Factorization Algebras in Quantum Field Theory Volume 1 (8 May 2016)
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Factorization Algebras in Quantum Field Theory Volume 1 (8 May 2016) Kevin Costello and Owen Gwilliam To Lauren & To Sophie Contents Chapter 1. Introduction1 1. The motivating example of quantum mechanics2 2. A preliminary definition of prefactorization algebras6 3. Prefactorization algebras in quantum field theory7 4. Comparisons with other formalizations of quantum field theory9 5. Overview of this volume 13 6. Acknowledgements 14 Part 1. Prefactorization algebras 17 Chapter 2. From Gaussian measures to factorization algebras 19 1. Gaussian integrals in finite dimensions 20 2. Divergence in infinite dimensions 22 3. The prefactorization structure on observables 26 4. From quantum to classical 28 5. Correlation functions 30 6. Further results on free field theories 32 7. Interacting theories 33 Chapter 3. Prefactorization algebras and basic examples 37 1. Prefactorization algebras 37 2. Associative algebras from prefactorization algebras on R 42 3. Modules as defects 43 4. A construction of the universal enveloping algebra 49 5. Some functional analysis 52 6. The factorization envelope of a sheaf of Lie algebras 61 7. Equivariant prefactorization algebras 65 Part 2. First examples of field theories and their observables 71 Chapter 4. Free field theories 73 1. The divergence complex of a measure 73 2. The prefactorization algebra of a free field theory 76 3. Quantum mechanics and the Weyl algebra 87 4. Pushforward and canonical quantization 92 5. Abelian Chern-Simons theory 94 6. Another take on quantizing classical observables 101 iii iv CONTENTS 7. Correlation functions 106 8. Translation-invariant prefactorization algebras 108 9. States and vacua for translation invariant theories 114 Chapter 5. Holomorphic field theories and vertex algebras 119 1. Holomorphically translation-invariant prefactorization algebras 122 2. A general method for constructing vertex algebras 129 3. The βγ system and vertex algebras 140 4. Kac-Moody algebras and factorization envelopes 154 Part 3. Factorization algebras 167 Chapter 6. Factorization algebras: definitions and constructions 169 1. Factorization algebras 169 2. Factorization algebras in quantum field theory 175 3. Variant definitions of factorization algebras 176 4. Locally constant factorization algebras 180 5. Factorization algebras from cosheaves 184 6. Factorization algebras from local Lie algebras 187 Chapter 7. Formal aspects of factorization algebras 191 1. Pushing forward factorization algebras 191 2. Extension from a basis 191 3. Pulling back along an open immersion 197 4. Descent along a torsor 198 Chapter 8. Factorization algebras: examples 201 1. Some examples of computations 201 2. Abelian Chern-Simons theory and quantum groups 206 Appendix A. Background 225 1. Simplicial techniques 226 2. Operads and algebras 232 3. Lie algebras and Chevalley-Eilenberg complexes 240 4. Sheaves, cosheaves, and their homotopical generalizations 244 5. Elliptic complexes 250 Appendix B. Functional analysis 255 1. Introduction 255 2. Differentiable vector spaces 256 3. Locally convex topological vector spaces 267 4. Bornological vector spaces 270 5. Convenient vector spaces 273 6. The relevant enrichment of DVS over itself 277 7. Vector spaces arising from differential geometry 285 Appendix C. Homological algebra in differentiable vector spaces 289 CONTENTS v 1. Introduction 289 2. Linear algebra and homological algebra in DVS 290 3. Spectral sequences 292 4. Differentiable pro-cochain complexes 294 5. Homotopy colimits 296 Appendix D. The Atiyah-Bott Lemma 307 Bibliography 309 Index 315 CHAPTER 1 Introduction This two-volume book will provide the analog, in quantum field theory, of the deformation quantization approach to quantum mechanics. In this introduction, we will start by recalling how deformation quantization works in quantum mechanics. The collection of observables in a quantum mechanical system forms an asso- ciative algebra. The observables of a classical mechanical system form a Poisson algebra. In the deformation quantization approach to quantum mechanics, one starts with a Poisson algebra Acl and attempts to construct an associative algebra Aq, which is an algebra flat over the ring C[[~]], together with an isomorphism of q cl cl associative algebras A =~ A . In addition, if a; b 2 A , and ea;eb are any lifts of a; b to Aq, then 1 cl lim [ea;eb] = fa; bg 2 A : ~!0 ~ Thus, Acl is recovered in the ~ ! 