Symmetric theta divisors of Klein surfaces

TL;DR

This paper computes determinant index bundles and their topological invariants for families of Real Dirac operators on Klein surfaces, linking algebraic geometry with gauge theory and real algebraic structures.

Contribution

It provides explicit calculations of determinant index bundles and their Stiefel-Whitney classes on Klein surfaces, connecting theta divisors with Real gauge theory.

Findings

Computed first Stiefel-Whitney classes of fixed point bundles.

Described determinant index bundles as theta line bundles.

Linked topological invariants to orientability of moduli spaces.

Abstract

This is a slightly expanded version of the talk given by Ch.O. at the conference "Instantons in complex geometry", at the Steklov Institute in Moscow. The purpose of this talk was to explain the algebraic results of our paper "Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces". In this paper we 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.