Open Virtual Structure Constants and Mirror Computation of Open Gromov-Witten Invariants of Projective Hypersurfaces

TL;DR

This paper extends the computation of open Gromov-Witten invariants to a broader class of hypersurfaces using open virtual structure constants, proposing a generalized mirror transformation and discussing modifications for higher dimensions.

Contribution

It introduces the concept of open virtual structure constants for Fano and Calabi-Yau hypersurfaces and proposes a generalized mirror transformation, expanding the computational framework for open Gromov-Witten invariants.

Findings

Generalization of Walcher's invariants to new hypersurfaces

Proposed and partially proved a generalized mirror transformation

Showed integrality of disk invariants after re-summation

Abstract

In this paper, we generalize Walcher's computation of the open Gromov-Witten invariants of the quintic hypersurface to Fano and Calabi-Yau projective hypersurfaces. Our main tool is the open virtual structure constants. We also propose the generalized mirror transformation for the open Gromov-Witten invariants, some parts of which are proven explicitly. We also discuss possible modification of the multiple covering formula for the case of higher dimensional Calabi-Yau manifolds. The generalized disk invariants for some Calabi-Yau and Fano manifolds are shown and they are certainly integers after re-summation by the modified multiple covering formula. This paper also contains the direct integration method of the period integrals for higher dimensional Calabi-Yau hypersurfaces in the appendix.