ریاضی-رشته ۲۰۱۶: ریاضی-رشته ۲۰۱۶، ۲۷ ژوئن تا ۲ ژوئیه ۲۰۱۶، کالج فرانسه، پاریس، فرانسه

This volume contains the proceedings of the conference String-Math 2016, which was held from June 27–July 2, 2016, at Collége de France, Paris, France. String-Math is an annual conference covering the most significant progress at the interface of string theory and mathematics. The two fields have had a very fruitful dialogue over the last thirty years, with string theory contributing key ideas which have opened entirely new areas of mathematics and modern mathematics providing powerful concepts and tools to deal with the intricacies of string and quantum field theory. The papers in this volume cover topics ranging from supersymmetric quantum field theories, topological strings, and conformal nets to moduli spaces of curves, representations, instantons, and harmonic maps, with applications to spectral theory and to the geometric Langlands program. Cover 1 Title page 2 Contents 4 Preface 6 Three-dimensional \cN=4 gauge theories in omega background 18 1. Introduction 18 2. Setup 19 3. Hilbert space 25 4. Monopole operators 29 5. Boundary conditions and overlaps 32 6. Vortex quantum mechanics 35 Acknowledgements 35 References 35 3d supersymmetric gauge theories and Hilbert series 38 1. Introduction 38 2. Moduli space of supersymmetric vacua and chiral ring 39 3. The Hilbert series 41 4. 3d \cN=2 gauge theories vs 4d \cN=1 gauge theories 46 5. ’t Hooft monopole operators 47 6. Monopole formula for the Hilbert series of 3d \cN≥2 gauge theories 52 7. Coulomb branch of 3d \cN=4 gauge theories 53 8. Moduli spaces of 3d \cN=2 theories and Hilbert series 57 9. Conclusion 60 Acknowledgements 61 References 61 Quantized Coulomb branches of Jordan quiver gauge theories and cyclotomic rational Cherednik algebras 66 1. Introduction 66 1(i). 66 1(ii). 68 Acknowledgments 69 2. The case with no framing 70 3. Poisson brackets 70 4. Cherednik algebras 71 4(i). Definitions 71 4(ii). Dunkl operators 72 4(iii). Rational Demazure-Lusztig operators 72 5. Cyclotomic rational Cherednik algebras 74 6. Affine Yangian of \gl(1) 81 6(i). Presentation 81 6(ii). From Yangian to difference operators 83 6(iii). Proof 83 6(iv). Automorphism 86 6(v). Shifted Yangian 88 Appendix A. 89 Appendix B. 93 References 94 Supersymmetric field theories and geometric Langlands: The other side of the coin 96 1. Introduction 96 2. Review 97 3. Defects of co-dimension two and surface operators 101 4. Partition functions versus conformal blocks 106 5. Another type of surface operators 109 6. Recovering the geometric Langlands correspondence 111 7. Sigma model interpretation? 114 Appendix A. Hitchin’s moduli spaces 120 References 121 A journey from the Hitchin section to the oper moduli 124 1. Introduction 124 2. Enumeration of ribbon graphs 125 3. A walk into the woods of Higgs bundles and connections 131 4. From Higgs bundles to quantum curves 135 5. The metamorphosis of quantum curves into opers 137 6. Hitchin moduli spaces for the Lie group G=SLr(C) 143 7. The limit oper of Gaiotto’s correspondence and the quantum curve 150 8. Conclusion 152 Acknowledgments 153 References 154 S-duality of boundary conditions and the Geometric Langlands program 156 1. Introduction and conclusions 156 2. Neumann-like boundary conditions and matter interfaces 161 3. A rich example: Particle-vortex duality in U(1) gauge theory 167 4. Bifundamental and fundamental interfaces 169 5. General NS5 and D5 interfaces for unitary groups 172 6. Ortho-symplectic examples 174 7. Tri-fundamental SU(2)×SU(2)×SU(2) interface 175 8. More examples with gauge group reductions 177 9. D-modules 180 10. Sheafs on the moduli space of local systems 180 Appendix A. Lagrangian submanifolds and generating functions 181 Appendix B. Supersymmetric Berry connections for \CN=4 SQM 182 Appendix C. The category of BBB branes 193 Appendix D. Gauge group reductions 194 Acknowledgements 195 References 196 Pure SU(2) gauge theory partition function and generalized Bessel kernel 198 1. Introduction 198 2. Isomonodromy and Riemann-Hilbert setup 201 3. Fredholm determinant representation 208 4. Series over Young diagrams 214 References 220 Reduction for SL(3) pre-buildings 224 1. Introduction 224 2. Spectral networks in X 227 3. The initial construction 228 4. Structures 230 5. The refraction property 232 6. Reduction 233 7. The new construction 236 8. Scholium 241 9. Further questions 242 Acknowledgments 243 References 243 Conformal nets are factorization algebras 246 1. Introduction 246 2. Factorization algebras 247 3. Proofs 249 4. An application 252 5. Appendix 255 Acknowledgement 256 References 256 Contracting the Weierstrass locus to a point 258 Introduction 258 1. Rational maps \forg2 and φ 260 2. Curves of genus 2 267 References 273 Spectral theory and mirror symmetry 276 1. Introduction 276 2. A problem in spectral theory 277 3. From topological strings to spectral theory 283 4. From spectral theory to topological strings 300 5. Outlook 306 Acknowledgments 307 References 307 Back Cover 314 Cover......Page 1 Title page......Page 2 Contents......Page 4 Preface......Page 6 1. Introduction......Page 18 2. Setup......Page 19 3. Hilbert space......Page 25 4. Monopole operators......Page 29 5. Boundary conditions and overlaps......Page 32 References......Page 35 1. Introduction......Page 38 2. Moduli space of supersymmetric vacua and chiral ring......Page 39 3. The Hilbert series......Page 41 4. 