Integral transforms in non-abelian Hodge theory

In order to construct integral transforms and Fourier-Mukai functors for variations of twistor structures one must have very strong functoriality properties of the non-abelian Hodge correspondence. I will discuss the problem of compatibility of non-abelian Hodge theory with Grothendieck's six operations and will report on a recent joint work with R. Donagi and C. Simpson. Our main result is an explicit algebraic formula which, in the tamely ramified case, captures the interaction of the Hodge correspondence with pushforwards.