In this chapter we give a formal definition of the notion of prefactorization algebra. With the definition in hand, we proceed to examine several examples that arise naturally in mathematics. In particular, we explain how associative algebras can be viewed as prefactorization algebras on the real line, and when the converse holds.

We also explain how to construct a prefactorization algebra from a sheaf of Lie algebras on a manifold M. This construction is called the factorization envelope, and it is related to the universal enveloping algebra of a Lie algebra as well as to Beilinson–Drinfeld's notion of a chiral envelope. Although the factorization envelope construction is very simple, it plays an important role in field theory. For example, the factorization algebra for any free theories is a factorization envelope, as is the factorization algebra corresponding to the Kac– Moody vertex algebra. More generally, factorization envelopes play an important role in our formulation of Noether's theorem for quantum field theories.

Finally, when the manifold M is equipped with an action of a group G, we describe what a G-equivariant prefactorization algebra is. We will use this notion later in studying translation-invariant field theories (see Section 4.8 in Chapter 4) and holomorphically translation-invariant field theories (see Chapter 5).

Prefactorization Algebras

In this section we give a formal definition of the notion of a prefactorization algebra, starting concretely and then generalizing. In the first subsection, using plain language, we describe a prefactorization algebra taking values in vector spaces. Readers are free to generalize by replacing “vector space” and “linear map” with “object of a symmetric monoidal category C” and “morphism in C.” (Our favorite target category is cochain complexes.) The next subsections give a concise definition using the language of multicategories (also known as colored operads) and allow an arbitrary multicategory as the target. In the final subsections, we describe the category (and multicategory) of such prefactorization algebras.

The Definition in Explicit Terms

Let M be a topological space. A prefactorization algebra F onM, taking values in vector spaces, is a rule that assigns a vector space F(U) to each open set along with the following maps and compatibilities.