Notes on factorization algebras, factorization homology and applications

TL;DR

This paper introduces factorization algebras and homology theory for manifolds, highlighting their applications in studying $E_n$-algebras, mapping spaces, and stratified spaces, with origins in quantum field theory.

Contribution

It provides an expanded introduction to factorization algebras and homology, emphasizing their role in manifold invariants and applications to $E_n$-algebras and mapping spaces.

Findings

Homology theory for manifolds yields invariants of manifolds and $E_n$-algebras.

Factorization algebras serve as sheaf-like structures in quantum field theory.

Examples include applications to stratified spaces and mapping spaces.

Abstract

These notes are an expanded version of two series of lectures given at the winter school in mathematical physics at les Houches and at the Vietnamese Institute for Mathematical Sciences. They are an introduction to factorization algebras, factorization homology and some of their applications, notably for studying $E_n$-algebras. We give an account of homology theory for manifolds (and spaces), which give invariant of manifolds but also invariant of $E_n$-algebras. We particularly emphasize the point of view of factorization algebras (a structure originating from quantum field theory) which plays, with respect to homology theory for manifolds, the role of sheaves with respect to singular cohomology. We mention some applications to the study of mapping spaces and study several examples, including some over stratified spaces.