Algebras over not too little discs

TL;DR

This paper establishes a mathematical framework connecting local observables in topological field theories to algebras over little disc operads, extending to theories with defects and including quantization of Poisson structures.

Contribution

It introduces locally constant prefactorization algebras at fixed scales and proves their equivalence to little n-disc operad algebras, extending to defect models and quantization.

Findings

Prefactorization algebras over al R^n are equivalent to little n-disc operad algebras.

Propagation of observables across scales is formalized mathematically.

Quantization of constant Poisson structures demonstrated in 1D.

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 $\mathbb{R}^p\times\mathbb{R}^{q}_{\geq 0}$. As a toy example in $1d$, we quantize, once more, constant Poisson structures.