We prove an extension of the nonabelian Hodge theorem [Sim92] 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 Beĭlinson and Bernstein [BB93]; 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 [NSS12a]—to reduce our statement to classical, untwisted Hodge theory [Sim02].
"A twisted nonabelian Hodge correspondence." J. Differential Geom. 107 (3) 455 - 518, November 2017. https://doi.org/10.4310/jdg/1508551223