theory, to the study of knot invariants and their categorification via topological field theory, to progress in the geometric Langlands program inspired by the study of supersymmetric gauge theories. Conversely, mathematics developed for mathematics’ sake has deeply influenced how physicists think about string theory, ranging from re-interpreting D-branes in terms of derived categories of sheaves or Lagrangian submanifolds, analyzing string compactifications using notions of generalized geometry, to the computation of elliptic genera and string amplitudes by invoking techniques from number theory. It is a safe bet that the interaction between the two fields will continue to flourish and lead to stunning new results (and this is independent of what surprises lie in store at the Large Hadron Collider of CERN in the next few years).

The String-Math 2016 conference gathered more than 200 mathematicians and physicists to discuss new, exciting developments at the interface between high en- ergy theory and mathematics. All talks were plenary, with 17 longer talks (50 min) and 18 shorter talks (25 min) given in equal numbers by physicists and mathemati- cians working at the interface of high-energy theory and mathematics (with some uncertainty owing to the thin line that separates the two fields in some subdomains). The talks covered a wide range of topics, including moduli spaces of various struc- tures such as curves, meromorphic connections and differentials, coherent sheaves,

v vi PREFACE and quiver representations; BPS states and wall-crossing phenomena; topological string amplitudes in compact and non-compact geometries; exact results in super- symmetric gauge theories in various dimensions; 4D/2D correspondence; applica- tions to the geometric Langlands program; integrability in N=4 super Yang-Mills theory; and assorted topics such as moonshine, topological quantum field theory, or analytic aspects of black hole mergers. In addition to the main scientific program, a special session comprising four lectures targeting a general scientific audience was organized in collaboration with S´eminaire Poincar´e on July 2nd, 2016.

All talks, including the general public session, were recorded and broadcast live. The recordings and slides can be found at the conference website: indico.cern.ch/e/string-math2016.

This volume collects the written contributions from some (but unfortunately not all) of the speakers. Below we record the complete list of talks, with a one-sentence summary which hopefully will convey the scope of each talk:

Plenary talks

(1) Mina Aganagic: Two mathematical applications of little string theory: In this talk, new variants of the AGT correspondence and of the quan- tum Langlands correspondence relating q-deformed W algebra conformal blocks and K-theoretic instanton counting were presented. (2) Benjamin Basso: Hexagons and 3-point functions: This talk described how the correlators of three single trace operators in N = 4 SYM theory can be computed at finite ’t Hooft coupling, thanks to the integrability in the planar limit, by using hexagonal patches as building blocks. (3) Mikhail Bershtein: Plane partitions and W algebras: This talk described a new example of W algebras labelled by three integers n, m, k,whose category of representations is equivalent to that of the product of three quantum groups gln|k, glk|m, glm|n. (4) Gaetan Borot: Chern-Simons theory on S3/G and topological strings: This talk discussed matrix models computing the SU(N) Chern-Simons partition function on spherical Seifert manifolds. (5) Mathew Bullimore: Monopoles, Vortices, and Vermas:Inthistalk,the notion of monopole operators creating and destroying vortices in 3d gauge theories was explored, and the implications for a finite version of the AGT correspondence discussed. (6) Stefano Cremonesi: 3D supersymmetric gauge theories and Hilbert series: In this talk, closely related to Nakajima’s talk, the Hilbert series of 3d N = 2 gauge theory was obtained via the counting of dressed ’t Hooft monopole operators. (7) Thibault Damour: Analytical Approaches to Coalescing Binary Black Holes: This talk reviewed the analytical and numerical methods used to predict the waveform emitted by coalescing binary black holes, with particular emphasis on the Effective One-Body method. PREFACE vii

