The Yang-Mills equations over Klein surfaces
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.
Cite
@article{arxiv.1109.5164,
title = {The Yang-Mills equations over Klein surfaces},
author = {Chiu-Chu Melissa Liu and Florent Schaffhauser},
journal= {arXiv preprint arXiv:1109.5164},
year = {2015}
}
Comments
Final version, 72 pages; formulae in the quaternionic, n>0 case corrected; proof of Theorem 1.3 revised; references added