Classification of the automorphism and isometry groups of Higgs bundle moduli spaces

Abstract

Let Mn,d be the moduli space of semi-stable rank n, trace-free Higgs bundles with fixed determinant of degree d on a Riemann surface of genus at least 3. We determine the following automorphism groups of Mn,d: (i) the group of automorphisms as a complex analytic variety, (ii) the group of holomorphic symplectomorphisms, (iii) the group of Kähler isomorphisms, (iv) the group of hypercomplex automorphisms, (v) the group ofhyper-Kähler isomorphisms. When n and d are coprime we show that Mn,d admits an anti-holomorphic isomorphism if and only if the corresponding Riemann surface admits such a map. We then use this to determine the isometry group of Mn,d.