Categorical proof of Holomorphic Atiyah-Bott formula

TL;DR

This paper develops a categorical framework to prove the Atiyah-Bott formula using advanced concepts in higher category theory and derived algebraic geometry, providing a new proof of a classical fixed point theorem.

Contribution

It introduces a functorial trace construction in symmetric monoidal (,2)-categories and applies it to derive a categorical proof of the Atiyah-Bott formula.

Findings

Categorical trace functor is constructed for (,2)-categories.

The proof connects higher category theory with classical fixed point formulas.

Provides a new perspective on the Atiyah-Bott formula through derived algebraic geometry.

Abstract

Given a symmetric monoidal $(\infty,2)$-category $\mathscr E$ we promote the trace construction to a functor. We then apply this formalism to the case when $\mathscr{E}$ is the $(\infty,2)$-category of $k$-linear presentable categories which in combination of various calculations in the setting of derived algebraic geometry gives a categorical proof of the classical Atiyah-Bott formula (also known as the Holomorphic Lefschetz fixed point formula).