University of Heidelberg Master Thesis Spectral Networks - A story of Wall-Crossing in Geometry and Physics Author: Supervisor: Sebastian Schulz Prof. Dr. Anna Wienhard A thesis submitted in fulfilment of the requirements for the degree of Master of Science in the Faculty of Mathematics and Computer Sciences Mathematical Institute November 24, 2015 Declaration of Authorship I, Sebastian Schulz, declare that this thesis titled, ’Spectral Networks - A story of Wall-Crossing in Geometry and Physics’ and the work presented in it are my own. I confirm that: • This work was done wholly while in candidature for a research degree at this University. • Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. • Where I have consulted the published work of others, this is always clearly at- tributed. • Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. Signed: Date: iii UNIVERSITY OF HEIDELBERG Abstract Faculty of Mathematics and Computer Sciences Mathematical Institute Differential Geometry Research Group Master of Science Spectral Networks - A story of Wall-Crossing in Geometry and Physics by Sebastian Schulz This thesis deals with the phenomenon of wall-crossing for BPS indices in d = 4 N = 2 supersymmetric gauge theories with gauge group SU(K). Compactification over S1 yields a three-dimensional σ-model with target space M a fiber bundle over the Coulomb branch B of the four-dimensional theory. We demonstrate how the wall- crossing is captured by smoothness conditions on the Hyperkähler metric of M. Three ways of determining the 4d BPS spectrum are explained, drawing on the work of Gaiotto, Moore and Neitzke. Firstly, a twistor space construction reduces the problem to finding holomorphic Darboux coordinates which are obtained as solutions to a Riemann-Hilbert problem for large radii R of the circle. Secondly, for a subclass of theories obtained by compactifying a six-dimensional theory over a surface C, the Darboux coordinates can be computed from Fock-Goncharov coordinates on certain triangulations of C for gauge group SU(2). Thirdly, a codimension one sublocus of C called a Spectral Network cap- tures the BPS degeneracies in a more efficient way. Die vorliegende Abschlussarbeit beschäftigt sich mit dem Phänomen des Wall-Crossing für BPS Indizes vierdimensionaler Eichtheorien mit N = 2 Supersymmetrie und Eich- gruppe SU(K). Kompaktifizierung über einer S1 liefert ein dreidimensionales σ-Modell, dessen Zielraum M ein Faserbündel über dem Coulombzweig B der vierdimensionalen Theorie ist. Wir zeigen auf, wie das Wall-Crossing geometrisch aufgefasst werden kann als Glattheit der Hyperkählermetrik von M. Das Kernstück ist die Beschreibung dreier Wege das BPS Spektrum der 4d-Theorie zu bestimmen, was maßgeblich die Arbeit von Gaiotto, Moore und Neitzke war. Erstens kann durch eine Twistorraum-Konstruktion das Problem dazu reduziert werden, holomorphe Darboux-Koordinaten für das Faser- bündel M → B zu finden. Diese lassen sich aus der Lösung eines Riemann-Hilbert- Problems bestimmen, die zumindest für große Radien des Kompaktifizierungskreises ex- istiert. Zweitens beschreiben wir eine Unterklasse von Theorien, die ihrerseits als Kom- paktifzierung einer sechsdimensionalen Theorie über einer gepunkteten Riemannschen Fläche entsteht. Die holomorphen Darboux-Koordinaten können für den SU(2)-Fall aus Fock-Goncharov-Koordinaten bestimmter Triangulierungen von C berechnet werden. Drittens und letztens erläutern wir, wie bestimmte Weben von Linien auf C, genannt Spektrale Netzwerke, die BPS Indizes sehr effizient kodieren. v Acknowledgements I would like to thank my advisors Anna Wienhard and Daniel Roggenkamp for intro- ducing me to this fascinating topic, along with their endless patience and the guidance I have received both academically and personally. My thanks go to the Differential Geometry Research Group for providing a warm and welcoming atmosphere. I am particularly grateful to Daniele Alessandrini for the helpful discussions. I would like to thank my friends and family for helping me to keep my head where it belongs: Above this thesis. My particular thanks go to my friends Beatriz Tapia and Sebastian Nill for the careful proofreading and the positive criticism. vii Contents Declaration of Authorship iii Abstract v Acknowledgements vii 1 Introduction 1 2 Functorial Quantum Field Theories 3 2.1 Classical TQFT’s . .3 2.2 Higher categories and the Cobordism Hypothesis . .7 2.3 Towards Functorial Quantum Field Theories . 13 2.4 Compactification of QFTs . 19 3 Four-dimensional N = 2 Gauge theories 21 3.1 Gauge theory and σ-models . 21 3.2 BPS states . 24 3.3 The BPS index and the Wall-Crossing Formula . 25 3.4 Seiberg-Witten theory . 29 3.5 Dimensional reduction to three dimensions . 30 3.6 Construction of the Hyperkähler metric . 32 4 Theories of class S 35 4.1 The Theory X ................................. 