DOCSLIB.ORG

String-Math 2016

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
String-Math 2016 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
Load more

Recommended publications
  • Some Mathematical Applications of Little String Theory
    Some mathematical applications of little string theory Mina Aganagic UC Berkeley Based on joint works with Edward Frenkel and Andrei Okounkov, to appear Andrei Okounkov, arXiv:1604.00423 [math.AG] , Nathan Haouzi, arXiv:1506.04183 [hep-th] , String theory predicts existence of a remarkable quantum field theory in six dimensions, the (2,0) superconformal field theory. The theory is labeled by a simply-laced Lie algebra, . It is remarkable, in part, because it is expected to play an important role in pure mathematics: Geometric Langlands Program Knot Categorification Program The fact that the theory has no classical limit, makes it hard to extract its predictions. AGT correspondence, after Alday, Gaiotto and Tachikawa, serves well to illustrate both the mathematical appeal of the theory and the difficulty of working with it. The AGT correspondence states that the partition function of the -type (2,0) SCFT, on a six manifold of the form is a conformal block on the Riemann surface of a vertex operator algebra which is also labeled by : -algebra The correspondence further relates defects of the 6d theory to vertex operators of the -algebra, inserted at points on . x x x x x x If one is to take the conjecture at its face value, it is hard to make progress on it. Since we do not know how to describe the (2,0) SCFT, we cannot formulate or evaluate its partition function in any generality. (Exceptions are g=A_1 , or special choices of defects.) I will argue in this talk that one can make progress by replacing the 6-dimensional conformal field theory, which is a point particle theory, by the 6-dimensional string theory which contains it, the (2,0) little string theory.
    [Show full text]
  • Higher AGT Correspondences, W-Algebras, and Higher Quantum
    Higher AGT Correspon- dences, W-algebras, and Higher Quantum Geometric Higher AGT Correspondences, W-algebras, Langlands Duality from M-Theory and Higher Quantum Geometric Langlands Meng-Chwan Duality from M-Theory Tan Introduction Review of 4d Meng-Chwan Tan AGT 5d/6d AGT National University of Singapore W-algebras + Higher QGL SUSY gauge August 3, 2016 theory + W-algebras + QGL Higher GL Conclusion Presentation Outline Higher AGT Correspon- dences, Introduction W-algebras, and Higher Quantum Lightning Review: A 4d AGT Correspondence for Compact Geometric Langlands Lie Groups Duality from M-Theory A 5d/6d AGT Correspondence for Compact Lie Groups Meng-Chwan Tan W-algebras and Higher Quantum Geometric Langlands Introduction Duality Review of 4d AGT Supersymmetric Gauge Theory, W-algebras and a 5d/6d AGT Quantum Geometric Langlands Correspondence W-algebras + Higher QGL SUSY gauge Higher Geometric Langlands Correspondences from theory + W-algebras + M-Theory QGL Higher GL Conclusion Conclusion 6d/5d/4d AGT Correspondence in Physics and Mathematics Higher AGT Correspon- Circa 2009, Alday-Gaiotto-Tachikawa [1] | showed that dences, W-algebras, the Nekrasov instanton partition function of a 4d N = 2 and Higher Quantum conformal SU(2) quiver theory is equivalent to a Geometric Langlands conformal block of a 2d CFT with W2-symmetry that is Duality from M-Theory Liouville theory. This was henceforth known as the Meng-Chwan celebrated 4d AGT correspondence. Tan Circa 2009, Wyllard [2] | the 4d AGT correspondence is Introduction Review of 4d proposed and checked (partially) to hold for a 4d N = 2 AGT conformal SU(N) quiver theory whereby the corresponding 5d/6d AGT 2d CFT is an AN−1 conformal Toda field theory which has W-algebras + Higher QGL WN -symmetry.
