In the last decade there has been a flurry of interest in arithmetic quantum field theories​. Since the 1960s, researchers have identified an analogy between various objects of arithmetic geometry and low-dimensional manifolds. For example, Spec of a number ring “looks like” an open 3-manifold, and primes therein “are” embedded knots. This story has become known as arithmetic topology​. The idea of arithmetic QFT is to enrich that analogy by importing tools from physics, just as with low-dimensional topology. One even dreams of using these tools to study number-theoretic questions (the behavior of L-functions, Langlands dualities, etc.).

But the objects of arithmetic geometry are not​ manifolds. The tools of topology and differential geometry do not work directly in arithmetic. So it’s unclear how to translate physical concepts to arithmetic settings.

To this end, we introduce a minimalist framework for factorization algebras, where the role of the spacetime manifold can be played by a geometric object of a very general sort. In retrospect, the main idea amounts to a categorification of Borcherds’ approach to vertex algebras.