35 4.2 QFTs with defects . 37 4.3 Compactify to 4d: Theories of class S ................... 39 4.4 Connection to the Hitchin integrable system . 41 4.5 String webs: The geometry of BPS states . 43 4.6 The sl(2, C) case . 46 5 Spectral Networks 51 5.1 2d-4d Wall-Crossing . 52 5.2 Surface operators and line operators . 55 5.3 The Formal Parallel Transport Theorem . 59 5.4 Spectral Networks at generic ϑ ....................... 62 5.5 Spectral Networks at critical ϑ ....................... 65 6 Conclusion and outlook 67 A Some category theory 69 A.1 Symmetric monoidal categories . 69 A.2 Internal categories . 71 A.3 Pseudo-categories . 73 ix B Supermathematics 75 B.1 Supermanifolds . 75 B.2 Super Lie algebras . 76 B.3 Representations of the super Poincaré algebra . 81 Bibliography 87 x Chapter 1 Introduction Supersymmetric gauge theories enjoy an intimate relationship to geometry, a fact that has led to a fruitful interplay of mathematics and physics. A particular interesting class of gauge theories are those in four dimensions and with N = 2 supersymmetry. The physical content of this thesis deals with certain properties of these theories. The main factors contributing are the Low Energy Effective Action (LEEA) and their BPS spectrum, the spectrum of distinguished states, called BPS states, that are very stable. While the former can be computed explicitely through the acclaimed Seiberg-Witten theory, the latter have only been determined in very particular cases. We aim at de- scribing different methods to derive this spectrum for a large class of theories, drawing on powerful methods from complex geometry. Let us explain the structure of this thesis. We start in chapter 2 by explaining a functorial approach to Quantum Field Theories. Its spirit is to circumvent the direct notion of path integrals by indirectly encoding its properties in a (higher) functor, a method that has been put to great success for Topological and Conformal Field Theories. Our use for these methods are to describe two important concepts in modern theoretical physics: Compactification and defects. While the former is a technique to obtain a lower dimensional QFT from a higher dimensional one, the latter encode ways of embedding lower dimensional QFTs inside higher dimensional ones. The concepts of this chapter depend crucially on abstract notions from category theory which we have gathered in appendix A. The main content of this thesis can be found in chapters 3-5.Each of these follow a similar pattern: The early sections introduce novel structures from physics through ax- iom systems that we aim to make as precise as possible. The mathematical implications are derived in the consecutive sections. Chapter 3 introduces the notions of four-dimensional gauge theories with N = 2 supersymmetry required for further understanding. We have attached a primer to su- persymmetry in appendix B which some readers might find useful. We review the notion of BPS states: Those are distinguished states of the theory which are very stable but can form or decay at certain walls in a manner governed by a wall-crossing formula. An index Ω is introduced in order to count these states. However, its computation can be infeasible due to the lack of proper tools in the strong coupling regime. A key insight is that Ω is locally constant over the moduli space of vacua B but jumps at the walls at which BPS states are generated or annihilated. Hence knowing the walls and how the index jumps, one can determine Ω globally by computing its value in a single region together with the jumps it is subject to. The question then remains how to determine these jumps. We start in section 3.5 by presenting one way in which geometry can enter this pic- 1 ture. Upon compactification over a circle SR of radius R, one obtains an N = 4 theory in three-dimensions with moduli space a Hyperkähler manifold M in the IR limit. M 1 2 Chapter 1. Introduction is naturally a fiber bundle over B with generic fiber an abelian variety, but these fibers degenerate over the singular loci of B. The wall-crossing formula is intimately related to the Hyperkähler metric g of M: It is equivalent to saying that g is smooth. Hence, the problem of determining the indices Ω has been translated into computing the Hyperkäh- ler metric g. We review a twistor space construction of this metric from [GMN10] in 3.6 in which holomorphic Darboux coordinates arise as solutions to a certain Riemann- Hilbert problem. The downside of this approach is that it only works for large radii R of the circle. A different approach is possible for a subclass of theories that arise as compactifica- tion limits from a six-dimensional theory X. The theory X is not very well-understood due to its lack of a proper action functional, hence compactifications are an important way to shed some light on it.