A diffeological perspective on non-Abelian Hodge theory

Abstract

We construct diffeological moduli stacks $\mathscr{M}_{Dol}(X)$ and $\mathscr{M}_{dR}(X)$ parametrizing smooth families of Higgs bundles and those of flat bundles, respectively, on a compact Kähler manifold $X$. We then establish an equivalence of stacks between diffeological substacks $\mathscr{M}^\mathscr{H}_{Dol} \subset \mathscr{M}_{Dol}(X)$ and $\mathscr{M}^\mathscr{H}_{dR}(X) \subset \mathscr{M}_{dR}(X)$, whose fibres over the point are the categories of semistable Higgs bundles, with the usual condition on Chern classes, and of flat bundles on $X$, respectively. $\mathscr{M}^\mathscr{H}_{Dol}(X)$ contains families of semistable Higgs bundles to points of which the classical correspondence of coarse moduli spaces does not extend continuously. This shows that the equivalence we provide is, in a sense, a common extension, in the context of diffeological moduli stacks, of both the homeomorphism of coarse moduli spaces, and of the equivalence of categories between semistable Higgs bundles and arbitrary flat bundles.