Low Dimensional Gauge Theories and Quantum Geometry Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨at Bonn von Urmi Ninad aus Lucknow (Indien) Bonn, July 2020 Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨at Bonn 1. Gutachter: Prof. Dr. Albrecht Klemm 2. Gutachter: Priv. Doz. Dr. Stefan Foerste Tag der Promotion: 02.10.2020 Erscheinungsjahr: 2020 Abstract This thesis explores the connection between supersymmetric gauge theories on curved spaces of dimensions two and three and geometries that arise in string theory. It does so by studying quantum field theoretic objects such as the partition function and correlation functions on the gauge theory and finding links to geometric information of its target space. An exact calculation of the path integral is made possible using the localisation technique for supersymmetric gauge theories. A 2d gauge theory with N = (2; 2) supersymmetry on the Ω-deformed two-sphere subject to the A-twist is the first object of study. Here we find that the correlation functions of the twisted chiral field fulfil certain universal and non-trivial relations. These relations can be interpreted as quantum operators that govern the moduli dependence of the ground state of the gauge theory in a Hilbert-space picture. Furthermore, the relations can be imparted a representation as differential operators that are shown to annihilate Givental's cohomology-valued I-function on the target space of the gauge theory. This is a consequence of the fact that the 2d gauge theory provides an ultraviolet model for quantum cohomology on a manifold. In particular, for gauge theories with Calabi-Yau target spaces, these operators coincide with Picard-Fuchs operators in algebraic geometry. For a certain class of Calabi-Yau manifolds, we turn the argument around and express the Picard-Fuchs operators in terms of a finite number of correlators in the gauge theory. 2 1 In 3d we study N = 2 gauge theories on the solid torus D ×q S , where q is the twist in the fibration of D2 over S1, with Grassmannian manifolds as target spaces. These theories are ultraviolet models for quantum K-theory on their target spaces. We compute the partition function and extract from it Givental's I-function of permutation symmetric quantum K-theory. This facilitates a calculation of the algebra of Wilson loops which, for different values of the Chern-Simons levels, is shown to be isomorphic to either the quantum K-theoretic ring of Schubert structure sheaves on the Grassmannian or the Verlinde algebra. Additionally, we evaluate difference equations that annihilate this I-function. A limit where the S1 contracts to a point, all the computed quantities are shown to coincide with the corresponding objects encountered in 2d. iii Acknowledgements I am immensely thankful to Dr. Hans Jockers for providing a thoughtful and tireless supervision, constant approachability and an ever-present support in all matters alike. I will be forever grateful for the many interesting avenues of research I was introduced to and for all that I learnt as a doctoral student under his supervision. I would also like to thank Prof. Albrecht Klemm for admitting me into his research group and acquainting me with several fascinating ideas in physics and mathematics during seminars and informal discussions. I am very grateful for the fruitful and engaging collaborations I had with Dr. Andreas Gerhardus, Prof. Peter Mayr, Dr. Mauricio Romo and Alexander Tabler. My interactions with all the present and past members of the Klemm group have been helpful and educational, for which I am thankful. I would especially like to thank Fabian Fischbach and Christoph Nega for assistance with the manuscript as well as Dr. Andreas Gerhardus and Dr. Thorsten Schimannek for scientific guidance. I would like to thank the Bonn-Cologne Graduate School of Physics and Astronomy whose financial support made several valuable academic trips possible. I express my gratitude to members and alumni of the BCTP who have provided levity in times of stress and motivation in times of inactivity. I am particularly grateful to Christa B¨orsch, Dagmar Fassbender, Petra Weiss, Dr. Andreas Wisskirchen and Patricia Z¨undorffor easing the many logistical hurdles one faces as a doctoral student. I also owe a lot to the friends I have made while studying physics and who have been pillars, mentors and educators alike. I am indebted to my uncle Dr. Ravi Sinha whose infectious love for fundamental physics has been instrumental in the direction my education and research took. Finally, an inexpressible thanks to Aai, Baa and Paul for much more, everything. v List of Publications The original work presented in this thesis is based on the following publications of the author: 1. A. Gerhardus, H. Jockers, and U. Ninad, The Geometry of Gauged Linear Sigma Model Correlation Functions, Nucl. Phys. B 933 (2018) 65-133, arXiv:1803.10253 [hep-th]. 