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Supersymmetric Field Theories and Isomonodromic Deformations” and the Work Presented in It Are My Own Scuola Internazionale Superiore di Studi Avanzati - Trieste DOCTORAL THESIS Supersymmetric Field Theories and Isomonodromic Deformations CANDIDATE : Fabrizio Del Monte ADVISORS : Prof. Giulio Bonelli Prof. Alessandro Tanzini OPPONENTS : Dr. Alba Grassi Prof. Andrei Marshakov ACADEMIC YEAR 2019 – 2020 SISSA - Via Bonomea 265 - 34136 TRIESTE - ITALY iii Declaration of Authorship I, Fabrizio DEL MONTE, declare that this thesis titled, “Supersymmetric Field Theories and Isomonodromic Deformations” and the work presented in it are my own. Where I have consulted the published work of others, this is always clearly attributed. The thesis is based on the three research papers • G. Bonelli, F. Del Monte, P. Gavrylenko and A. Tanzini, "N = 2∗ gauge theory, free fermions on the torus and Painlevé VI", Commun. Math. Phys. 377 (2020) no.2, 1381-1419, https://doi.org/10.1007/s00220-020-03743-y; • G. Bonelli, F. Del Monte, P. Gavrylenko and A. Tanzini, "Circular quiver gauge theories, isomonodromic deformations and WN fermions on the torus", https://arxiv.org/abs/1909.07990; • G. Bonelli, F. Del Monte and A. Tanzini, "BPS quivers of five-dimensional SCFTs, Topological Strings and q-Painlevé equations", https://arxiv. org/abs/2007.11596 as well as the still unpublished work • G. Bonelli, F. Del Monte and A. Tanzini, "Higher genus class S theories and isomonodromy", work in progress v SISSA Abstract Doctor of Philosophy Supersymmetric Field Theories and Isomonodromic Deformations by Fabrizio DEL MONTE The topic of this thesis is the recently discovered correspondence between su- persymmetric gauge theories, two-dimensional conformal field theories and isomonodromic deformation problems. Its original results are organized in two parts: the first one, based on the papers [1], [2], as well as on some further unpublished results, provides the extension of the correspondence between four-dimensional class S theories and isomonodromic deformation problems to Riemann Surfaces of genus greater than zero. The second part, based on the results of [3], is instead devoted to the study of five-dimensional super- conformal field theories, and their relation with q-deformed isomonodromic problems. vii Acknowledgements I express my gratitude to my advisors Prof. Giulio Bonelli and Prof. Alessan- dro Tanzini for their important guidance and support, that always helped me to find a direction without ever imposing one. I also thank my collabo- rator and friend Pasha Gavrylenko, with whom I had the fortune of having frequent and prolific discussions that resulted in much of the work in this thesis. I express also my sincerest gratitude to my parents Antonella and Giuseppe that have always supported me and nourished my interests and passions, whether long or short-lived. I want also to thank my past mentors, Kenichi Konishi and Dario Francia: they played a major role in shaping my academic self and my scientific views. I thank all the people that I had the pleasure of discussing during these years, that helped me grow as a person and researcher. Special thanks go to my colleagues and friends at SISSA, that contributed to make sure that my life in Trieste was never boring. I want also to remember all the friends that I left in Lucca, with whom I always look forward to spend time every time I go back. Finally, I thank SISSA and INFN for their continued financial support through all my travels. ix Contents Abstractv Acknowledgements vii I Coulomb Branches and Integrable Systems7 1 Seiberg-Witten Theory and Integrable Systems9 1.1 (Extended) Supersymmetry Algebra in four dimensions....9 1.2 Low-Energy Effective Actions and Prepotentials........ 11 1.3 The Seiberg-Witten Solution.................... 13 1.3.1 Example: Pure Super Yang-Mills............. 14 1.4 Intermezzo: a brief overview of Integrable Systems...... 17 1.5 Seiberg-Witten Theory as an Integrable System......... 20 2 String Theory Embeddings 23 2.1 Witten’s construction of linear quiver theories......... 23 2.2 Class S theories........................... 26 2.3 Intermezzo: Hitchin Systems................... 28 2.4 Hitchin moduli space and Seiberg-Witten theory........ 29 2.5 Geometric Engineering....................... 32 3 Instanton Counting and AGT Correspondence 35 3.1 The instanton moduli space.................... 35 3.2 W-background and instanton partition functions........ 37 3.2.1 Five-dimensional partition functions........... 39 3.3 The AGT correspondence..................... 41 3.3.1 Pants decompositions and conformal blocks...... 44 3.3.2 Topological Vertex, five-dimensional gauge theory and q − WN conformal blocks................. 49 3.4 Surface Operators.......................... 50 3.4.1 M5-brane defects and Kac-Moody Conformal Blocks. 