(8) Olivia Dumitrescu: From the Hitchin component to opers:Inthistalk,a holomorphic description of the limiting oper appearing in Gaiotto’s con- jecture, proved by the speaker and collaborators, weas presented. (9) Abhijit Gadde: Conformal constraints on defects: In this talk, the con- straints imposed by conformal invariance on defects of arbitrary codimen- sion in any conformal field theory were explored. (10) Davide Gaiotto: Geometric Langlands applications of boundary condi- tions for maximally supersymmetric Yang Mills theory: General classes of branes in the two-dimensional sigma model on Hitchin moduli space were constructed, and their relation via the Geometric Langlands corre- spondence were established. (11) Jaume Gomis: Correlation Functions in Superconformal Field Theories: This talk discussed new exact methods for computing correlation func- tions of local operators in the Coulomb branch in four-dimensional N=2 superconformal field theories. (12) Nikolay Gromov: Quantum Spectral Curve for AdS/CFT and its applica- tions: In this talk, a Riemann-Hilbert type equation known as the quan- tum spectral curve was introduced and applications to the computation of scaling dimensions in planar N = 4 SYM were discussed. (13) Sarah Harrison: Umbral symmetry groups and K3 CFTs:Inthistalk,the extension of umbral moonshine to the case of singular CFTs and under the inclusion of worldsheet parity were discussed. (14) Andr´e Henriques: What Chern-Simons theory assigns to a point?:This talk proposed that Chern-Simons theory assigns the representations of based loop groups to points and discussed the implications of this point of view. (15) Min-xin Huang: Derivation of modular anomaly equation in compact elliptic Calabi-Yau spaces: In this talk, complementary to Katz’ talk, the modular anomaly equation for elliptically fibered Calabi-Yau spaces was derived from the BCOV holomorphic anomaly equations. (16) Kohei Iwaki: Exact WKB analysis, cluster algebras and Painlev´eequa- tions: The relation between Voros symbols in exact WKB analysis and cluster variables was described, and the notion of Voros symbols was gen- eralized to Painlev´e equations. (17) Sheldon Katz: Elliptically fibered Calabi-Yau threefolds: mirror symmetry and Jacobi forms: This talk explained how to compute the topological string partition function on elliptically fibered Calabi-Yau threefolds using a combination of B-model, homological mirror symmetry, and geometric techniques. (18) Maxim Kontsevich: Resurgence and exact quantization via holomorphic Floer cohomology: In this talk, the Riemann-Hilbert correspondence and wall-crossing problem was revisited by considering the Fukaya categories associated with a holomorphic symplectic manifold and a possibly singular analytic Lagrangian subvariety. viii PREFACE

(19) Oleg Lisovyy: Fredholm determinant and Nekrasov type representations for isomonodromic tau functions: In this talk, Fredholm determinant rep- resentations for isomonodromic tau functions of Fuchsian systems with n regular singular points on the Riemann sphere and general monodromy in GL(N,C) were derived. (20) Marcos Mari˜no: Spectral theory and topological strings:Thistalkpre- sented a conjectural correspondence between topological string theory on toric Calabi-Yau manifolds and the spectral theory of certain trace class operators, obtained by quantizing the mirror curves. (21) Gregory Moore: Framed BPS States In Two And Four Dimensions:This talk gave a broad review of wall-crossing phenomena in two and four dimensions, with applications to the categorification of knot invariants. (22) Hiraku Nakajima: Quantized Coulomb branches of 3d N=4 gauge theories and difference operators: This talk gave a mathematical construction of the Coulomb branch of these theories and their quantization. (23) Rahul Pandharipande: Moduli spaces of holomorphic and meromorphic differentials: This talk introduced a new moduli space of holomorphic/ meromorphic differentials on Riemann surfaces and made the connection to Pixton’s formulas and Witten’s r-spin class. (24) Alexander Polishchuk: Moduli spaces of curves with non-special divisors: This talk discussed the moduli spaces of pointed curves with possibly non- nodal singularities, in terms of a quotient of an affine scheme by a torus action. (25) Leonardo Rastelli: Higgs branches, vertex operator algebras and modular differential equations: This talk discussed the relationship between the Higgs branch of a D=4, N=2 SCFT and the associated vertex operator algebra, obtaining modular equations for the Schur index. (26) Jørgen Rennemo: Derived equivalences from a duality of non-abelian gauge linear sigma models: In this talk, the equivalence between the category of B-branes of GLSM duals proposed by Kentaro Hori was proved, and new examples of varieties exhibiting this duality presented. (27) Laura Schaposnik: Higgs bundles, branes and applications:Inthistalk, Higgs bundles for complex Lie groups and the associated Hitchin fibration were introduced, a natural construction of families of subspaces corre- sponding to different types of branes discussed, and applications to Lang- lands duality and representations of 3-manifolds presented. (28) Olivier Schiffmann: Cohomological Hall algebra actions and Kac polyno- mials: This talk related the cohomological Hall algebras associated to quivers with the Yangians constructed by Maulik and Okounkov, and showed that their Hilbert series are encoded by the Kac polynomials of the underlying quiver. (29) Pavol Severa: Poisson-Lie T-duality: Poisson-Lie T-duality was reviewed and explained in terms of Chern-Simons theory and its generalizations with appropriate boundary conditions. PREFACE ix

