Factorization algebras and free field theories

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.