NORTHWESTERN UNIVERSITY The Holomorphic σ-model and its Symmetries A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree DOCTOR OF PHILOSOPHY Field of Mathematics By Brian R. Williams EVANSTON, ILLINOIS June 2018 2 c Copyright by Brian R. Williams 2018 All Rights Reserved 3 ABSTRACT The Holomorphic σ-model and its Symmetries Brian R. Williams The holomorphic σ-model is a field theory that exists in any complex dimension that describes the moduli space of holomorphic maps from one complex manifold to another. We introduce the general notion of a holomorphic field theory, which is one that is sensi- tive to the underlying complex structure of the spacetime manifold (and potentially other geometric input data). Throughout, we rigorously study perturbative quantum field the- ory in a way that combines the Batalin-Vilkovisky formalism and the effective approach to renormalization. In addition to computing the one-loop anomaly of the holomorphic σ-model, we study the local operators of the theory using factorization algebras and com- pute the local index. The final part of this thesis investigates the symmetries present in a general holomorphic theories. We characterize the local central extensions of the two fundamental symmetry algebras, which provide higher dimensional generalizations of the Kac-Moody and Virasoro vertex algebras. 4 Acknowledgements My journey to pursue mathematics started well before entering the Ph.D. program at Northwestern. Throughout this, I've relied on the constant support of my family, especially my parents. I thank my mother for being a great listener and for giving me the freedom to follow my passions. I thank my high school calculus teacher, who also happens to be my father, for kindling my interest in math. He's urged me to always think critically; he is my first, and most important, teacher. To my brother, Craig, I am grateful for the sustained camaraderie and competitiveness that we've shared ever since I can remember. His visits to Chicago over the past few years were a welcome distraction filled with unforgettable memories. At the core of all of this is my wife, Icon. She has been there for me as a partner, a mentor, and so much more. I thank her for being the exemplification of hard work for which I know no parallels. Throughout many anxiety-ridden periods during this program, her warmth and display of equanimity has unquestionably made the completion of this thesis possible. Finally, words can't begin to explain how grateful I am for her willingness to drop her life in Chicago and move across the country so that I may continue to pursue my career in mathematics. Each day we grow closer to each other, and being married to her is by far my proudest achievement. Next, I'd like to thank my two Ph.D. advisors, Kevin Costello and John Francis. Kevin has been pivotal in molding my mathematical taste and interests. I am grateful for 5 his constant encouragement and patient explanation of essentially all facets of quantum field theory that I have learned in grad school. His generosity to share his ideas has bestowed the inspiration for many ideas that lie at the backbone of this thesis. John has provided me valuable guidance at crucial points throughout my time as a graduate student. He pushed me to pursue difficult problems, and I continue to be inspired by his dedication and focus. At the University of Florida, my undergraduate mentor, David Groisser, devoted countless hours of his time to provide me with my first introduction to research level mathematics. Your selflessness is not forgotten, and serves as a model that I strive to emulate. I'd like to single out Owen Gwilliam as an extremely influential friend and mentor. I met Owen at a critical juncture in my graduate career as I transitioned from coursework to research mathematics. His willingness to collaborate and his support for my ideas instilled in me the confidence and energy I needed to pursue new, and often overwhelming, projects. I thank him for the running invitation to visit him in Germany, where our discussions and collaborations influenced, and greatly improved, large chunks of this thesis. The generosity of his family, especially his wife, Sophie, made these visits possible. I'll miss the hills of Niederbrombach, but I hope to visit the Laszlo Center for Mathematical Research (LCMR) following its upcoming relocation. I'd like to thank the rest of my Costello-family Dylan Butson, Chris Elliott, and Philsang Yoo for the fun and always stimulating collaborations. I've learned a lot from each of them. I thank Ryan Grady for teaching me about jets and stacks, as well as his friendship and guidance at important junctures during grad school. Thanks to