Wilson loops, instantons and quantum mechanics Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn von Marc Schiereck aus Herford Bonn 2014 Dieser Forschungsbericht wurde als Dissertation von der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Bonn angenommen und ist auf dem Hochschulschriftenserver der ULB Bonn http://hss.ulb.uni-bonn.de/diss_online elektronisch publiziert. 1. Gutachter: Prof. Dr. Albrecht Klemm 2. Gutachter: PD Dr. Stefan Förste Tag der Promotion: 23.5.2014 Erscheinungsjahr: 2014 Abstract In this thesis we will examine two different problems. The first is the computation of vacuum expectation values of Wilson loop operators in ABJM theory, the other problem is finding the instanton series of the refined topological string on certain local Calabi–Yau geometries in the Nekrasov–Shatashvili limit. Based on the description of ABJM theory as a matrix model, it is possible to find a description of it in terms of an ideal Fermi gas with a non–trivial one–particle Hamiltonian. The vacuum–expectation–values of Wilson loop operators in ABJM theory correspond to averages of operators in the statistical–mechanical problem. Using the WKB expansion, it is possible to extract the full 1/N expansion of the vevs, up to exponentially small contributions, for arbitrary Chern–Simons coupling. We will compute these vevs for the 1/6 and 1/2 BPS Wilson loops at any winding number. These can be written in terms of the Airy function. The expressions we found reproduce the low genus results previously obtained in the ’t Hooft expansion. In another problem we use mirror symmetry, quantum geometry and modularity properties of elliptic curves to calculate the refined free energies, given in terms of an instanton sum, in the Nekrasov–Shatashvili limit on non–compact toric Calabi–Yau manifolds, based on del Pezzo surfaces. Quantum geometry here is to be understood as a quantum deformed version of rigid special geometry, which has its origin in the quantum mechanical behavior of branes in the topological string B-model. We will argue that in the Seiberg–Witten picture only the Coulomb parameters lead to quantum corrections, while the mass parameters remain uncorrected. In certain cases we will also compute the expansion of the free energies at the orbifold point and the conifold locus. We will compute the quantum corrections order by order on ~ by deriving second order differential operators, which act on the classical periods. iii Danksagungen Mein besonderer Dank gilt Prof. Dr. Albrecht Klemm, der es mir ermöglichte, auf diesem inter- essanten Gebiet arbeiten zu können. Weiterhin bedanke ich mich bei Dr. Min–xin Huang, Prof. Dr. Marcos Mariño, Dr. Masoud Soroush und Jonas Reuter für die Zusammenarbeit bei meinen Veröffentlichungen. PD Dr. Stefan Förste danke ich für die Übernahme der Zweitkorrektur. Während meiner Zeit als Doktorand habe ich mit Dr. Hans Jockers, Dr. Denis Klevers, Dr. Daniel Lopes, Dr. Jan Manschot, Dr. Marco Rauch, Dr. Thomas Wotschke, Navaneeth Gaddam, Jie Gu, Maximilian Poretschkin, Jose Miguel Zapata Rolon und Thorsten Schimannek viele anregende und erhellende Diskussionen gehabt. Vielen Dank dafür. Nicht zuletzt bedanke ich mich bei meiner Familie und meinen Freunden für die Unter- stützung. v Contents 1 Introduction 1 1.1 Fermi gas . .4 1.2 Quantum Geometry . .5 1.3 Structure . .6 2 Preliminaries 9 2.1 Matrix Models . .9 2.1.1 Definitions . .9 2.1.2 Perturbative solutions of matrix models . 11 2.1.2.1 Spectral curves . 12 2.1.3 The topological recursion of Eynard and Orantin . 14 1 1 2.1.3.1 local P × P ............................. 16 2.2 ABJM theory . 19 2.3 Localization . 20 2.3.1 ABJM as a Matrix model . 25 2.3.2 The geometry of ABJM theory . 28 2.3.3 Wilson loops in the geometric description . 33 3 ABJM Wilson loops in the Fermi gas approach 37 3.1 Introduction . 37 3.2 The Fermi gas approach . 38 3.2.1 Quantum mechanics in phase space . 38 3.2.2 Quantum Statistical Mechanics in phase space . 39 3.2.3 Large N expansion from the grand canonical ensemble . 43 3.2.4 Quantum corrections . 43 3.2.4.1 Wigner–Kirkwood expansion . 44 3.2.4.2 Sommerfeld expansion . 45 3.2.5 Fermi Gas for Chern Simons Matter theories . 46 3.2.5.1 The Fermi Surface . 49 3.2.5.2 Derivation of the N 3/2 behaviour of ABJM theory . 52 3.3 Wilson loops in the Fermi gas approach . 53 3.3.1 Incorporating Wilson loops . 53 3.3.2 Quantum Hamiltonian and Wigner–Kirkwood corrections . 54 3.3.3 Integrating over the Fermi surface . 61 3.3.4 Genus expansion . 65 vii 4 Quantum Geometry of del Pezzo surfaces in the Nekrasov–Shatashvili limit 71 4.1 Introduction . 71 4.2 Geometric setup . 73 4.2.1 Branes and Riemann surfaces . 