2. H. Jockers, P. Mayr, U. Ninad, and A. Tabler, Wilson loop algebras and quantum K-theory for Grassmannians, arXiv:1911.13286 [hep-th]. vii Contents 1 Introduction 1 2 Localisation of Gauge Theories9 2.1 The Main Idea.....................................9 2.2 Localisation in Non-Linear Sigma Models...................... 10 2.2.1 A-Model..................................... 12 2.2.2 B-Model..................................... 13 2.3 Localisation in Supersymmetric Gauge Theories................... 14 2.3.1 2d........................................ 15 2.3.2 3d........................................ 19 3 2d Gauge Theories and Relations of Correlators 23 3.1 GLSM to NLSM.................................... 23 3.1.1 Infrared Dynamics of GLSMs......................... 24 3.1.2 Illustrative Example: Hori-Tong GLSM................... 32 3.2 Partition Functions and Correlators of GLSMs................... 35 2 3.2.1 A-twisted GLSM correlators on SΩ ...................... 36 3.3 Relations of GLSM Correlators............................ 40 3.3.1 Derivation of Relations............................ 42 3.3.2 Correlator Relations and the Differential Ideal............... 44 3.3.3 Examples.................................... 46 3.4 Picard-Fuchs Operators from Correlators...................... 52 3.4.1 Calabi-Yau Manifolds............................. 53 3.4.2 Relations of one parameter CY threefolds.................. 54 3.4.3 Examples.................................... 56 3.5 Givental's Cohomological I-function and GLSMs.................. 59 3.5.1 Gromov-Witten Invariants and the I-function................ 60 3.5.2 I-function and GLSM correlators....................... 63 3.5.3 Examples.................................... 64 4 3d Gauge Theories and Wilson Loop Algebras 67 4.1 Lagrangian in 3d.................................... 67 2 1 4.2 Partition Function on D ×q S ............................ 69 4.2.1 Grassmannian Gr(M; N) Target Space.................... 72 4.3 Quantum K-theoretic I- and J-functions....................... 77 4.3.1 Wilson Loop Algebra.............................. 79 4.3.2 q-Difference Equations............................. 88 4.3.3 The 2d Limit.................................. 91 ix 5 Summary and Outlook 95 A The Grassmannian 101 A.1 Basics of the Grassmannian Gr(M; N)........................ 101 A.1.1 Introductory Definitions............................ 101 A.1.2 Characteristic Classes on Gr(M; N)..................... 103 A.2 Quantum Cohomology Ring on Gr(M; N)...................... 106 Bibliography 109 x CHAPTER 1 Introduction Point Particle Physics Einstein's theory of gravity is encapsulated in the Einstein-Hilbert action, 1 Z p S ∼ d4x g R ; (1.1) GN expressed in natural units, where GN is the Newton's gravitational constant. Here g is the determinant of the metric tensor of spacetime gµν and R is a dynamical scalar that captures the curvature of spacetime, known as the Ricci scalar. This theory was considered a complete description of the gravitational force and inspired landmark findings at the cosmological scale until a microscopic description of the electromagnetic force came along. For an exposition to general relativity, see for instance [1,2]. Quantum field theory is a formalism that elevates classical fields to quantum fields, going beyond a (first) quantisation of particles and earning itself the term second quantisation. The architects of quantum field theory set out trying to unify quantum mechanics of multi-particle states with special relativity and were faced with explaining particle creation and annihilation. This ultimately led them to postulate the wave-functions associated to particles as quantum- mechanical objects themselves and gave birth to a vast network of ideas that we today call quantum field theory. The first consistently constructed quantum field theory was that of electrodynamics which paved way for the quantisation of weak and strong interactions. Some comprehensive and pedagogical reviews include [3{6]. Soon after its conception, quantum field theory was faced with the problem of infinities. Assuming that the principles of quantum field theory are applicable for infinite energy scales, physical observables such as the mass and strength of interaction are also rendered infinite by the underlying mathematical framework. This led to the birth of renormalisation, an idea whereby the fields are renormalised in a way that restores the finiteness of physical observables. While the original idea was