2013/08/06 by Penka Georgieva, Georgieva, Penka, Aleksey Zinger +1
Computer Science · Mathematics · #14N35 #53D45 #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric and Algebraic Topology #Symplectic Geometry (math.SG) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1308.1345
openalex publication_date 2013/08/06 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28
Just as a symmetric surface with separating fixed locus halves into two\noriented bordered surfaces, an arbitrary symmetric surface halves into two\noriented symmetric half-surfaces, i.e. surfaces with crosscaps. Motivated in\npart by the string theory view of real Gromov-Witten invariants, we previously\nintroduced moduli spaces of maps from surfaces with crosscaps, developed the\nrelevant Fredholm theory, and resolved the orientability problem in this\nsetting. In this paper, we determine the relative signs of the automorphisms of\nthese moduli spaces induced by interchanges of boundary components of the\ndomain and by the anti-symplectic involution on the target manifold, without\nany global assumptions on the latter. As immediate applications, we describe\nsufficient conditions for the moduli spaces of real genus 1 maps and for real\nmaps with separating fixed locus to be orientable; we treat the general genus\n2+ case in a separate paper. Our sign computations also lead to an extension of\nrecent Floer-theoretic applications of anti-symplectic involutions and to a\nrelated reformulation of these results in a more natural way.\n