    [Show full text]
  • Arxiv:1404.2323V1 [Math.AG] 8 Apr 2014 Membranes and Sheaves
    Membranes and Sheaves Nikita Nekrasov and Andrei Okounkov April 2014 Contents 1 A brief introduction 2 1.1 Overview.............................. 2 1.2 MotivationfromM-theory . 3 1.3 Planofthepaper ......................... 7 1.4 Acknowledgements ........................ 7 2 Contours of the conjectures 8 2.1 K-theorypreliminaries ...................... 8 2.2 Theindexsheaf.......................... 10 2.3 Comparison with Donaldson-Thomastheory . 15 2.4 Fields of 11-dimensional supergravity and degree zero DT counts 19 3 The DT integrand 24 3.1 Themodifiedvirtualstructuresheaf. 24 3.2 The interaction term Φ ...................... 28 arXiv:1404.2323v1 [math.AG] 8 Apr 2014 4 The index of membranes 33 4.1 Membranemoduli......................... 33 4.2 Deformationsofmembranes . 36 5 Examples 37 5.1 Reducedlocalcurves ....................... 37 5.2 Doublecurves........................... 39 5.3 Single interaction between smooth curves . 42 5.4 HigherrankDTcounts. .. .. 43 5.5 EngineeringhigherrankDTtheory . 45 1 6 Existence of square roots 49 6.1 Symmetricbundlesonsquares . 49 6.2 SquarerootsinDTtheory . 50 6.3 SquarerootsinM-theory. 52 7 Refined invariants 54 7.1 Actionsscalingthe3-form . 54 7.2 Localization for κ-trivialtori................... 56 7.3 Morsetheoryandrigidity . 60 8 Index vertex and refined vertex 61 8.1 Toric Calabi-Yau 3-folds . 61 8.2 Virtualtangentspacesatfixedpoints . 63 8.3 Therefinedvertex ........................ 66 A Appendix 70 A.1 Proofofthebalancelemma . 70 1 A brief introduction 1.1 Overview Our goal in this paper is to discuss a conjectural correspondence between enumerative geometry of curves in Calabi-Yau 5-folds Z and 1-dimensional sheaves on 3-folds X that are embedded in Z as fixed points of certain C×- actions. In both cases, the enumerative information is taken in equivariant K-theory, where the equivariance is with respect to all automorphisms of the problem.
    [Show full text]
  • Arxiv:1512.05388V2 [Hep-Th] 2 Jan 2016 1
    BPS/CFT CORRESPONDENCE: NON-PERTURBATIVE DYSON-SCHWINGER EQUATIONS AND qq-CHARACTERS NIKITA NEKRASOV Abstract. We study symmetries of quantum field theories involving topologically dis- tinct sectors of the field space. To exhibit these symmetries we define special gauge invariant observables, which we call the qq-characters. In the context of the BPS/CFT correspondence, using these observables, we derive an infinite set of Dyson-Schwinger- type relations. These relations imply that the supersymmetric partition functions in the presence of Ω-deformation and defects obey the Ward identities of two dimen- sional conformal field theory and its q-deformations. The details will be discussed in the companion papers. Contents 1. Introduction 4 1.1. Dyson-Schwinger equations 4 1.2. Non-perturbative Dyson-Schwinger identities 6 1.3. Organization of the presentation 8 1.4. Acknowledgements 10 2. The BPS/CFT correspondence 12 2.1. = 2 partition functions 12 2.2. DefectN operators and lower-dimensional theories 13 2.3. The Y- and X-observables 14 2.4. The physics of X-observables 14 arXiv:1512.05388v2 [hep-th] 2 Jan 2016 2.5. Hidden symmetries 17 2.6. Some notations. 17 2.7. Equivariant virtual Chern polynomials 22 3. Supersymmetric gauge theories 22 3.1. Quivers 22 3.2. Quivers with colors 22 3.3. The symmetry groups 22 3.4. The parameters of Lagrangian 24 3.5. The group H 25 3.6. Perturbative theory 26 3.7. Realizations of quiver theories 28 4. Integration over instanton moduli spaces 30 1 2 NIKITA NEKRASOV 4.1. Instanton partition function 30 4.2.
