Moduli spaces of vector bundles over a Klein surface

TL;DR

This paper explores the relationship between dianalytic vector bundles on Klein surfaces and holomorphic vector bundles on their complex covers, constructing special submanifolds in moduli spaces that reveal new geometric insights.

Contribution

It introduces a method to relate vector bundles over Klein surfaces to those over Riemann surfaces via moduli space submanifolds, extending previous work to surfaces with boundary and non-orientability.

Findings

Construction of totally real, totally geodesic, Lagrangian submanifolds in moduli spaces.

Establishment of a correspondence between dianalytic and holomorphic vector bundles.

Extension of known results to Klein surfaces with boundary and non-orientable structures.

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) .