2015/12/22 by Penka Georgieva, Georgieva, Penka, Aleksey Zinger +1
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1512.07220
The present note overviews our recent construction of real Gromov-Witten\ntheory in arbitrary genera for many real symplectic manifolds, including the\nodd-dimensional projective spaces and the renowned quintic threefold, its\nproperties, and its connections with real enumerative geometry. Our\nconstruction introduces the principle of orienting the determinant of a\ndifferential operator relative to a suitable base operator and a real setting\nanalogue of the (relative) spin structure of open Gromov-Witten theory.\nOrienting the relative determinant, which in the now-standard cases is\ncanonically equivalent to orienting the usual determinant, is naturally related\nto the topology of vector bundles in the relevant category. This principle and\nits applications allow us to endow the uncompactified moduli spaces of real\nmaps from symmetric surfaces of all topological types with natural orientations\nand to verify that they extend across the codimension-one boundaries of these\nspaces, thus implementing a far-reaching proposal from C.-C. Liu's thesis.\n