    [Show full text]
  • Geometric Langlands from 4D N=2 Theories
    Geometric Langlands from 4d N = 2 theories Aswin Balasubramanian DESY & U. Hamburg September 11, 2017 IMSc Seminar, Chennai based on 1702.06499 w J. Teschner + work with Ioana Coman-Lohi, J.T Aswin Balasubramanian Geometric Langlands from 4d N = 2 theories 1 / 46 Goals for my talk Part A : Light introduction to Geometric Langlands (Kapustin-Witten, Beilinson-Drinfeld, Our Motivating Questions) Part B : Review N = 2 Class S theories (Hitchin system, AGT correspondence) Part C: Aspects of Geometric Langlands from Class S Aswin Balasubramanian Geometric Langlands from 4d N = 2 theories 2 / 46 Goals for my talk Part A : Light introduction to Geometric Langlands (Kapustin-Witten, Beilinson-Drinfeld, Our Motivating Questions) Part B : Review N = 2 Class S theories (Hitchin system, AGT correspondence) Part C: Aspects of Geometric Langlands from Class S Aswin Balasubramanian Geometric Langlands from 4d N = 2 theories 2 / 46 Goals for my talk Part A : Light introduction to Geometric Langlands (Kapustin-Witten, Beilinson-Drinfeld, Our Motivating Questions) Part B : Review N = 2 Class S theories (Hitchin system, AGT correspondence) Part C: Aspects of Geometric Langlands from Class S Aswin Balasubramanian Geometric Langlands from 4d N = 2 theories 2 / 46 Part A : Approaches to Geometric Langlands Aswin Balasubramanian Geometric Langlands from 4d N = 2 theories 3 / 46 Historical Background The Langlands program has its roots in Number Theory, specifically the classification of Automorphic forms for G(Z). These generalize modular forms of SL(2; Z) . Fourier co-efficients of Automorphic forms encode very interesting Number Theoretic information. For several reasons, it was interesting to ask if there was a geometric analog.
    [Show full text]
  • Probabilistic Conformal Blocks for Liouville Cft on the Torus
    PROBABILISTIC CONFORMAL BLOCKS FOR LIOUVILLE CFT ON THE TORUS PROMIT GHOSAL, GUILLAUME REMY, XIN SUN, AND YI SUN Abstract. Virasoro conformal blocks are a family of important functions defined as power series via the Virasoro algebra. They are a fundamental input to the conformal bootstrap program for 2D conformal field theory (CFT) and are closely related to four dimensional supersymmetric gauge theory through the Alday- Gaiotto-Tachikawa correspondence. The present work provides a probabilistic construction of the 1-point toric Virasoro conformal block for central change greater than 25. More precisely, we construct an analytic function using a probabilistic tool called Gaussian multiplicative chaos (GMC) and prove that its power series expansion coincides with the 1-point toric Virasoro conformal block. The range (25; 1) of central charges corresponds to Liouville CFT, an important CFT originating from 2D quantum gravity and bosonic string theory. Our work reveals a new integrable structure underlying GMC and opens the door to the study of non-perturbative properties of Virasoro conformal blocks such as their analytic continuation and modular symmetry. Our proof combines an analysis of GMC with tools from CFT such as Belavin-Polyakov- Zamolodchikov differential equations, operator product expansions, and Dotsenko-Fateev type integrals. 1. Introduction A conformal field theory (CFT) is a way to construct random functions on Riemannian manifolds that transform covariantly under conformal (i.e. angle preserving) mappings. Since the seminal work of Belavin- Polyakov-Zamolodchikov in [BPZ84], two dimensional (2D) CFT has grown into one of the most prominent branches of theoretical physics, with applications to 2D statistical physics and string theory, as well as far reaching consequences in mathematics; see e.g.
