Factorization algebras and free field theories

Description

We study the Batalin-Vilkovisky (BV) formalism for quantization of field theories in several contexts. First, we extract the essential homological procedure and study it from the perspective of derived algebraic geometry. Our main result here is that the BV formalism provides a natural determinant functor we call “cotangent quantization,” sending a perfect R-module to an invertible R-module and quasi-isomorphisms to quasi-isomorphisms, where R is an artinian commutative differential graded algebra over a field of characteristic zero. Second, we introduce the formalism of factorization algebras, a local-to-global object much like a sheaf, and describe several perspectives on how the BV formalism makes the observables of a free quantum field theory into a factorization algebra. We study in detail the free βγ system, a holomorphic field theory living on any Riemann surface, and we recover the βγ vertex algebra from the factorization algebra of quantum observables. We also construct the factorization algebras on a Riemann surface that recover the vertex algebras arising from affine Kac-Moody Lie algebras. Finally, we study quantization of families of elliptic complexes. Our main result here is an index theorem relating the associated family of factorization algebras to the determinant line of the family of elliptic complexes. At the heart of our work is the formalism for perturbative quantum field theory developed by Costello [Cos11] and for the associated observables by Costello-Gwilliam [CG], and this thesis provides an exposition of the ideas and techniques in an accessible context. Acknowledgements Chapter 1. Introduction 1.1. An overview of the chapters 1.2. Notations Chapter 2. Motivation and algebraic techniques 2.1. Classical BV formalism: the derived critical locus 2.2. Quantum BV formalism: the twisted de Rham complex 2.3. Wick's lemma and Feynman diagrams, homologically 2.4. A compendium of essential definitions and constructions 2.5. The homological perturbation lemma in the BV formalism 2.6. Global observables and formal Hodge theory Chapter 3. BV formalism as a determinant functor 3.1. Cotangent quantization of k-vector spaces 3.2. Recollections 3.3. Properties of cotangent quantization over any commutative dg algebra 3.4. Invertibility survives over artinian dg algebras Chapter 4. Factorization algebras 4.1. Definitions 4.2. Associative algebras as factorization algebras on R 4.3. Associative algebras and the bar complex 4.4. The category of factorization algebras 4.5. General construction methods for factorization algebras 4.6. A novel construction of the universal enveloping algebra 4.7. Extension from a factorizing basis 4.8. Pushforward and Pullback Chapter 5. Free fields and their observables 5.1. Introduction 5.2. Elliptic complexes and free BV theories 5.3. Observables as a factorization algebra 5.4. BV quantization as a Heisenberg Lie algebra construction 5.5. BV quantization as a determinant functor 5.6. Implications for interacting theories 5.7. Theories with a Poincaré lemma Chapter 6. Free holomorphic field theories and vertex algebras 6.1. The system 6.2. The quantum observables of the system 6.3. Recovering a vertex algebra 6.4. Vertex algebras from Lie algebras 6.5. Definitions and a conjecture Chapter 7. An index theorem 7.1. A motivating example 7.2. A precise statement of the theorem 7.3. Setting up the problem 7.4. Background about BV theories and renormalization 7.5. The proof 7.6. Global statements Bibliography