The Yang–Mills equations over Klein surfaces. (1st March 2013)

• Record Type: Journal Article
• Authors: Liu, Chiu-Chu Melissa; Schaffhauser, Florent
• Is Part Of: Journal of topology. Volume 6:Part 3(2013)
• Page Start: 569
• Page End: 643
• Publication Date: 2013-03-01
• Subjects: Topology -- Periodicals; 514.05
• DOI: 10.1112/jtopol/jtt001
• Languages: English
• ISSNs: 1753-8416

Abstract

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 Poincaré polynomials of these moduli spaces in the coprime case. We also compute the mod 2 Poincaré 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.