Factorization algebra

TL;DR

This survey introduces factorization algebras as local-to-global structures on manifolds, highlighting their role in mathematics and physics, especially in quantum field theory, and discusses their definitions, examples, and key results.

Contribution

It provides a comprehensive overview of factorization algebras, comparing them with other formalisms and explaining how they encode higher symmetries in field theories.

Findings

Defines and illustrates key examples of factorization algebras

Shows how they formalize observables in quantum field theory

Explains encoding of higher symmetries

Abstract

Factorization algebras are local-to-global objects living on manifolds, and they arise naturally in mathematics and physics. Their local structure encompasses examples like associative algebras and vertex algebras; in these examples, their global structure encompasses Hochschild homology and conformal blocks. In the setting of quantum field theory, factorization algebras articulate a minimal set of axioms satisfied by the observables of a theory, and they capture concepts like the operator product and correlation functions. In this survey article for the Encyclopedia of Mathematical Physics, 2nd Edition, we give the definitions and key examples, compare this approach with other approaches to mathematically formalizing field theory, describe key results, and explain how higher symmetries can be encoded in this framework.