Holomorphic Higgs bundles over the Teichmüller space

We study which representations $\rho$ of the fundamental group of a compact oriented surface $X$ admit Higgs data that depend holomorphically on the Riemann surface $\mathrm{\Sigma }=(X,J)$ via non-abelian Hodge correspondence. For representations $\rho$ into $\mathrm{S}\mathrm{L}(2,\mathbb{C})$ we show that holomorphic dependency is equivalent to $\rho$ being unitary. For higher ranks this equivalence fails – we show the existence of non-unitary and irreducible representations of the fundamental group into $\mathrm{S}\mathrm{L}(n,\mathbb{C})$ admitting Higgs data that are holomorphic in $\mathrm{\Sigma }$, for $n$ large enough.