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Factorization Algebras in Quantum Field Theory Volume 2 (28 April 2016)

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Factorization algebras in quantum field theory Volume 2 (28 April 2016) Kevin Costello and Owen Gwilliam Contents Chapter 1. Overview1 1.1. Classical field theory and factorization algebras1 1.2. Quantum field theory and factorization algebras2 1.3. The quantization theorem3 1.4. The rigid quantization conjecture4 Chapter 2. Structured factorization algebras and quantization7 2.1. Structured factorization algebras7 2.2. Commutative factorization algebras9 2.3. The P0 operad9 2.4. The Beilinson-Drinfeld operad 13 Part 1. Classical field theory 17 Chapter 3. Introduction to classical field theory 19 3.1. The Euler-Lagrange equations 19 3.2. Observables 20 3.3. The symplectic structure 20 3.4. The P0 structure 21 Chapter 4. Elliptic moduli problems 23 4.1. Formal moduli problems and Lie algebras 24 4.2. Examples of elliptic moduli problems related to scalar field theories 28 4.3. Examples of elliptic moduli problems related to gauge theories 30 4.4. Cochains of a local L¥ algebra 34 4.5. D-modules and local L¥ algebras 36 Chapter 5. The classical Batalin-Vilkovisky formalism 45 5.1. The classical BV formalism in finite dimensions 45 5.2. The classical BV formalism in infinite dimensions 47 5.3. The derived critical locus of an action functional 50 5.4. A succinct definition of a classical field theory 55 5.5. Examples of field theories from action functionals 57 5.6. Cotangent field theories 58 Chapter 6. The observables of a classical field theory 63 iii iv CONTENTS 6.1. The factorization algebra of classical observables 63 6.2. The graded Poisson structure on classical observables 64 6.3. The Poisson structure for free field theories 65 6.4. The Poisson structure for a general classical field theory 68 Part 2. Quantum field theory 73 Chapter 7. Introduction to quantum field theory 75 7.1. The quantum BV formalism in finite dimensions 76 7.2. The “free scalar field” in finite dimensions 78 7.3. An operadic description 80 7.4. Equivariant BD quantization and volume forms 81 7.5. How renormalization group flow interlocks with the BV formalism 82 7.6. Overview of the rest of this Part 83 Chapter 8. Effective field theories and Batalin-Vilkovisky quantization 85 8.1. Local action functionals 86 8.2. The definition of a quantum field theory 87 8.3. Families of theories over nilpotent dg manifolds 96 8.4. The simplicial set of theories 102 8.5. The theorem on quantization 105 Chapter 9. The observables of a quantum field theory 109 9.1. Free fields 109 9.2. The BD algebra of global observables 113 9.3. Global observables 120 9.4. Local observables 123 9.5. Local observables form a prefactorization algebra 124 9.6. Local observables form a factorization algebra 128 9.7. The map from theories to factorization algebras is a map of presheaves 134 Chapter 10. Further aspects of quantum observables 141 10.1. Translation-invariant factorization algebras from translation-invariant quantum field theories 141 10.2. Holomorphically translation-invariant theories and their factorization algebras 144 10.3. Renormalizability and factorization algebras 150 10.4. Cotangent theories and volume forms 163 10.5. Correlation functions 171 Part 3. A factorization enhancement of Noether’s theorem 173 Chapter 11. Noether’s theorem in classical field theory 175 11.1. Symmetries of a classical field theory 177 11.2. Examples of classical field theories with an action of a local L¥ algebra 183 CONTENTS v 11.3. The factorization algebra of equivariant observable 188 11.4. Inner actions 188 11.5. Classical Noether’s theorem 191 11.6. Conserved currents 194 11.7. Examples of classical Noether’s theorem 196 Chapter 12. Noether’s theorem in quantum field theory 201 12.1. Quantum Noether’s theorem 201 12.2. Actions of a local L¥-algebra on a quantum field theory 205 12.3. Obstruction theory for quantizing equivariant theories 209 12.4. The factorization algebra associated to an equivariant quantum field theory 212 12.5. Quantum Noether’s theorem 213 12.6. The quantum Noether theorem and equivariant observables 221 12.7. Noether’s theorem and the local index 226 12.8. The partition function and the quantum Noether theorem 230 Appendix A. Background 233 A.1. Lie algebras and L¥ algebras 233 A.2. Derived deformation theory 237 Appendix B. Functions on spaces of sections 247 B.1. Classes of functions on the space of sections of a vector bundle 247 B.2. Derivations 253 Bibliography 257 CHAPTER 1 Overview In this chapter, we give an overview of the contents of the book. 1.1. Classical field theory and factorization algebras The main aim of this book is to present a deformation-quantization approach to quan- tum field theory. In this section we will