On Cohomological Algebras in Supersymmetric Quantum Field Theories

On Cohomological Algebras in Supersymmetric Quantum Field Theories by Nafiz Ishtiaque A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Physics Waterloo, Ontario, Canada, 2019 c Nafiz Ishtiaque 2019 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner: Erich Poppitz Professor, Dept. of Physics, University of Toronto Supervisor: Jaume Gomis Faculty, Perimeter Institute for Theoretical Physics Adjunct Professor, Dept. of Physics and Astronomy, University of Waterloo Internal Member: Davide Gaiotto Faculty, Perimeter Institute for Theoretical Physics Adjunct Professor, Dept. of Physics and Astronomy, University of Waterloo Internal Member: Niayesh Afshordi Associate Faculty, Perimeter Institute for Theoretical Physics Associate Professor, Dept. of Physics and Astronomy, University of Waterloo Internal-External Member: Ruxandra Moraru Associate Professor, Dept. of Pure Mathematics, University of Waterloo ii Author's Declaration This thesis consists of material all of which I authored or co-authored: see Statement of Contributions included in the thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Statement of Contributions This thesis is based on the following papers: 1. Jaume Gomis and Nafiz Ishtiaque. Kählerpotential and ambiguities in 4d N = 2 SCFTs. JHEP, 04:169, 2015, 1409.5325. 2. Efrat Gerchkovitz, Jaume Gomis, Nafiz Ishtiaque, Avner Karasik, Zohar Komar- godski, and Silviu S. Pufu. Correlation Functions of Coulomb Branch Operators. JHEP, 01:103, 2017, 1602.05971. 3. Nafiz Ishtiaque. 2D BPS Rings from Sphere Partition Functions. JHEP, 04:124, 2018, 1712.02551. 4. Nafiz Ishtiaque, Seyed Faroogh Moosavian, and Yehao Zhou. Topological Holog- raphy: The Example of The D2-D4 Brane System. 2018, 1809.00372. iv Abstract In this thesis we compute certain supersymmetric subsectors of the algebra of observables in some Quantum Field Theories (QFTs) and demonstrate an application of such computation in checking an instance of Holographic duality. Computing the algebra of observables beyond perturbative approximation in weakly coupled field theories is far from a tractable problem. In some special, yet interesting large classes of supersymmetric theories, supersymmetry can be used to extract exact nonperturbative information about certain subsets of observables. This is an old idea which we advance in this thesis by introducing new techniques of computations, computing certain observalbes for the first time, and re- producing earlier results about some other observables. We also propose a new toy model of holographic duality involving topological/holomorphic theories, demonstrating the power of exact computations in supersymmetric subsectors. To be more specific, the subject of this thesis includes the following: 1. Computing the algebra of chiral and twisted chiral operators in 2d N = (2; 2) theories { while these algebras were previously known, we demonstrate how they can be computed using relatively modern techniques of supersymmetric localization. 2. Computing the chiral rings of 4d N = 2 Superconformal Field Theories (SCFTs) { we compute this algebra for the first time. We use the same method of supersymmetric localization that we use in the 2d case.1 3. Computing the algebra of operators on a defect in the topological 2d BF theory, along with its holographic dual. This is a new toy model of holographic duality set in the world of 6d topological string theory. We also argue that this setup is in fact a certain supersymmetric subsector of the holographic duality involving 4d N = 4 Super Yang-Mills (SYM) theory and its 10d supergravity dual { both involving some defects. In order to be able to discuss these different theories in different dimensions with different symmetries without sounding disparate and ad hoc, we employ the language of cohomological algebra. Since this is perhaps not a language most commonly used in the standard physics literature, we would like to emphasize that this is not a novel idea, it is merely a convenient thematic and linguistic umbrella that covers all the topics of this thesis. In 1Chronologically this is the first work demonstrating the use of localization in computing supersymmetry invariant operator algebra. Application of this method to the 2d case came later. However, pedagogically it seems more relevant to present the 2d case first and then the 4d case. v the Batalin-Vilkovisky (BV) formulation of a Quantum Field Theory (QFT), the algebra of observables is presented as the cohomology algebra of a certain complex consisting of fields and anti-fields. In this language restriction to supersymmetric subsectors