The Yang-Mills equations over Klein surfaces

TL;DR

This paper studies moduli spaces of real and quaternionic vector bundles over Klein surfaces, presenting their structure as Lagrangian quotients, extending Atiyah-Bott methods, and computing their mod 2 Poincare polynomials to identify new maximal real algebraic varieties.

Contribution

It generalizes Atiyah-Bott techniques to real and quaternionic settings and computes mod 2 Poincare polynomials for these moduli spaces and stacks.

Findings

Computed mod 2 Poincare polynomials of moduli spaces in the coprime case.

Identified connected components of real points in the moduli variety.

Provided new examples of maximal real algebraic varieties.

Abstract

Moduli spaces of semi-stable real and quaternionic vector bundles of a fixed topological type admit a presentation as Lagrangian quotients, and can be embedded into the symplectic quotient corresponding to the moduli variety of semi-stable holomorphic vector bundles of fixed rank and degree on a smooth complex projective curve. From the algebraic point of view, these Lagrangian quotients are connected sets of real points inside a complex moduli variety endowed with a real structure; when the rank and the degree are coprime, they are in fact the connected components of the fixed-point set of the real structure. This presentation as a quotient enables us to generalize the methods of Atiyah and Bott to a setting with involutions, and compute the mod 2 Poincare polynomials of these moduli spaces in the coprime case. We also compute the mod 2 Poincare series of moduli stacks of all real and quaternionic vector bundles of a fixed topological type. As an application of our computations, we give new examples of maximal real algebraic varieties.