A twisted nonabelian Hodge correspondence

TL;DR

This paper extends the nonabelian Hodge correspondence to twisted torsors on complex projective varieties, using homotopy-theoretic methods to relate twisted vector bundles with twisted Higgs data.

Contribution

It introduces a formal, homotopy-theoretic approach to establish a twisted nonabelian Hodge correspondence, expanding the classical theory to twisted objects.

Findings

Extended nonabelian Hodge theorem to twisted torsors

Used principal ∞-bundles to reduce to classical Hodge theory

Provided a formal proof avoiding complex analysis

Abstract

We prove an extension of the nonabelian Hodge theorem in which the underlying objects are twisted torsors over a smooth complex projective variety. In the prototypical case of $GL_n$-torsors, one side of this correspondence consists of vector bundles equipped with an action of a sheaf of twisted differential operators in the sense of Beilinson and Bernstein; on the other side, we endow them with appropriately defined twisted Higgs data. The proof we present here is formal, in the sense that we do not delve into the analysis involved in the classical nonabelian Hodge correspondence. Instead, we use homotopy-theoretic methods ---chief among them the theory of principal $\infty$-bundles--- to reduce our statement to classical (untwisted) Hodge theory.