3 \cN=2 gauge theories vs 4 \cN=1 gauge theories......Page 46 5. ’t Hooft monopole operators......Page 47 6. Monopole formula for the Hilbert series of 3 \cN≥2 gauge theories......Page 52 7. Coulomb branch of 3 \cN=4 gauge theories......Page 53 8. Moduli spaces of 3 \cN=2 theories and Hilbert series......Page 57 9. Conclusion......Page 60 References......Page 61 1(i).......Page 66 1(ii).......Page 68 Acknowledgments......Page 69 3. Poisson brackets......Page 70 4(i). Definitions......Page 71 4(iii). Rational Demazure-Lusztig operators......Page 72 5. Cyclotomic rational Cherednik algebras......Page 74 6(i). Presentation......Page 81 6(iii). Proof......Page 83 6(iv). Automorphism......Page 86 6(v). Shifted Yangian......Page 88 Appendix A.......Page 89 Appendix B.......Page 93 References......Page 94 1. Introduction......Page 96 2. Review......Page 97 3. Defects of co-dimension two and surface operators......Page 101 4. Partition functions versus conformal blocks......Page 106 5. Another type of surface operators......Page 109 6. Recovering the geometric Langlands correspondence......Page 111 7. Sigma model interpretation?......Page 114 Appendix A. Hitchin’s moduli spaces......Page 120 References......Page 121 1. Introduction......Page 124 2. Enumeration of ribbon graphs......Page 125 3. A walk into the woods of Higgs bundles and connections......Page 131 4. From Higgs bundles to quantum curves......Page 135 5. The metamorphosis of quantum curves into opers......Page 137 6. Hitchin moduli spaces for the Lie group =r(C)......Page 143 7. The limit oper of Gaiotto’s correspondence and the quantum curve......Page 150 8. Conclusion......Page 152 Acknowledgments......Page 153 References......Page 154 1. Introduction and conclusions......Page 156 2. Neumann-like boundary conditions and matter interfaces......Page 161 3. A rich example: Particle-vortex duality in (1) gauge theory......Page 167 4. Bifundamental and fundamental interfaces......Page 169 5. General NS5 and D5 interfaces for unitary groups......Page 172 6. Ortho-symplectic examples......Page 174 7. Tri-fundamental (2)×(2)×(2) interface......Page 175 8. More examples with gauge group reductions......Page 177 10. Sheafs on the moduli space of local systems......Page 180 Appendix A. Lagrangian submanifolds and generating functions......Page 181 Appendix B. Supersymmetric Berry connections for \CN=4 SQM......Page 182 Appendix C. The category of BBB branes......Page 193 Appendix D. Gauge group reductions......Page 194 Acknowledgements......Page 195 References......Page 196 1. Introduction......Page 198 2. Isomonodromy and Riemann-Hilbert setup......Page 201 3. Fredholm determinant representation......Page 208 4. Series over Young diagrams......Page 214 References......Page 220 1. Introduction......Page 224 2. Spectral networks in ......Page 227 3. The initial construction......Page 228 4. Structures......Page 230 5. The refraction property......Page 232 6. Reduction......Page 233 7. The new construction......Page 236 8. Scholium......Page 241 9. Further questions......Page 242 References......Page 243 1. Introduction......Page 246 2. Factorization algebras......Page 247 3. Proofs......Page 249 4. An application......Page 252 5. Appendix......Page 255 References......Page 256 Introduction......Page 258 1. Rational maps \forg2 and ......Page 260 2. Curves of genus 2......Page 267 References......Page 273 1. Introduction......Page 276 2. A problem in spectral theory......Page 277 3. From topological strings to spectral theory......Page 283 4. From spectral theory to topological strings......Page 300 5. Outlook......Page 306 References......Page 307 Back Cover......Page 314 "This volume contains the proceedings of the conference String-Math 2016, which was held from June 27-July 2, 2016, at Collége de France, Paris, France. String-Math is an annual conference covering the most significant progress at the interface of string theory and mathematics. The two fields have had a very fruitful dialogue over the last thirty years, with string theory contributing key ideas which have opened entirely new areas of mathematics and modern mathematics providing powerful concepts and tools to deal with the intricacies of string and quantum field theory. The papers in this volume cover topics ranging from supersymmetric quantum field theories, topological strings, and conformal nets to moduli spaces of curves, representations, instantons, and harmonic maps, with applications to spectral theory and to the geometric Langlands program."-- Provided by publisher Contains the proceedings of the conference String-Math 2016, held in June 2016 at College de France, Paris. The papers in this volume cover topics ranging from supersymmetric quantum field theories, topological strings, and conformal nets to moduli spaces of curves, representations, instantons, and harmonic maps, with applications to spectral theory and to the geometric Langlands program. Amir-kian Kashani-poor, Ruben Minasian, Nikita Nekrasov, Boris Pioline, Editors. Includes Bibliographical References. Amir-kian Kashani-poor, Ruben Minasian, Nikita Nekrasov, Boris Pioline, Editors.