Costello K., Gwilliam O. Factorization Algebras in Quantum Field Theory. Volume 1

Cambridge University Press, UK, 2016. — 324 p. — (New Mathematical Monographs 31-1) — ISBN10: 1107163102.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 This work is split into two volumes. Volume 1 develops the theory of factorization 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 between the concept of perturbative quantum field theory as developed in Costello (2011b) and the theory of factorization algebras.
Factorization algebras are local-to-global objects that play a role in classical and quantum field theory which is similar to the role of sheaves in geometry: they conveniently organize complicated information. Their local structure encompasses examples like associative and vertex algebras; in these examples, their global structure encompasses Hochschild homology and conformal blocks. In this first volume, the authors develop the theory of factorization algebras in depth, but with a focus upon examples exhibiting their use in field theory, such as the recovery of a vertex algebra from a chiral conformal field theory and a quantum group from Abelian Chern-Simons theory. Expositions of the relevant background in homological algebra, sheaves and functional analysis are also included, thus making this book ideal for researchers and graduates working at the interface between mathematics and physics.Prefactorization algebras
From Gaussian measures to factorization algebras
Prefactorization algebras and basic examples
First examples of field theories and their observables
Free field theories
Holomorphic field theories and vertex algebras
Factorization algebras
Factorization algebras: definitions and constructions
Formal aspects of factorization algebras
Factorization algebras: examples
Appendixes
Background
Functional analysis
Homological algebra in differentiable vector spaces
The Atiyah-Bott Lemma