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Monopoles and Fueter Sections on Three-Manifolds MONOPOLESANDFUETERSECTIONSONTHREE-MANIFOLDS aleksander doan A dissertation presented for the degree of Doctor of Philosophy Department of Mathematics Stony Brook University 2019 Well-versed in the expanses that stretch from earth to stars, we get lost in the space from earth up to our skull. Intergalactic reaches divide sorrow from tears. En route from false to true you wither and grow dull. — Wisława Szymborska, To My Friends1 Dedicated to my parents and grandparents, who taught me to navigate ’the space from earth up to my skull’ 1 Poems New and Collected, transl. Stanisław Bara´nczakand Clare Cavanagh, Harcourt Inc., 1998. ABSTRACT The subject of this dissertation is a relationship between two classes of solutions to elliptic differential equations on three-manifolds: monopoles and Fueter sections. The main result is a description of a wall-crossing phenomenon for a signed count of monopoles. As a corollary, we prove the existence of Fueter sections with singularities. We also study monopoles and Fueter sections using methods of complex geometry, on three-manifolds which are the product of a circle and a surface. Finally, we discuss the relevance of our results to gauge theory on higher-dimensional Riemannian manifolds with special holonomy. DECLARATION This dissertation is based on the articles [Doa17a; Doa17b;DW 17a;DW 17b; DW17c], the last three of which were written in collaboration with Thomas Wal- puski. In Section 1.6 and at the beginning of each chapter, we explain which parts were written in collaboration, and refer to the relevant sections of the articles. DISSERTATIONCOMMITTEE Prof. Simon Donaldson, Simons Center for Geometry and Physics (Ph.D. Advisor) Prof. Mohammed Abouzaid, Department of Mathematics, Columbia University Prof. Kenji Fukaya, Simons Center for Geometry and Physics Prof. John Morgan, Simons Center for Geometry and Physics iii ACKNOWLEDGMENTS I owe a huge debt of gratitude to my advisor Simon Donaldson, for all his teaching, guidance, and generosity in sharing his ideas with me. For the last five years, his kindness and encouragement have kept me motivated and excited about mathematics, and our meetings have been something to look forward to every week. I could not have asked for a better advisor. I am deeply grateful to Thomas Walpuski, whose influence on my research will be obvious to anyone reading this dissertation. I have been inspired not only by his deep understanding of mathematics, but also by his work ethics, honesty, and professional integrity. It is my great pleasure to thank Andriy Haydys, Rafe Mazzeo, Clifford Taubes, and Richard Thomas for many illuminating discussions and their patience in answering my questions. I am also grateful to my friends and colleagues from the Stony Brook math department for creating a stimulating research environment and for everything they have taught me. I want to mention, in particular, Kenji Fukaya, Blaine Lawson, Mark McLean, Ben Sibley, Song Sun, Alex Waldron, and Aleksei Zinger. I was fortunate to have had many inspiring teachers and mentors during my studies in Cambridge, Bonn, and Warsaw. They gave me the background and encouragement necessary to pursue a Ph.D. in mathematics. I’m especially indebted to Ivan Smith and Chris Wendl, who introduced me to the beautiful world of symplectic geometry, and to Matthias Kreck, whose fascinating lectures on 4–manifolds were a turning point in my mathematical education. I am grateful to the Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics for many exciting meetings and conferences. In years 2017—2019 my research was supported by a Simons Collaboration grant. Finally, I would like to thank my family for all their love and support. This dissertation is dedicated to them. iv CONTENTS 1 introduction1 1.1 Gauge theory and invariants of manifolds . 1 1.2 Monopoles and quaternionic representations . 2 1.3 Fueter sections . 4 1.4 A conjectural picture . 6 1.5 Gauge theory on G2–manifolds . 12 1.6 Summary of the results . 17 2 monopoles and fueter sections 19 2.1 Hyperkähler quotients of quaternionic representations . 19 2.2 Seiberg–Witten equations . 21 2.3 Deformation theory of monopoles . 25 2.4 Orientations . 31 2.5 Fueter sections and the Haydys correspondence . 33 2.6 Deformation theory of Fueter sections . 39 2.7 Gluing Kuranishi models . 42 2.8 A wall-crossing phenomenon . 50 2.9 ADHM monopoles . 53 3 fueter sections with singularities 57 3.1 Existence of harmonic Z2 spinors . 57 3.2 The Seiberg–Witten equation with two spinors . 61 3.3 Computation of the hyperkähler quotient . 68 3.4 Compactness of the moduli space . 69 3.5 Wall-crossing and the spectral flow . 72 3.6 Compactification of the cobordism . 75 3.7 Orientation at infinity . 76 3.8 Proof of the wall-crossing formulae . 78 3.9 Transversality for paths . 80 3.10 Proof of the existence of singular harmonic Z2 spinors . 84 3.11 Relation to gauge theory on G2–manifolds . 91 4 adhm monopoles on riemann surfaces 94 4.1 A dimensional reduction . 