0 limit, i.e., the classical limit. We will describe an analogous approach to studying perturbative quantum field theory. In order to do this, we need to explain the following. • The structure present on the collection of observables of a classical field theory. This structure is the analog, in the world of field theory, of the commutative algebra that appears in classical mechanics. We call this structure a commutative factorization algebra. • The structure present on the collection of observables of a quantum field theory. This structure is that of a factorization algebra. We view our definition of factorization algebra as a differential geometric ana- log of a definition introduced by Beilinson and Drinfeld. However, the definition we use is very closely related to other definitions in the literature, in particular to the Segal axioms. • The additional structure on the commutative factorization algebra as- sociated to a classical field theory that makes it “want” to quantize. This structure is the analog, in the world of field theory, of the Poisson bracket on the commutative algebra of observables. • The deformation quantization theorem we prove. This states that, pro- vided certain obstruction groups vanish, the classical factorization al- gebra associated to a classical field theory admits a quantization. Fur- ther, the set of quantizations is parametrized, order by order in ~, by the space of deformations of the Lagrangian describing the classical theory. 1 2 1. INTRODUCTION This quantization theorem is proved using the physicists’ techniques of perturba- tive renormalization, as developed mathematically in Costello(2011b). We claim that this theorem is a mathematical encoding of the perturbative methods developed by physicists. This quantization theorem applies to many examples of physical interest, in- cluding pure Yang-Mills theory and σ-models. For pure Yang-Mills theory, it is shown in Costello(2011b) that the relevant obstruction groups vanish and that the deformation group is one-dimensional; thus there exists a one-parameter family of quantizations. In Li and Li(2016), the topological B-model with target a complex manifold X is constructed; the obstruction to quantization is that X be Calabi-Yau. Li and Li show that the observables and correlations functions recovered by their quantization agree with well-known formulas. In Grady et al.(n.d.), Grady, Li, and Li describe a 1-dimensional σ-model with target a smooth symplectic manifold and show how it recovers Fedosov quantization. Other examples are considered in Gwilliam and Grady(2014), Costello(2010, 2011a), and Costello and Li(2011). We will explain how (under certain additional hypotheses) the factorization algebra associated to a perturbative quantum field theory encodes the correlation functions of the theory. This fact justifies the assertion that factorization algebras encode a large part of quantum field theory. This work is split into two volumes. Volume 1 develops the theory of fac- torization algebras, and explains how the simplest quantum field theories — free theories — fit into this language. We also show in this volume how factorization algebras provide a convenient unifying language for many concepts in topological and quantum field theory. Volume 2, which is more technical, derives the link be- tween the concept of perturbative quantum field theory as developed in Costello (2011b) and the theory of factorization algebras. 1. The motivating example of quantum mechanics The model problems of classical and quantum mechanics involve a particle moving in some Euclidean space Rn under the influence of some fixed field. Our main goal in this section is to describe these model problems in a way that makes the idea of a factorization algebra (Section1) emerge naturally, but we also hope to give mathematicians some feeling for the physical meaning of terms like “field” and “observable.” We will not worry about making precise definitions, since that’s what this book aims to do. As a narrative strategy, we describe a kind of cartoon of a physical experiment, and we ask that physicists accept this cartoon as a friendly caricature elucidating the features of physics we most want to emphasize. 1.1. A particle in a box. For the general framework we want to present, the details of the physical system under study are not so important. However, for con- creteness, we will focus attention on a very simple system: that of a single particle confined to some region of space. We confine our particle inside some box and occasionally take measurements of this system. The set of possible trajectories of the particle around the box constitute all the imaginable behaviors of this particle; we might describe this space of behaviors mathematically as Maps(I; Box), where 1. THE MOTIVATING EXAMPLE OF QUANTUM MECHANICS 3 I ⊂ R denotes the time interval over which we conduct the experiment. We say the set of possible behaviors forms a space of fields on the timeline of the particle. The behavior of our theory is governed by an action functional, which is a function on Maps(I; Box). The simplest case typically studied is the massless free field theory, whose value on a trajectory f : I ! Box is Z S ( f ) = ( f (t); f¨(t))dt: t2I Here we use (−; −) to denote the usual inner product on Rn, where we view the box as a subspace of Rn, and f¨ to denote the second derivative of f in the time variable t. The aim of this section is to outline the structure one would expect the observ- ables — that is, the possible measurements one can make of this system — should satisfy. 1.2. Classical mechanics. Let us start by considering the simpler case where our particle is treated as a classical system.