73 4.2.1.1 Mirror symmetry for non-compact Calabi-Yau spaces . 74 4.3 The refinement . 75 4.3.1 The Nekrasov-Shatashvili limit . 76 4.3.2 Schrödinger equation from the β-ensemble . 77 4.3.3 Special geometry . 79 4.3.4 Quantum special geometry . 81 4.3.5 Genus 1-curves . 84 4.3.5.1 Elliptic curve mirrors and closed modular expressions . 84 4.3.5.2 Special geometry . 86 4.3.5.3 Quantum Geometry . 87 4.4 Examples . 88 4.4.1 The resolved Conifold . 88 4.4.2 local F0 ..................................... 89 4.4.2.1 Difference equation . 90 4.4.2.2 Operator approach . 92 4.4.2.3 Orbifold point . 93 2 4.4.3 O(−3) → P .................................. 97 4.4.3.1 Orbifold point . 98 4.4.3.2 Conifold point . 99 4.4.4 local F1 ..................................... 100 4.4.4.1 Operator approach . 101 4.4.4.2 Difference equation . 101 F 4.4.5 O(−KF2 ) → 2 ................................. 103 4.4.6 O(−KB2 ) → B2 ................................. 106 4.4.7 local B1(F2) ................................... 108 4.4.8 A mass deformation of the local E8 del Pezzo . 111 4.5 Relation to the Fermi Gas . 113 5 Conclusions and Outlook 117 5.1 Fermi gas . 117 5.2 Quantum Geometry of del Pezzo surfaces in the Nekrasov–Shatashvili limit . 119 A Matrix Models 121 A.1 Schwinger Dyson . 121 A.1.1 Correlators of the β-ensemble . 122 B Fermi Gas 123 B.1 1/6 BPS Wilson loops at arbitrary winding number . 123 B.2 Results at g = 3 and g = 4 ............................... 124 C Quantum Geometry 127 C.1 Eisenstein series . 127 viii C.2 local F0 ......................................... 127 C.2.1 Orbifold point . 128 2 C.3 O(−3) → P ...................................... 129 C.3.1 Orbifold point . 130 C.3.2 Conifold point . 130 C.4 local F1 ......................................... 131 C.5 local F2 ......................................... 132 Bibliography 135 List of Figures 145 List of Tables 147 Acronyms 149 ix CHAPTER 1 Introduction Instantons in gauge theories are of long standing interest to physicists. One well known example was given by Belavin, Polyakov, Schwartz and Tyupkin in [1], where the authors analyzed the solutions of the pure Yang–Mills Lagrangian. This is relevant for the correct definition of QCD, because it is a solution to the U(1)–problem [2], which describes the apparent existence of an U(1) symmetry in QCD that is not realized in the real world. The relevant solutions are called instantons [3], which are topologically non–trivial stationary points of the Yang–Mills action. They give nonperturbative contributions to the functional integral, which means they are not visible in the perturbative expansion in the coupling. The result described above shows us, that it is necessary to consider the nonperturbative structure of a quantum field theory in order to properly analyze it. This problem is generally very difficult, but the introduction of supersymmetry restricts the structure of gauge theories in a manner so that their exploration is often possible in an exact way. Especially N = 2 supersymmetric gauge theories in four dimensions attracted a lot of interest, because they exhibit non–perturbative effects, while it is possible to find exact results due to the work of Seiberg and Witten [4, 5]. At first their work was only about SU(2) theories, but it was extended to different gauge groups and those of higher rank since then. The information about the instanton sum is encoded in an elliptic curve C which we call Seiberg–Witten curve, and a meromorphic differential λ, defined on this curve. More precisely, the prepotential of such a gauge theory is encoded in the tuple (C, λ). Later the relation to topological string theory was studied. The Seiberg–Witten curve also appears in the topological string theory as a mirror geometry in the B–model [6, 7]. It is possible to construct a topological string theory on certain local Calabi–Yau manifolds, which, when taking a suitable limit, correspond to Seiberg–Witten theories. The genus zero amplitudes of the topological B–model amplitudes on this local Calabi–Yau manifold is captured by the Seiberg–Witten curve. Furthermore the topological B–model on local Calabi–Yau threefolds is dual to matrix models [8, 9] in the large N limit. In this approach the spectral curve corresponds to the curve C and the differential λ encodes the filling fractions and the one point function. One way to obtain the instanton sum is a localization computation on the moduli space of instantons, invented by Nekrasov [10]. He proposed the Ω–background, which introduced the two deformation parameters 1 and 2 in order to regularize the moduli space of instantons in N = 2 supersymmetric gauge theories.