Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces

TL;DR

This paper computes the topological invariants of determinant line bundles associated with Real Dirac operators on Klein surfaces, linking them to theta line bundles and their orientability properties in Real gauge theory.

Contribution

It provides explicit calculations of Stiefel-Whitney classes of fixed point bundles on Klein surfaces, advancing understanding of orientability in Real moduli spaces.

Findings

Computed first Stiefel-Whitney classes of fixed point bundles.

Linked determinant index bundles to theta line bundles.

Analyzed implications for orientability in Real gauge theory.

Abstract

The purpose of this paper is to compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. On a Klein surface these determinant index bundles have a natural holomorphic description as theta line bundles. In particular we compute the first Stiefel-Whitney classes of the corresponding fixed point bundles on the real part of the Picard torus. The computation of these classes is important, because they control to a large extent the orientability of certain moduli spaces in Real gauge theory and Real algebraic geometry.