The orientability problem in open Gromov-Witten theory

Abstract

We give an explicit formula for the holonomy of the orientation bundle of a family of real Cauchy-Riemann operators. A special case of this formula resolves the orientability question for spaces of maps from Riemann surfaces with Lagrangian boundary condition. As a corollary, we show that the local system of orientations on the moduli space of J-holomorphic maps from a bordered Riemann surface to a symplectic manifold is isomorphic to the pull-back of a local system defined on the product of the Lagrangian and its free loop space. As another corollary, we show that certain natural bundles over these moduli spaces have the same local systems of orientations as the moduli spaces themselves (this is a prerequisite for integrating the Euler classes of these bundles). We will apply these conclusions in future papers to construct and compute open Gromov-Witten invariants in a number of settings.

23 pages, 1 figure

References in corpus (4)

• Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions
• Disk enumeration on the quintic 3-fold
• Orientations for pseudoholomorphic quilts
• Evidence for Tadpole Cancellation in the Topological String

Cited by in corpus (10)

• Counting genus zero real curves in symplectic manifolds
• The Lagrangian Floer-quantum-PSS package and canonical orientations in Floer theory
• Enumeration of real curves in CP^{2n-1} and a WDVV relation for real Gromov-Witten invariants
• Orientability in real Gromov-Witten theory
• Solomon-Tukachinsky's vs. Welschinger's Open Gromov-Witten Invariants of Symplectic Sixfolds
• Localization computation of one-point disk invariants of projective Calabi-Yau complete intersections
• Non-orientable surfaces and electric-magnetic duality
• Spin/Pin-Structures and Real Enumerative Geometry
• A Geometric Depiction of Solomon-Tukachinsky's Construction of Open GW-Invariants
• Degree One Contributions and Open Gromov-Witten Invariants