    [Show full text]
  • Model Building and SUSY Breaking on D-Branes
    Model building and SUSY breaking on D-branes Dmitry Malyshev a dissertation presented to the faculty of princeton university in candidacy for the degree of doctor of philosophy recommended for acceptance by the department of physics Adviser: Herman Verlinde September 2008 c Copyright by Dmitry Malyshev, 2008. All rights reserved. Abstract In the recent years there has been an increase of the interest in applying the String theory to construct viable extensions of the Standard Model and to find stringy analogs of the known field theory models for supersymmetry breaking. In my dissertation I will focus on the constructions involving the D-branes at the Calabi-Yau singularities in type IIB string theory. The main motivations for this choice are: the decoupling of the D-brane field theory from the supergravity, the well established tools for deriving the field theories from the configurations of the D-branes, and the possibility of using the gauge-gravity correspondence in solving some non-perturbative aspects in the field theory. A construction of an extended supersymmetric Standard Model on a D3-brane at the del Pezzo 8 singularity is presented in the first part of the dissertation. In the second part we discuss the possible representations of the SUSY breaking models in String Theory and obtain the metastable SUSY breaking vacuum in a system of D-branes on the suspended pinch point singularity. The gauge mediation of this SUSY breaking model is described in details. In the simplest model the spectrum of particles in the visible sector has a split SUSY breaking, the sfermions are one order of magnitude heavier than the gauginos.
    [Show full text]
  • Nikita Nekrasov 講義録
    Nikita Nekrasov 講義録 December, 1998 講義録作成1 ：橋本幸士, 岸本 功 （京大理） 目次 第 I 部 超共形ゲージ理論と AdS 超重力理論 2 1 Introduction と motivation 3 2 Orbifold 理論 6 3 時空の状況 12 4 Orbifold から conifold への変形 17 5 まとめ 19 第 II 部 N =2ゲージ理論における状態のカウント 21 6 導入 22 7 D3-プローブ上の理論 22 8M理論への持ち上げ/引き落とし 27 9 結論 39 1この講義録は「Workshop ”ゲージ理論の力学と弦双対性” 1998 年 12 月 16 日（水）–12月 18 日（金）東京工業大学国際交流会館」における Nekrasov 氏の講義に基づくものです。第 I 部が 16 日に、第 II 部が 18 日に行われました。 1 第 I 部 超共形ゲージ理論とAdS超重力理論 (SUPER)CONFORMAL GAUGE THEORIES AND ANTI-DE-SITTER SUPERGRAVITY2 Nikita Nekrasov 1. Introduction と motivation ——哲学, D-brane と付随した場の理論, 特異な幾何からの conformal 理論 2. Orbifold 理論 ——理論の構築（：gauge 群, matter の内容, （超）対称性）, β 関数の計算, 摂動論の解析 3. 時空の状況 ——near-horizon limit, 超重力における変形 4. Orbifold から conifold への変形 ——場の理論の実現, Higgs branch の幾何, 時空解釈（conjecture） 2文献 [1, 2] に基づく。[3, 4, 5, 6, 7, 8] も見よ。 2 第 1 章 Introduction と motivation 双対性に関する動きの中でおそらく最も（そしておそらく唯一つの）重要な教訓は、面白い物理は面白い幾何に関係 しているという発想に立ち返るということである。Einstein の時代に比べての違いは、勿論量子的な物理量が古典的な 幾何を通じて表現されうるという仮定である。例えば、N =2SU(2) gauge 理論の有効結合定数（と長距離相関関数） は補佐的な楕円曲線 Eu の modular 変数 τ(u) を用いて表される： y2 =(x − u)(x2 − Λ4). (1) ここで u = hTrφ2i は真空の moduli 空間を label する。 u-plane θ(u) 4πi Eu: τ(u)= 2π + e2(u) vacuum 図 1: N =2SU(2) gauge 理論の moduli 空間。 物理から幾何、また逆に幾何から物理への変換を理解するのに必要不可欠な道具は D-brane である。D-brane に関し ての良い点は以下の二つである: (1) D-brane の上に弦が端点を持つので、いくつかの D-brane がお互いに重なるとその worldvolume 上に非可換 gauge boson が発生する。(2) D-brane は Ramond-Ramond 場の電荷を持っており、（適切な条 件の下で）弦理論における soliton として記述されうる。 (1) (2) 図 2: D-brane の二つの描像。(1) 開弦の端点としての D-brane、(2) 弦理論の soliton としての D-brane。 一般的な発想は、gauge 理論とそれに対応する soliton の背景を伝播する弦の理論を等しくするように、D-brane を用 いる、ということである。もし（IIB 型超弦理論で）N 枚の D3-brane が平坦な空間内でお互いに重なっていると、この 重なった brane は二つの記述法を持つ: 1.
