Holomorphic Higgs bundles over the Teichm\"uller space

TL;DR

This paper investigates which surface group representations admit Higgs data that vary holomorphically with the complex structure, revealing that for SL(2,C) this implies unitarity, but for higher ranks non-unitary examples exist.

Contribution

It characterizes the conditions under which Higgs data depend holomorphically on the Riemann surface, showing a distinction between rank 2 and higher ranks.

Findings

Holomorphic Higgs data imply unitarity for SL(2,C) representations.

Existence of non-unitary, irreducible representations with holomorphic Higgs data for large n.

Holomorphic dependence of Higgs data does not always correspond to unitarity in higher ranks.

Abstract

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 $\Sigma\,=\, (X,\, J)$ via non-abelian Hodge correspondence. For representations $\rho$ into $\mathrm{SL}(2,\mathbb C)$ 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 $\mathrm{SL}(n,\mathbb C)$ admitting Higgs data that are holomorphic in $\Sigma$, for $n$ large enough.