Fixed Points of Involutions of G-Higgs Bundle Moduli Spaces over a Compact Riemann Surface with Classical Complex Structure Group

Abstract

Let X be a compact and connected Riemann surface of genus g ≥ 2. In this paper, moduli spaces of Higgs bundles over X with structure group SL(n, ℂ), for n ≥ 3, and Spin(2n, ℂ), for n ≥ 4, are considered. In the case of structure group SL(n, ℂ), two involutions of the Higgs bundle moduli space are defined and an alternative proof is given to show, using specific properties of the structure group, that the stable fixed points can be described as certain Higgs pairs with structure group SO(n, ℂ) or Sp(n, ℂ). Moreover, the notions of stability, semistability, and polystability of the obtained Higgs pairs are established. Also, two involutions of the moduli space of Higgs bundles with structure group Spin(2n, ℂ) are defined for n ≥ 4 analogously and it is shown, also using specific properties of the group, that their stable fixed points can be described as certain Higgs pairs whose structure group is of the form Spin(2r + 1, ℂ) × Spin(2n − 2r − 1, ℂ) for some 0 ≤ r ≤ n − 1.

Keywords

Higgs bundle / Higgs pair / spin / automorphism / involution / fixed point