The Atiyah-Bott formula and connectivity in chiral Koszul duality

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.

References in corpus (3)

• Higher Topos Theory
• Rational Homotopy Theory
• Free factorization algebras and homology of configuration spaces in algebraic geometry

Cited by in corpus (1)

• Koszul duality in exact categories