Moduli spaces of vector bundles over a Klein surface

Abstract

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the original surface S. In this paper, we compare dianalytic vector bundles over S and holomorphic vector bundles over M, devoting special attention to the implications that this has for moduli varieties of semistable vector bundles over M. We construct, starting from S, totally real, totally geodesic, Lagrangian submanifolds of moduli varieties of semistable vector bundles of fixed rank and degree over M. This relates the present work to the constructions of Ho and Liu over non-orientable compact surfaces with empty boundary (arXiv:math/0605587) .

19 pages, 1 figure

References in corpus (3)

• The moduli space of flat SU(2)-bundles over a nonorientable surface
• The Yang-Mills stratification for surfaces revisited
• On real moduli spaces over M-curves

Cited by in corpus (5)

• Involutions of the moduli spaces of $G$-Higgs bundles over elliptic curves
• Homotopy Decompositions of Gauge Groups over Real Surfaces
• Parabolic vector bundles on Klein surfaces
• Principal bundles on 2-dimensional CW-complexes with disconnected structure group
• A Torelli theorem for moduli spaces of principal bundles on curves defined over $\mathbb{R}$