51 3.4.2 M2-brane defects, loop operators and degenerate fields 52 3.5 Fusion, braiding and loop operators............... 53 3.5.1 Fusion algebra of degenerate fields............ 53 3.5.2 Gauge theory loop operators from Verlinde loop oper- ators............................. 57 x II Isomonodromic Deformations and Class S Theories 59 4 Isomonodromic deformations 61 4.1 Painlevé equations......................... 61 4.2 Linear systems of ODEs and isomonodromy.......... 63 4.2.1 Painlevé VI from isomonodromic deformations.... 64 4.3 Isomonodromic deformations and CFT on the Riemann Sphere 66 4.3.1 Free fermion reformulation................ 71 4.4 The Painlevé-Gauge theory correspondence........... 75 5 One-punctured torus, N = 2∗ theory 77 5.1 Isomonodromic deformations and torus conformal blocks.. 77 5.1.1 Linear systems on the torus................ 77 5.1.2 Monodromies of torus conformal blocks......... 80 5.1.3 CFT solution to the Riemann-Hilbert problem and tau function........................... 83 5.2 Tau function and Nekrasov functions.............. 88 5.2.1 Periodicity of the tau functions.............. 88 5.2.2 Asymptotic calculation of the tau function....... 89 Results............................ 90 5.2.3 Asymptotic computation of monodromies....... 92 5.2.4 Initial conditions of Painlevé and Coulomb branch co- ordinates........................... 96 5.2.5 A self-consistency check.................. 96 5.3 Gauge theory, topological strings and Fredholm determinants 99 5.3.1 Line operators and B-branes................ 103 5.3.2 The autonomous/Seiberg-Witten limit.......... 104 6 Torus with many punctures, circular quiver gauge theories 109 6.1 General Fuchsian system on the torus.............. 111 6.2 Kernel and tau function from free fermions........... 113 6.3 Torus monodromies with Verlinde loop operators....... 118 6.3.1 General setup........................ 118 6.3.2 B-cycle monodromy operator............... 120 6.3.3 Fourier transformation................... 122 6.4 Relation to Krichever’s connection................ 124 6.5 Solution of the elliptic Schlesinger system............ 127 7 Higher genus Class S Theories and isomonodromy 131 7.1 Hamiltonian approach to isomonodromic deformations.... 131 7.1.1 Relation to class S theories................. 137 7.1.2 Isomonodromic deformations from flatness of a uni- versal connection...................... 137 7.1.3 Poisson brackets and canonical variables........ 139 7.2 Linear systems and free fermions on Riemann surfaces.... 140 7.3 Conclusions and Outlook..................... 147 xi III Discrete Equations and five-dimensional theories 149 8 Five-dimensional SCFTs and q-Painlevé equations 151 8.1 BPS spectrum of 5d SCFT on S1 and quiver mutations..... 154 8.1.1 Super Yang-Mills, k = 0.................. 155 8.1.2 Super Yang-Mills, k = 1.................. 158 8.1.3 Nf = 2, k = 0........................ 159 8.2 Discrete quiver dynamics, cluster algebras and q-Painlevé.. 165 8.2.1 Cluster algebras and quiver mutations.......... 165 8.2.2 Pure gauge theory and q-Painlevé III3 .......... 168 8.2.3 Super Yang-Mills with two flavors and qPIII1 ...... 171 8.2.4 Super Yang-Mills with two flavours, q-Painlevé IV and q-Painlevé II......................... 174 8.2.5 Summary of dP3 bilinear equations........... 176 8.3 Solutions............................... 176 8.3.1 Local F0 and qPIII3 ..................... 177 8.3.2 q-Painlevé III1 and Nekrasov functions......... 179 8.4 Degeneration of cluster algebras and 4d gauge theory..... 183 8.4.1 From local F0 to the Kronecker quiver.......... 184 8.4.2 Local F1 ........................... 185 8.4.3 Local dP3 .......................... 186 8.5 Conclusions and Outlook..................... 191 IV Appendices 195 A Elliptic and theta functions 197 B Schottky uniformization and twisted differentials 201 C q-Painlevé III and IV in Tsuda’s parametrization 205 C.1 qPIII.................................. 205 C.2 qPIV................................. 206 Bibliography 209 1 Introduction and Summary Quantum Field Theory is the framework allowing to describe physical phe- nomena at all scales, ranging from the astronomical scales of General Rela- tivity, to the everyday physics of conducting metals, all the way to the funda- mental physics at the microscopic level of fundamental particles. In all these applications, however, we can see a common thread: while we can perform approximate computations that are often in spectacular agreement with ex- perimental results, we still have a very poor grasp of the nonperturbative and strongly-coupled aspects of Quantum Field Theory. We can predict to incredible accuracy the anomalous magnetic moment of the electron, but we are still unable to give any quantitative argument for confinement in a real- istic case. A lot of progress, however, has been achieved
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