The Atiyah-Bott formula and connectivity in chiral Koszul duality

TL;DR

This paper explores the connection between the Atiyah-Bott formula and chiral Koszul duality, demonstrating how certain dualities and equivalences simplify the proof of the Atiyah-Bott formula under specific conditions.

Contribution

It establishes a new equivalence of categories induced by Koszul duality in the context of sheaves on the Ran space, providing a simpler approach to a key step in the Atiyah-Bott formula proof.

Findings

Koszul duality induces an equivalence of categories under connectivity assumptions.

The equivalence interacts well with Verdier duality and factorization homology.

Provides a simpler alternative to a main step in the Atiyah-Bott formula proof.

Abstract

The $\otimes^\star$-monoidal structure on the category of sheaves on the $\mathrm{Ran}$ space is not pro-nilpotent in the sense of Francis-Gaitsgory. However, under some connectivity assumptions, we prove that Koszul duality induces an equivalence of categories and that this equivalence behaves nicely with respect to Verdier duality on the $\mathrm{Ran}$ space and integrating along the $\mathrm{Ran}$ space, i.e. taking factorization homology. Based on ideas sketched by Gaitsgory, we show that these results also offer a simpler alternative to one of the two main steps in the proof of the Atiyah-Bott formula given in Gaitsgory-Lurie and Gaitsgory.