Scuola Internazionale Superiore di Studi Avanzati - Trieste Doctoral Thesis Developments in Quantum Cohomology and Quantum Integrable Hydrodynamics via Supersymmetric Gauge Theories Supervisors: Author: Giulio Bonelli Antonio Sciarappa Alessandro Tanzini A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in the Theoretical Particle Physics group SISSA SISSA - Via Bonomea 265 - 34136 TRIESTE - ITALY Contents Contents ii 1 Introduction1 2 Supersymmetric localization7 2.1 Supersymmetric localization: an overview..................7 2.2 Supersymmetric localization: the S2 case.................. 10 2.2.1 = (2; 2) gauge theories on S2 .................... 10 N 2.2.2 Localization on S2 - Coulomb branch................ 11 2.2.3 Localization on S2 - Higgs branch.................. 13 3 Vortex counting and Gromov-Witten invariants 15 3.1 Gromov-Witten theory from ZS2 ....................... 15 3.2 Abelian GLSMs................................. 23 3.2.1 Projective spaces............................ 23 3.2.1.1 Equivariant projective spaces................ 24 3.2.1.2 Weighted projective spaces................. 26 3.2.2 Quintic................................. 28 1 3.2.3 Local Calabi{Yau: (p) ( 2 p) P ............. 30 O ⊕ O − − ! 3.2.3.1 Case p = 1......................... 31 − 3.2.3.2 Case p = 0.......................... 32 3.2.3.3 Case p 1.......................... 34 ≥ 3.2.4 Orbifold Gromov-Witten invariants.................. 34 n 3.2.4.1 KPn−1 vs. C =Zn ...................... 35 3 3.2.4.2 Quantum cohomology of C =Zp+2 and crepant resolution 36 3.3 Non-abelian GLSM............................... 41 3.3.1 Grassmannians............................. 41 3.3.1.1 The Hori-Vafa conjecture.................. 42 3.3.2 Holomorphic vector bundles over Grassmannians.......... 42 3.3.3 Flag manifolds............................. 51 3.3.4 Quivers................................. 52 4 ADHM quiver and quantum hydrodynamics 55 4.1 Overview.................................... 55 4.1.1 6d theories and ADHM equivariant quantum cohomology..... 55 4.1.2 Quantum hydrodynamics and gauge theories............ 57 4.2 The ADHM Gauged Linear Sigma Model.................. 58 4.2.1 Reduction to the Nekrasov partition function............ 60 ii 4.2.2 Classification of the poles....................... 61 4.3 Equivariant Gromov-Witten invariants of k;N ............... 63 M 4.3.1 Cotangent bundle of the projective space.............. 64 4.3.2 Hilbert scheme of points........................ 67 4.3.3 A last example............................. 69 4.3.4 Orbifold cohomology of the ADHM moduli space.......... 70 4.4 Donaldson-Thomas theory and stringy corrections to the Seiberg-Witten prepotential................................... 72 4.5 Quantum hydrodynamic systems....................... 76 4.5.1 The Intermediate Long Wave system................. 77 4.5.2 ILW as hydrodynamic limit of elliptic Calogero-Moser....... 79 4.6 Landau-Ginzburg mirror of the ADHM moduli space and quantum Inter- mediate Long Wave system.......................... 81 4.6.1 Derivation via large r limit and norm of the ILW wave-functions. 85 4.6.2 Quantum ILW Hamiltonians..................... 86 4.6.3 Quantum KdV............................. 89 5 ∆ILW and elliptic Ruijsenaars: a gauge theory perspective 91 5.1 Introduction................................... 91 5.2 The quantum Ruijsenaars-Schneider integrable systems.......... 92 5.3 Ruijsenaars systems from gauge theory.................... 94 5.4 Free field realization of Ruijsenaars systems................. 98 5.4.1 The trigonometric case......................... 98 5.4.2 The elliptic case............................ 102 5.5 The finite-difference ILW system....................... 103 5.6 Finite-difference ILW from ADHM theory on S2 S1 ........... 109 × 5.7 ∆ILW as free field Ruijsenaars: the gauge theory side........... 112 5.7.1 The trigonometric case: ∆BO from tRS............... 112 5.7.2 The elliptic case: ∆ILW from eRS.................. 114 5.7.3 The gauge theory correspondence................... 118 A A and D-type singularities 119 A.1 Ap−1 singularities................................ 119 A.1.1 Analysis of the Coulomb branch................... 121 A.1.2 Equivariant quantum cohomology of (~k; N;~ p).......... 124 M A.1.2.1 The N = 1, k = 1 sector.................. 125 A.1.2.2 The N = 1, k = 2 sector.................. 130 A.1.2.3 The N = 2 sector, p = 2................... 134 A.2 Dp singularities................................. 136 A.2.1 Instanton partiton function...................... 138 A.2.2 Equivariant quantum cohomology.................. 138 A.2.3 Analysis of the Coulomb branch................... 140 B Equivariant quantum cohomology in oscillator formalism 143 C Hydrodynamic limit of elliptic Calogero-Moser 147 C.1 Details on the proof of (4.77) and (4.82)................... 