Real Higgs pairs and non-abelian Hodge correspondence on a Klein surface
Abstract
. We introduce real structures on L-twisted Higgs pairs over a
compact connected Riemann surface X equipped with an antiholomorphic involution, where L is a holomorphic line bundle on
X with a real structure, and prove a Hitchin–Kobayashi correspondence for the L-twisted Higgs pairs. Real GR-Higgs bundles, where
GR is a real form of a connected semisimple complex affine algebraic group G, constitute a particular class of examples of these
pairs. In this case, the real structure of the moduli space of G-Higgs
pairs is defined using a conjugation of G that commutes with the
one defining the real form GR and a compact conjugation of G preserving GR. We establish a homeomorphism between the moduli
space of real GR-Higgs bundles and the moduli space of representations of the fundamental group of X in GR that can be extended to
a representation of the orbifold fundamental group of X into a certain enlargement of GR with quotient Z/2Z. Finally, we show how
real GR-Higgs bundles appear naturally as fixed points of certain
anti-holomorphic involutions of the moduli space of GR-Higgs bundles, constructed using the real structures on G and X. A similar
result is proved for the representations of the orbifold fundamental
group.
Real Higgs pairs and non-abelian Hodge correspondence on a Klein surface
Tipo de Actividad
Artículos en revistasISSN
1019-8385Palabras Clave
.Structures Higgs pairs Real GR-Higgs bundles Moduli space Orbifold fundamental group