(30) Carlos Simpson: Reduction for SL(3) pre-buildings: This talk discussed the reduction of SL(3) spectral curves and their relations to harmonic maps and the WKB problem. (31) J¨org Teschner: SUSY field theories and geometric Langlands: The other side of the coin: In this talk, the implications of the AGT correspon- dence in the presence of surface operators for the geometric Langlands programme and its quantum version were discussed. (32) Richard Thomas: A Vafa-Witten invariant for projective surfaces:Inthis talk, the Vafa-Witten invariants for algebraic surfaces were discussed and related to the DT-invariants on non-compact Calabi-Yau threefolds. (33) Daisuke Yamakawa: Meromorphic connections and quivers:Inthistalk, the proof of Boalch’s generalization of the Crawley-Boevey result relating meromorphic connections on the Riemann sphere and quivers was pre- sented. (34) Shing-Tung Yau: Period integrals of algebraic manifolds and their differ- ential equations: This talk addressed the question of which solutions of Picard-Fuchs equations correspond to periods of algebraic manifolds. (35) Dimitri Zvonkine: The Chern character of the Verlinde bundle:Inthis talk, the Chern character of the Verlinde bundle was computed by invoking Teleman’s classification of semi-simple cohomological field theories.

General Public Session “Maths & Cordes”

(1) Hirosi Ooguri (Caltech and IPMU): What is gravity?: This talk introduced the general public to the challenges posed by the unification of gravity with the other fundamental forces in nature. (2) Andrei Okounkov (Columbia U.): Catching monodromy: The remarkable analytic properties of a certain class of special functions which play a central role in mathematical physics were explained and generalized. (3) Robbert Dijkgraaf (IAS Princeton): Quantum Geometry:Theinterac- tions between mathematics and physics over the last century were put in perspective, and ideas for unifying algebra and geometry via stringy geometry were outlined. (4) Nima Arkani-Hamed (IAS Princeton): Physics and Mathematics for the End of Spacetime: The difficulty of defining observables in a theory of quantum gravity were discussed, and a new geometrical approach for com- puting scattering amplitudes was presented.

The 2016 conference was organized by Boris Pioline (CERN, Gen`eve and LPTHE, Paris), Ruben Minasian (Commissariata ` l’Energie Atomique, Saclay), Amir-Kian Kashani-Poor (Ecole Normale Sup´erieure, Paris), Nikita Nekrasov (Simons Center for Geometry and Physics, Stony Brook), Philip Boalch (Universit´e Paris Sud- Orsay), Miranda Cheng (Amsterdam University), Alessandro Chiodo (Universit´e Pierre et Marie Curie - Paris 6), Maxim Kontsevich (Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette), and Don Zagier (Max-Planch Institut f¨ur Mathe- matik, Bonn). xPREFACE

Support from the following institutions is gratefully acknowledged: Clay Mathemat- ics Institute, Agence Nationale de la Recherche, Centre National de la Recherche Scientifique, Fondation Math´ematique Jacques Hadamard, F´ed´eration de Recherche Interactions Fondamentales, Fondation Meyer pour le d´eveloppement culturel et artistique, Institut Henri Poincar´e, Institut des Hautes Etudes Scientifiques, Insti- tutdePhysiqueTh´eorique du CEA, Institut de Math´ematiques de Jussieu - Paris Rive Gauche, Laboratoire de Physique Th´eorique et Hautes Energies, UPMC, Lab- oratoire de Physique Th´eorique de l’Ecole Normale Sup´erieure, National Science Foundation, Universit´e Paris Sud - Orsay, and Universit´e Pierre et Marie Curie - Paris 6.

After Paris, the next String-Math conferences are due to take place in Hamburg (2017), Sendai (2018), Uppsala (2019), and Stellenbosch (2020).

Amir-Kian Kashani-Poor Ruben Minasian Nikita Nekrasov Boris Pioline List of Participants

Aganagic Mina Beisert Niklas University of California at Berkeley ETH Z¨urich Aghaei Nezhla Bena Iosif DESY and Hamburg University IPhT, CEA, Saclay Alexandrov Sergey Benetti Genolini Pietro LCC, Montpellier Oxford University Anderson Louise Benvenuti Sergio Imperial College London SISSA, Trieste Arkani-Hamed Nima Bershtein Mikhail Institute for Advanced Study, Princeton Landau Institute for Theoretical Artamonov Semen Physics, Moscow Rutgers University Bertolini Marco Ashfaque Johar Duke University University of Liverpool Bettadapura Kowshik Ashmore Anthony Australian National University, Imperial College London Canberra Babalic Mirela Biquard Olivier IBS Center for Geometry and Physics, Ecole Normale Sup´erieure, Paris Pohang Boalch Philip Bachas Constantin Universit´e Paris Sud, Orsay Ecole Normale Sup´erieure, Paris Bonechi Francesco Balasubramanian Aswin INFN Firenze University of Hamburg and DESY Ballard Matthew Borot Gaetan University of South Carolina MPIM, Bonn Balzin Eduard Bossard Guillaume Laboratoire J.-A. Dieudonn´e, Nice CPHT, Ecole Polytechnique, Palaiseau Banerjee Sibasish Bouchard Vincent IPhT,CEA,Saclay University of Alberta Basso Benjamin Bousseau Pierrick LPTENS, Paris Imperial College London