147 C.1.1 Proof of (4.77)............................. 147 C.1.2 Proof of (4.82)............................. 152 Bibliography 154 Chapter 1 Introduction In the last sixty years, theoretical particle physicists dedicated themselves to uncovering the mysteries underlying Quantum Field Theories, in order to help the experimental particle physics colleagues understand the world of elementary particles. The combined efforts finally led to the formulation of the Standard Model, the greatest achievement in particle physics up to now, recently strengthened by the discovery of the Higgs boson at CERN. Nevertheless, although in excellent agreement with experiments, basic and commonly taken for granted properties of the Standard Model such as confinement, existence of a mass gap, and non-perturbative phenomena have not been completely understood yet and are still waiting for a satisfactory explanation at the theoretical level. The basic problem here is that the conventional approach to Quantum Field Theory based on Feynman diagrams relies on a perturbative weak-coupling expansion, which by definition cannot take into account strong-coupling or non-perturbative effects. Since at the moment we are lacking an alternative description of Quantum Field Theory more suitable for tackling these problems, what we can do now is to study those theories on which we have a better analytical control because of the high amount of symmetries they possess: in particular we consider theories which enjoy supersymmetry, conformal symmetry, or both. For these cases, many tools coming from string theory, integrability or geometry appear to be of great help in understanding their properties. In this sense the simplest non-trivial theory in four dimensions is the U(N) = 4 super Yang-Mills N theory, being the one which possesses all of the allowed symmetries. By considering theories with fewer number of supersymmetries we get closer to the phenomenological world, but we quickly lose analytical techniques to study them. Nevertheless many remarkable new ideas, such as for example Seiberg-Witten theory and Seiberg duality, arose from considering = 2 and = 1 theories; the hope is that some of these ideas N N may also be applied to non-supersymmetric cases. 1 Chapter 1. Introduction 2 Apart from possible applications in particle physics phenomenology, with time people realized that supersymmetric theories, because of their deep connection to geometrical structures, can also be useful as a different way of approaching problems in mathe- matics, based on Lagrangians and path integral techniques. Although path integrals were unfamiliar to many mathematicians in the past, especially because of their lack of mathematical rigorousness, they are extremely efficient and led to great discoveries in the theory of topological invariants of manifolds and in the context of integrable systems, as well as in many other different topics. Clearly, this also works in the other direction: common mathematical techniques unfamiliar to physicists can give an important alter- native point of view on physics problems and provide hints on how to affront them, thus allowing us to gain a deeper understanding of Quantum Field Theories. Nowadays, su- persymmetric theories and string theory are used as an additional source of inspiration in many other different contexts, such as cosmological models, statistical physics sys- tems and condensed matter systems. The interplay and interactions between the various disciplines will lead to many more developments in the future. In the past, one of the mathematical problems that have been studied with field and string theory methods was the enumerative problem of computing genus zero Gromov- Witten invariants (GW) for Calabi-Yau and K¨ahlermanifolds. Roughly speaking, genus zero GW invariants Nη count the number of holomorphic maps of degree η from a two- dimensional sphere S2 (a genus zero Riemann surface) to a K¨ahlermanifold M, which is usually denoted as target manifold. From the physics point of view, computing these invariants is especially important when M is a Calabi-Yau three-fold (manifold of com- plex dimension 3): in fact if one wants to construct supersymmetric generalizations of the Standard Model starting from a ten-dimensional string theory set-up one has to compactify six of the ten dimensions, and the easiest way to preserve some supersymme- try is to consider a Calabi-Yau three-fold as compactification space. In this context, the two-sphere is interpreted as the world-sheet of the strings, and the genus zero GW invari- ants enter in determining the world-sheet non-perturbative corrections to the Yukawa couplings of the resulting effective four-dimensional