The moduli space of maps with crosscaps: Fredholm theory and orientability

TL;DR

This paper develops a Fredholm theory for moduli spaces of maps from surfaces with crosscaps, addressing orientability issues crucial for real Gromov-Witten invariants, and provides explicit formulas for orientation bundle holonomy.

Contribution

It introduces the moduli space framework for maps with crosscaps, derives explicit holonomy formulas, and resolves orientability problems in genus 0, advancing the understanding of real Gromov-Witten invariants.

Findings

Explicit holonomy formulas for orientation bundles.

Proved orientability of real maps from spheres with involution.

Extended orientability results to genus 1 maps.

Abstract

Just as a symmetric surface with separating fixed locus halves into two oriented bordered surfaces, an arbitrary symmetric surface halves into two oriented symmetric half-surfaces, i.e. surfaces with crosscaps. Motivated in part by the string theory view of real Gromov-Witten invariants, we introduce moduli spaces of maps from surfaces with crosscaps, develop the relevant Fredholm theory, and resolve the orientability problem in this setting. In particular, we give an explicit formula for the holonomy of the orientation bundle of a family of real Cauchy-Riemann operators over Riemann surfaces with crosscaps. Special cases of our formulas are closely related to the orientability question for the space of real maps from symmetric Riemann surfaces to an almost complex manifold with an anti-complex involution and in fact resolve this question in genus 0. In particular, we show that the moduli space of real J-holomorphic maps from the sphere with a fixed-point free involution to a simply connected almost complex manifold with an even canonical class is orientable. In a sequel, we use the results of this paper to obtain a similar orientability statement for genus 1 real maps.