Hiro Lee Tanaka for officiating our wedding. He has been a model of discipline and mindfulness 6 both in and outside of mathematics. Most of what I know about complex geometry and string theory I've learned from Si Li. I thank him for answering all my silly questions and for feeding me a lot of delicious food. Thanks to Matt Szczesny for answering numerous questions about vertex algebras, and also for the consistent encouragement over the past few years. Early on in graduate school I was part of a vibrant community of graduate students at Northwestern who have undoubtedly shaped my interests in mathematics. I'd like to es- pecially thank those in this group: Lauren Bandklayder, Elden Elmanto, Aron Heleodoro, Ben Knudsen, Rob Legg, Johan Konter, Sean Pohorence, Paul Vankoughnett, and Dylan Wilson. I also thank Perimeter Institute for Theoretical Physics for hosting me numer- ous times over the past four years, including the entire Fall of 2015. Those visits were made more enjoyable and productive by great company including Theo Johnson-Freyd and David Svoboda. I have been fortunate enough to receive the support and guidance of many other faculty outside of Northwestern. I thank Stephan Stolz for his invested interest in my work and career, and for offering me an invaluable platform to share my ideas and results during my visits to Notre Dame. His suggestions and pointed questions have definitely improved the quality of this work. I thank Vassily Gourbonov for teaching me about CDO's. His support for my idea of using formal geometry to study σ-models in the BV formalism helped to cradle the initial work that eventually became this thesis. Thank you to David Ayala for his valuable advice and the invitation to visit Montana State University, where part of the research for this thesis took place. 7 I have received support from the National Science Foundation as a graduate student research fellow under Award DGE-1324585. In addition to being a visitor at Perimeter Institute and MSU, parts of this research were performed while a visitor at the Max Planck Institute f¨urMathematik (MPIM) and the Institut des Hautes Etudes` Scientifiques (IHES). I thank them for their support. 8 Table of Contents ABSTRACT 3 Acknowledgements 4 Table of Contents 8 List of Tables 10 List of Figures 11 Chapter 1. Introduction 12 1.1. Summary of the results 15 1.2. Overview 16 Chapter 2. Holomorphic quantum field theory 18 2.1. The definition of a quantum field theory 21 2.2. Holomorphic field theories 44 2.3. Renormalization of holomorphic theories 66 2.4. Equivariant BV quantization 91 Chapter 3. The holomorphic σ-model 101 3.1. Gelfand-Kazhdan formal geometry 107 3.2. Descent for extended pairs 142 9 3.3. The classical holomorphic σ-model 153 3.4. Deformations of the holomorphic σ-model 166 3.5. BV quantization of the holomorphic σ-model 180 3.6. The local operators 204 Chapter 4. Local symmetries of holomorphic theories 245 4.1. The current algebra on complex manifolds 249 4.2. Local structures of the Kac-Moody factorization algebra 267 4.3. The Kac-Moody factorization algebra on general manifolds 287 4.4. Universal Grothendieck-Riemann-Roch from BV quantization 299 4.5. Holomorphic diffeomorphisms 317 References 342 Appendix A. Appendix 354 A.1. The dg model for punctured affine space 354 10 List of Tables 2.1 From holomorphic to BV 55 11 List of Figures 3.1 The (d + 1)-vertex wheel contributing to the anomaly. The vertices are given by v0; : : : ; vd which are labeled by a formal vector field Xα. On the black internal edges are we place the propagator P<L. On the red edge labeled by e we place the heat kernel K. 188 3.2 Nesting spherical shells. The blue and green shaded regions represent the spherical shells Nr1,1 and Nr2,2 , respectively. The black spheres denote the inner and outer boundaries of the closure of the neighborhood Nr,. 221 g 4.1 The diagram representing the weight WΓ;e(P<L;K;I ) in the case d = 2. On the black internal edges are we place the propagator P<L of the βγ system. On the red edge labeled by e we place the heat (i) 0;∗ 2 kernel K. The external edges are labeled by elements α 2 Ωc (C ).306 12 CHAPTER 1 Introduction The topic of this thesis is a mathematically rigorous analysis of a class of quantum field theories that depend on the complex structure of a manifold in an analogous way that topological theories depend on the smooth structure. Topological field theories have gained much interest in the world of mathematics due to their elegant functorial descrip- tions, as well as their applications to geometry and topology. Holomorphic theories, as we will define them, often contain strictly more information than their topological coun- terparts.