  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.