xi xii PARTICIPANTS

Braghiroli Matteo Damour Thibault University of Rome La Sapienza IHES, Bures-sur-Yvette Brennan Theodore Davison Ben Rutgers, the State University of New EPFL, Lausanne Jersey Dijkgraaf Robbert Brini Andrea Institute for Advanced Study, Princeton Universit´e de Montpellier Donagi Ron Bugden Mark University of Pennsylvania Australian National University, Doryn Dmitry Canberra IBS, Center for Geometry and Physics, Bulgakova Daria Pohang IPHT, CEA, SACLAY Dumitrescu Olivia MPIM, Bonn Bullimore Mathew University of Oxford Durand Philippe CNAM, Paris Calvo Jackson Juan Carlos University of Oxford Elias Rebelo Jose Gustavo SISSA, Trieste Caorsi Matteo SISSA, Trieste Fang Ziqi King’s College London Cassani Davide LPTHE, UPMC Paris 6 FeldmanAndrey Weizmann Institute of Science, Rehovot Cheng Miranda University of Amsterdam Ferrari Francesca University of Amsterdam Chiodo Alessandro IMJ-PRG, UPMC Paris 6 Fiset Marc-Antoine University of Oxford Choi Jinwon Franchetti Guido Sookmyung Women’s University, Seoul Leibniz University Cirafici Michele Franco Sebastian IST, Lisbon City College of CUNY, New York Coimbra Andr´e Frenkel Edward IHES, Bures-sur-Yvette University of California at Berkeley Cosnier-Horeau Charles Gaberdiel Matthias CPHT, Palaiseau and LPTHE, Paris 6 ETH Z¨urich Cotti Giordano Gadde Abhijit SISSA, Trieste Institute for Advanced Study, Princeton Cremonesi Stefano Gahramanov Ilmar King’s College London Albert Einstein Institute, Potsdam Cromer Michael Gaiotto Davide Australian National University Perimeter Institute, Waterloo PARTICIPANTS xiii

Gautason Fridrik Huang Minxin IPhT,CEA,Saclay University of Science and Technology of China Geipel Jakob Leibniz Universit¨at, Hannover Hulik Ondrej Masaryk University, Brno Gomis Jaume Perimeter Institute, Waterloo Hull Christopher Imperial College London Goncharov Yegor Lebedev Physical Institute, Moscow Intriligator Kenneth University of California at San Diego Grana Mariana IPhT, CEA Saclay Israel Dan Grassi Alba LPTHE, UPMC Paris 6 ICTP, Trieste Iwaki Kohei Gromov Nikolay Nagoya University King’s College London Janda Felix Gu Jie IMJ-PRG, UPMC Paris 6 LPTENS, Paris Jefferson Patrick Guere Jeremy Harvard University Humboldt Universit¨at, Berlin Julia Bernard Gukov Sergei LPTENS Paris California Institute of Technology Katz Sheldon Gupta Rajesh University of Illinois at ICTP, Trieste Urbana-Champaign Gustafsson Henrik Keßler Enno Chalmers University of Technology MPI Leipzig Haehnel Philipp Kimura Takashi Trinity College Dublin Boston University Halmagyi Nicholas Kiritsis Elias LPTHE, UPMC Paris 6 APC, Universit´e Paris 7 Haouzi Nathan Kivinen Oscar University of California at Berkeley University of California at Davis Harrison Sarah Kontsevich Maxim Harvard University IHES, Bures-sur-Yvette Hatsuda Yasuyuki Kramer Reinier Universit´edeGen`eve Universiteit van Amsterdam Hemo Tamir Lasso Oscar Technion, Haifa IFT-UAM/CSIC, Madrid Henriques Andr´e Lazaroiu Calin Oxford University and Utrecht IBS Center for Geometry and Physics, University Pohang xiv PARTICIPANTS

