Details

Autor(en) / Beteiligte

• García‐Prada, Oscar
• Ramanan, S.

Titel

Involutions and higher order automorphisms of Higgs bundle moduli spaces

Ist Teil von

• Proceedings of the London Mathematical Society, 2019-09, Vol.119 (3), p.681-732

Erscheinungsjahr

2019

Link zum Volltext

Quelle

Wiley Online Library

Beschreibungen/Notizen

• We consider the moduli space M(G) of G‐Higgs bundles over a compact Riemann surface X, where G is a complex semisimple Lie group. This is a hyperkähler manifold homeomorphic to the moduli space R(G) of representations of the fundamental group of X in G. In this paper, we study finite‐order automorphisms of M(G) obtained by combining the action of an element of order n in H1(X,Z)⋊Out(G), where Z is the centre of G and Out(G) is the group of outer automorphisms of G, with the multiplication of the Higgs field by an nth‐root of unity, and describe the subvarieties of fixed points. We give special attention to the case of involutions, defined by the action of an element of order 2 in H1(X,Z)⋊Out(G) combined with the multiplication of the Higgs field by ±1. In this situation, the subvarieties of fixed points are hyperkähler submanifolds of M(G) in the (+1)‐case, corresponding to the moduli space of representations of the fundamental group in certain reductive complex subgroups of G defined by holomorphic involutions of G; while in the (‐1)‐case, they are Lagrangian subvarieties corresponding to the moduli space of representations of the fundamental group of X in real forms of G and certain extensions of these. We illustrate the general theory with the description of involutions for G= SL (n,C) and involutions and order three automorphism defined by triality for G= Spin (8,C).