    [Show full text]
  • Mini-Symposium on String Theory and M-Theory
    Mini-Symposium on String Theory and M-Theory Chair: Michael R. Douglas Duiliu-EmanuelDiaconescu JaumeGomis ChrisM.Hull AlbrechtKlemm Jos´eM.F.Labastida MarcosMarin˜ o NikitaNekrasov ChristophSchweigert AngelM.Uranga Duiliu-Emanuel Diaconescu Institute for Advanced Study, Princeton K-theory from M-theory Wednesday, 12.30 – 13.00, Room B I report on joint work with Gregory Moore and Edward Witten —“E8 Gauge Theory, and a Derivation of K-Theory from M-Theory”. It has become clear in recent years that type II string theories contain nonperturbative objects —called D-branes— which support gauge fields. From a more abstract point of view, D-brane charges are classified by K-theory classes of the space- time manifold X. Since D-branes are charged under Ramond-Ramond fields, E. Witten and then G. Moore and E. Witten proposed a K-theoretic interpretation of Ramond-Ramond fluxes themselves. This is more striking, as the RR fields are states in the perturbative closed string spectrum which a priori have no connection with vector bundles and K-theory. In the present work, we show that this proposal is consistent with M-theory. The main idea is a detailed comparison of the leading terms in the long distance partition function of M-theory and IIA string theory. The analysis relies heavily on index theory and homotopy theory techniques. It is found a precise agreement which confirms the internal consistency of M-theory, as well as the validity of the K-theoretic formalism in string theory. Jaume Gomis Caltech University D-branes in nongeometric phases of string theory Tuesday, 12.00 – 12.30, Room B Recent progress in string theory hinges upon our improved understanding of its nonperturbative states.
    [Show full text]
  • Pos(TASI2017)015
    The String Landscape, the Swampland, and the Missing Corner PoS(TASI2017)015 T. Daniel Brennana, Federico Cartab,∗ and Cumrun Vafayc a Department of Physics and Astronomy, Rutgers University 126 Frelinghuysen Rd., Piscataway NJ 08855, USA b Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain c Jefferson Physical Laboratory, Harvard University Cambridge, MA 02138, USA Email: [email protected], [email protected], [email protected] We give a brief overview of the string landscape and techniques used to construct string com- pactifications. We then explain how this motivates the notion of the swampland and review a number of conjectures that attempt to characterize theories in the swampland. We also compare holography in the context of superstrings with the similar, but much simpler case of topological string theory. For topological strings, there is a direct definition of topological gravity based on a sum over a “quantum gravitational foam.” In this context, holography is the statement of an identification between a gravity and gauge theory, both of which are defined independently of one another. This points to a missing corner in string dualities which suggests the search for a direct definition of quantum theory of gravity rather than relying on its strongly coupled holographic dual as an adequate substitute (Based on TASI 2017 lectures given by C. Vafa). Theoretical Advanced Study Institute in Elementary Partical Physics (TASI): “Physics at the Fundamental Frontier” University of Colorado, Boulder Boulder, CO June, 5 – 30 2017 ∗La Caixa-Severo Ochoa Scholar ySpeaker. c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]