outline how a classical field theory gives rise to the classical algebraic structure we consider. We use the Lagrangian formulation throughout. Thus, classical field theory means the study of the critical locus of an action functional. In fact, we use the language of derived geometry, in which it becomes clear that functions on a derived critical locus (section 5.1) should form a P0 algebra (section 2.3), that is, a commutative algebra with a Poisson bracket of cohomological degree 1. (For an overview of these ideas, see the section 1.4.) Applying these ideas to infinite-dimensional spaces, such as the space of smooth func- tions on a manifold, one runs into analytic problems. Although there is no difficulty in constructing a commutative algebra Obscl of classical observabes, we find that the Poisson bracket on Obscl is not always well-defined. However, we show the following. 1.1.0.1 Theorem. For a classical field theory (section 5.4) on a manifold M, there is a sub- c commutative factorization algebra Obsg l of the commutative factorization algebra Obscl on which cl the Poisson bracket is defined, so that Obsg forms a P0 factorization algebra. Further, the inclusion cl Obsg ! Obscl is a quasi-isomorphism of factorization algebras. Remark: Our approach to field theory involves both cochain complexes of infinite-dimensional vector spaces and families over manifolds (and dg manifolds). The infinite-dimensional vector spaces that appear are of the type studied in functional analysis: for example, spaces of smooth functions and of distributions. One approach to working with such vector spaces is to treat them as topological vector spaces. In this book, we will instead treat them as differentiable vector spaces. In particular, Obscl will be a factorization alge- bra valued in differentiable vector spaces. For a careful discussion of differential vector 1 2 1. OVERVIEW spaces, see Appendix ??. The basic idea is as follows: a differentiable vector space is a vec- tor space V with a smooth structure, meaning that we have a well-defined set of smooth maps from any manifold X into V; and further, we have enough structure to be able to differentiate any smooth map into V. These notions make it possible to efficiently study cochain complexes of vector spaces in families over manifolds. 1.1.1. A gloss of the main ideas. In the rest of this section, we will outline why one would expect that classical observables should form a P0 algebra. More details are avail- able in section3. The idea of the construction is very simple: if U ⊂ M is an open subset, we will let EL(U) be the derived space of solutions to the Euler-Lagrange equation on U. Since we are dealing with perturbative field theory, we are interested in those solutions to the equations of motion which are infinitely close to a given solution. The differential graded algebra Obscl(U) is defined to be the space of functions on EL(U). (Since EL(U) is an infinite dimensional space, it takes some work to define Obscl(U). Details will be presented later (Chapter ??). On a compact manifold M, the solutions to the Euler-Lagrange equations are the criti- cal points of the action functional. If we work on an open subset U ⊂ M, this is no longer strictly true, because the integral of the action functional over U is not defined. However, fields on U have a natural foliation, where tangent vectors lying in the leaves of the foli- ation correspond to variations f ! f + df, where df has compact support. In this case, the Euler-Lagrange equations are the critical points of a closed one-form dS defined along the leaves of this foliation. Any derived scheme which arises as the derived critical locus (section 5.1) of a func- tion acquires an extra structure: it’s ring of functions is equipped with the structure of a P0 algebra. The same holds for a derived scheme arising as the derived critical locus of a closed one-form define along some foliation. Thus, we would expect that Obscl(U) is equipped with a natural structure of P0 algebra; and that, more generally, the commuta- cl tive factorization algebra Obs should be equipped with the structure of P0 factorization algebra. 1.2. Quantum field theory and factorization algebras Another aim of the book is to relate perturbative quantum field theory, as developed in [Cos11b], to factorization algebras. We give a natural definition of an observable of a quantum field theory, which leads to the following theorem. 1.3. THE QUANTIZATION THEOREM 3 1.2.0.1 Theorem. For a classical field theory (section 5.4) and a choice of BV quantization (section 8.2), the quantum observables Obsq form a factorization algebra over the ring R[[h¯ ]]. Moreover, the factorization algebra of classical observables Obscl is homotopy equivalent to Obsq modh ¯ as a factorization algebra.
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