The moduli space of maps with crosscaps: the relative signs of the natural automorphisms

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 previously introduced moduli spaces of maps from surfaces with crosscaps, developed the relevant Fredholm theory, and resolved the orientability problem in this setting. In this paper, we determine the relative signs of the automorphisms of these moduli spaces induced by interchanges of boundary components of the domain and by the anti-symplectic involution on the target manifold, without any global assumptions on the latter. As immediate applications, we describe sufficient conditions for the moduli spaces of real genus 1 maps and for real maps with separating fixed locus to be orientable; we treat the general genus 2+ case in a separate paper. Our sign computations also lead to an extension of recent Floer-theoretic applications of anti-symplectic involutions and to a related reformulation of these results in a more natural way.