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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 Color graphic policy. Any graphics created in color will be rendered in grayscale for the printed version unless color printing is authorized by the Publisher. In general, color graphics will appear in color in the online version. 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 select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2018 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains 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/ 10987654321 232221201918 Contents 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
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  • Gravity, Gauge Theory and Strings

    Gravity, Gauge Theory and Strings

    Les Houches - Ecole d'Ete de Physique Theorique 76 Unity from Duality: Gravity, Gauge Theory and Strings Les Houches Session LXXVI, July 30 - August 31, 2001 Bearbeitet von Constantin P. Bachas, Adel Bilal, Michael R. Douglas, Nikita A. Nekrasov, Francois David 1. Auflage 2003. Buch. xxxiv, 664 S. Hardcover ISBN 978 3 540 00276 5 Format (B x L): 15,5 x 23,5 cm Gewicht: 1220 g Weitere Fachgebiete > Technik > Verfahrenstechnik, Chemieingenieurwesen, Lebensmitteltechnik Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. Preface The 76th session of the Les Houches Summer School in Theoretical Physics was devoted to recent developments in string theory, gauge theories and quantum gravity. As frequently stated, Superstring Theory is the leading candidate for a unified theory of all fundamental physical forces and elementary parti- cles. This claim, and the wish to reconcile general relativity and quantum mechanics, have provided the main impetus for the development of the the- ory over the past two decades. More recently the discovery of dualities, and of important new tools such as D-branes, has greatly reinforced this point of view. On the one hand there is now good reason to believe that the underlying theory is unique. On the other hand, we have for the first time working (though unrealistic) microscopic models of black hole mechan- ics.
  • Instanton Counting in Class Sk

    Instanton Counting in Class Sk

    Prepared for submission to JHEP DESY 17-189 Instanton counting in Class Sk Thomas Bourton, Elli Pomoni DESY Theory Group, Notkestraße 85, 22607 Hamburg, Germany E-mail: [email protected], [email protected] Abstract: We compute the instanton partition functions of N = 1 SCFTs in class Sk. We obtain this result via orbifolding Dp/D(p-4) brane systems and calculating the partition function of the supersymmetric gauge theory on the worldvolume of K D(p-4) branes. Starting with D5/D1 setups probing a Z` × Zk orbifold singularity we obtain the K instanton partition 4 2 2 functions of 6d (1; 0) theories on R × T in the presence of orbifold defects on T via com- puting the 2d superconformal index of the worldvolume theory on K D1 branes wrapping the T 2. We then reduce our results to the 5d and to the 4d instanton partition func- tions. For k = 1 we check that we reproduce the known elliptic, trigonometric and rational Nekrasov partition functions. Finally, we show that the instanton partition functions of SU(N) quivers in class Sk can be obtained from the class S mother theory partition func- 2πij=k tions with SU(kN) gauge factors via imposing the ‘orbifold condition’ aA ! aAe with A = jA and A = 1;:::;N, j = 1; : : : ; k on the Coulomb moduli and the mass parameters. arXiv:1712.01288v1 [hep-th] 4 Dec 2017 Contents 1 Introduction2 2 String theory description4 2.1 Type IIB realisation4 2.2 Type IIA realisation6 2.3 A 6d uplift7 2.4 On supersymmetry of the D1/D5 system9 3 4d N = 2∗ Instantons from a 2d superconformal index computation 10