Additivity of Constructible Factorization Algebras

Factorization algebras, introduced by Costello and Gwilliam, encode the structure of observables in perturbative quantum field theory, capturing concepts like the operator product and correlation functions. Beyond this, their local structures encompass notions such as associative and $A_{\mathrm{\infty }}$-algebras, vertex algebras, bimodules, and $E_n$-algebras. Examples of the latter are (possibly braided) tensor categories and n-fold loop spaces.
In this talk, I will provide an overview of the surprisingly recently well-studied theory of locally constant factorization algebras. Building on this foundation, I will introduce a broader class of constructible factorization algebras, defined on manifolds with corners or, more generally, stratified manifolds. Examples are given by factorization homology. I will report on recent progress regarding the additivity of factorization algebras, including the additivity of the generalized Swiss cheese operad- a result inspired by Dunn’s theorem for the little disks operad. This is based on joint work with Victor Carmona. In our approach, we use the reformulation of factorization algebras within the framework of $\mathrm{\infty }$-operads, which offers powerful tools to study their behavior over product spaces.