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Factorization algebras in quite a lot of generality
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.
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