Algebras over not too little discs

Abstract

By the introduction of locally constant prefactorization algebras at a fixed scale, we show a mathematical incarnation of the fact that observables at a given scale of a topological field theory propagate to every scale over euclidean spaces. The key is that these prefactorization algebras over ${\mathbb{R}}^n$ are equivalent to algebras over the little $n$-disc operad. For topological field theories with defects, we get analogous results by replacing ${\mathbb{R}}^n$ with the spaces modelling corners ${\mathbb{R}}^p\times {\mathbb{R}}_{\ge 0}^q$. As a toy example in $1d$, we quantize, once more, constant Poisson structures.

Comments are welcome! v3/v4: Accepted version in Communications in Mathematical Physics. v2: New proof strategy for the main theorem that avoids the use of an incorrect description of the hammock localization in the literature ([21] in v1). A version with closed discs and cubes has been added