Le Floch Bruno Muteeb Muhammad Nouman Princeton University ICTP, Trieste Li Wei Nagoya Hajime ITP, Chinese Academy of Science, Kanazawa University Beijing Nakajima Hiraku Lisovyi Oleg RIMS, Kyoto LMPT, Universit´edeTours Nedelin Anton Lockhart Guglielmo University of Milano-Bicocca Universiteit van Amsterdam Negut Andrei Longhi Pietro MIT, Boston Uppsala University Nekrasov Nikita Mari˜no Marcos SCGP, Stony Brook Universit´edeGen`eve Nian Jun Markou Chrysoula IHES, Bures-sur-Yvette LPTHE, UPMC Paris 6 Nieri Fabrizio Matsuo Yutaka Uppsala University The University of Tokyo Norton Thomas Minasian Ruben California Institute of Technology IPhT,CEA,Saclay Okounkov Andrei Miramontes J Luis Columbia University University of Santiago de Compostela Ooguri Hirosi Mischi Alexander California Institute of Technology University of Innsbruck Pandharipande Rahul Mohaupt Thomas ETH Z¨urich University of Liverpool Paquette Natalie Stanford University Monnier Samuel Universit´edeGen`eve Pawelkiewicz Michal IPHT, Saclay Moore Gregory Rutgers University Petrini Michela LPTHE, UPMC, Paris 6 Moosavian Seyed Faroogh Perimeter Institute, Waterloo Pezzella Franco INFN, Napoli Morrison David University of California at Santa Piazzalunga Nicolo’ Barbara SISSA, Trieste Mulase Motohico Pioline Boris University of California at Davis LPTHE, Paris 6 and CERN, Geneva Musaev Edvard Plencner Daniel Albert Einstein Institute, Golm LPTENS/LPTHE, Paris PARTICIPANTS xv

Plesser M. Ronen Schmid Christian Duke University University of California at Berkeley Policastro Giuseppe Sciarappa Antonio LPTENS, Paris KIAS, Seoul Polishchuk Alexander Severa Pavol University of Oregon Universit´edeGen`eve Pradisi Gianfranco Shahbazi Carlos University of Rome Tor Vergata IPhT, CEA, Saclay Prins Daniel Sharpe Eric IPhT,CEA,Saclay Virginia Tech Quigley Callum Sheshmani Artan University of Toronto Massachussetts Institute of Technology Rabinovici Eliezer Simpson Carlos IHES, Bures-sur-Yvette CNRS, Universit´e Nice Sophia Antipolis Ramirez Pedro F. Solard Gautier IFT Madrid University Milano Bicocca Rastelli Leonardo Strickland-Constable Charles YITP, Stony Brook IPhT, Saclay and IHES, Reid-Edwards Ron Bures-sur-Yvette University of Hull Stromwall Joakim Rennemo Surrey University Jørgen All Souls College, Oxford Svanes Eirik Ronzani LPTHE, UPMC Paris 6 Massimiliano SISSA, Trieste Tavakol Mehdi Ros Camacho Ana IBS Center for Geometry and Physics, IMJ-PRG, Paris 6 Pohang Rose Simon Taylor Washington K¨obenhavns Universitet Massachusetts Institute of Technology Rossi Paolo Teschner Joerg Universit´e de Bourgogne DESY, Hamburg Sasmal Soumya Theisen Stefan Universit´e Paris Sud Albert Einstein Institute, Golm Savelli Raffaele Thomas Richard CEA Saclay Imperial College London Schaposnik Laura Tonita Valentin University of Illinois at Chicago Humboldt University, Berlin Schiffmann Olivier Troost Jan Universit´edeParis-SudOrsay LPTENS, Paris xvi PARTICIPANTS

Turton David Zhang Hong IPhT,CEA,Saclay Sogang University, Seoul Vaintrob Dmitry Zhang Ming Massachusetts Institute of Technology University of Michigan van Garrel Michel Zvonkine Dimitri KIAS, Seoul IMJ-PRG, Paris 6 vanLeuvenSam University of Amsterdam Voros Andr´e IPhT, CEA Saclay Wakimoto Yuki Tokyo Metropolitan University Weitsman Jonathan Northeastern University, Boston Wimmer Robert AMS (Physical Review D), Astoria Windey Paul LPTHE, UPMC Paris 6 Wrase Timm TU Vienna Yamakawa Daisuke Tokyo Institute of Technology Yang Zhiwei University of Pisa Yang Di SISSA, Trieste Yau Shing Tung Harvard University Ye Ke California Institute of Technology Zabzine Maxim Uppsala University Zagier Don MPIM, Bonn Zenkevich Yegor INR, Moscow Zerbini Federico MPIM, Bonn PSPUM/98 PSPUM String-Math 2016 • Kashani-Poor et al., Editors 98 AMS