95 4.2 ADHM bundles on Riemann surfaces . 102 4.3 Monopoles with two spinors . 105 4.4 A tale of two compactifications . 113 4.5 Harmonic Z2 spinors and complex geometry . 123 4.6 Moduli spaces of framed vortices . 129 4.7 Examples and computations . 134 bibliography 143 v INTRODUCTION 1 1.1 gauge theory and invariants of manifolds Over the last thirty years, the study of the equations of gauge theory has led to spectacular advances in low-dimensional topology. These developments trace back to Donaldson’s construction of invariants of smooth 4–manifolds using instantons—a special class of solutions to the Yang–Mills equations [Don83; Don87; Don90]. Floer extended Donaldson’s ideas to 4–manifolds with boundary and introduced invariants of 3–manifolds, which behave functorially with respect to cobordisms, i.e. define a 3 + 1–dimensional topological field theory [Flo88; Ati88a; Ati88b]. Surfaces can be incorporated in this theory as well, by considering flat connections in dimension two. dimension invariant type of invariant 4 Donaldson invariants number 3 Instanton Floer homology vector space 2 Fukaya category of moduli space category The basic ideas behind Donaldson–Floer theory are: 1. The Donaldson invariants of a closed 4–manifold X are defined by integrat- ing certain cohomology classes over the moduli space of instantons on X. In the simplest situation when the moduli space is zero-dimensional, the invariant is a signed count of instantons. 2. The instanton Floer homology groups of a closed 3–manifold Y are de- fined by formally applying the construction of Morse homology to the Chern–Simons functional—a functional on the infinite-dimensional space of connections on Y, whose critical points are flat connections on Y, and whose gradient flowlines correspond to instantons on the cylinder X = Y × R. 3. For a closed surface Z, the moduli space MZ of irreducible flat connec- tions on Z is naturally a symplectic manifold [AB83]. For every symplec- tic manifold there is an associated Fukaya category, whose objects are Lagrangian submanifolds and morphisms spaces are constructed using pseudo-holomorphic discs with boundary on such submanifolds; see, for example, [Aur14; Smi15]. The Atiyah–Floer conjecture and its various refine- ments relate the algebraic structures of the Fukaya category of MZ to the instanton invariants of 3–manifolds admitting a Heegaard splitting along Z [Ati88a; Sal95;DF 18]. The conjecture is based on the observation that 2 instantons on X = Z × R give rise to pseudo-holomorphic discs in MZ. There is a parallel topological field theory based on the Seiberg–Witten equation on 4–manifolds and its dimensional reductions to 3–manifolds and surfaces [Mor96;KM 07;OS 04]. It is natural to ask whether equations of gauge theory other than the instanton and Seiberg–Witten equations can be used to construct 1 1.2 monopoles and quaternionic representations2 invariants of manifolds. The subject of this dissertation fits into two new, quickly developing lines of research whose goal is to extend the scope of applications of gauge theory to geometry: 1. Beyond Yang–Mills & Seiberg–Witten. While most of research in mathematical gauge theory so far focused on the Yang–Mills and Seiberg–Witten equa- tions, physicists have long considered other interesting differential equations on low-dimensional manifolds. These include the Kapustin–Witten equation, which is expected to lead to new insights into the topology of 3–manifolds and knots [KW07; Wit18], and the Vafa–Witten equation, which has links to algebraic geometry [VW94]. There has been recently a surge in the study of these and related equations, initiated by deep work of Taubes [Tau13b; Tau13a; Tau14]. 2. Beyond low dimensions. Many attractive features of Yang–Mills theory in low dimensions generalize to manifolds of dimensions six, seven, and eight, pro- vided that they are equipped with a Riemannian metric of special holonomy, given by the Lie groups SU(3), G2, and Spin(7), respectively. Donaldson, Thomas, and E. Segal [DT98;DS 11] proposed to define invariants of special holonomy manifolds using Yang–Mills theory, mimicking Donaldson–Floer theory in low dimensions. Such invariants would be helpful in classifying the existing millions of examples of special holonomy manifolds. There is a rather surprising connection between these two topics, which was discovered by Walpuski [Wal13; Wal17], building on ideas of Donaldson–Segal [DS11] and Haydys [Hay12]. In the next three sections, we discuss two interesting classes of solutions to elliptic differential equations on low-dimensional manifolds and explain a conjectural relationship between them. We then relate this discus- sion to higher-dimensional Yang–Mills theory. At the end of the introduction, we summarize the main results proved in this dissertation. 1.2 monopoles and quaternionic representations The discovery of the Seiberg–Witten equation gave mathematicians powerful tools for studying low-dimensional manifolds.
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