correspond to modifying the BV differential by the addition of the relevant supersymmetry generator. We simply refer to this modification as reduction to cohomology (with respect to the choice of supersymmetry). vi Acknowledgements I would like to thank my supervisor Jaume Gomis for his patient guidance whenever needed and incredible support in helping me shape my own path in physics. I am also grateful to him for allowing me to focus entirely on my research without being burdened by other usual chores of academia. I would like to thank Kevin Costello, Davide Gaiotto, and Jaume Gomis for their generosity in sharing their valuable insights that helped making significant leaps in under- standing possible for me. In addition to my supervisor, I must thank Efrat Gerchkovitz, Avner Karasik, Zohar Komargodski, Seyed Faroogh Moosavian, Silviu Pufu, and Yeaho Zhou { this thesis stands on our collaborations. I thank Nima Doroud, Hee-Cheol Kim, Dalimil Mazac, Junya Yagi, and Yehao Zhou for being extremely kind in answering my uncountable questions on basic physics and mathematics throughout the years which I can not imagine being even remotely enjoyable for them { though they have never been anything but enthusiastically helpful, I could not have asked for better teachers. Perimeter Institute has been an extraordinary distraction-free workplace. I especially thank Debbie Guenther for always carefully taking care of all the formal affairs allowing me to not worry about them. Finally, I would like to express my gratitude toward Mahbub Majumdar and Arshad Momen. They were my first exposure to theoretical physics, they supported my hope for higher studies when they had no reason to, and without their encouraging and optimistic support I would never have had any chance of pursuing my studies in physics. vii Table of Contents Abbreviations xii 1 Introduction1 1.1 Operator Algebra................................1 1.2 Scalar Field Theory...............................4 1.3 Supersymmetry and Cohomology.......................6 1.3.1 A Toy Model: 0 Dimensional QFT..................7 1.4 Lessons from BV................................ 13 1.4.1 Supersymmetry and Cohomological Algebra............. 17 1.5 Cohomological Algebras and Dualities..................... 19 1.6 Organization of The Thesis.......................... 19 2 Chiral Rings in 2 Dimensions 21 2.1 Prologue..................................... 21 2.2 The BPS Rings................................. 22 2.3 Computing the Ring Structures........................ 27 2.3.1 Extremal Correlators on S2 ...................... 28 2.3.2 Chiral Ring Coefficients from Extremal Correlators on S2 ...... 32 2.4 Some Examples................................. 36 2.4.1 Twisted Chiral Ring of the Quintic GLSM.............. 36 2.4.2 Chiral Rings of the LG Minimal Models............... 43 2.5 Epilogue..................................... 51 viii 3 Chiral Rings in 4 Dimensions 53 3.1 Introduction and Conclusions......................... 53 3.1.1 Conformal Manifolds.......................... 57 3.1.2 The Chiral Ring of N = 2 SCFTs................... 58 3.1.3 Subtle Aspects of Conformal Field Theories on S4 .......... 63 3.2 The Chiral Ring in 4d N = 2 SCFTs and S4 ................. 64 3.2.1 Placing the Deformed Theory on S4 .................. 65 3.2.2 Chiral Primary Correlators from the Deformed Partition Function. 66 3.2.3 The Deformed Partition Function on S4 ............... 67 3.2.4 The Relation Between Correlators in R4 and S4 ........... 69 3.2.5 Summary of the Algorithm....................... 71 3.3 Examples.................................... 73 3.3.1 SU(2) Gauge Group.......................... 73 3.3.2 SU(N) Gauge Group.......................... 79 3.3.3 SU(3) and SU(4) SQCD........................ 84 4 An Application to Duality: Topological Holography 89 4.1 Introduction and Summary........................... 89 4.2 Isomorphic algebras from holography..................... 91 4.3 The dual theories................................ 93 4.3.1 Brane construction........................... 93 4.3.2 The closed string theory........................ 94 4.3.3 BF: The theory on D2-branes..................... 98 4.3.4 4d Chern-Simons: The theory on D4-branes............. 100 Op 4.4 A (Tbd) from BF ⊗ QM theory........................ 104 4.4.1 Free theory limit, O(~0)........................ 107 4.4.2 Loop corrections from BF theory................... 109 4.4.3 Large N limit: The Yangian...................... 116 ix Sc 4.5 A (Tbk) from 4d Chern-Simons Theory.................... 118 4.5.1 Relation to anomaly of Wilson line.................. 123 4.5.2 Classical

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