A Gamma Class Formula for Open Gromov-Witten Calculations

TL;DR

This paper introduces a gamma class formula for open Gromov-Witten invariants in toric Calabi-Yau threefolds, linking disk factors to gamma classes and combinatorial data, with applications to the quintic 3-fold and Lagrangian cycles from torus knots.

Contribution

It provides a new gamma class formula for open Gromov-Witten invariants under specific symmetry conditions, extending previous localization methods.

Findings

The formula encodes expected invariants verified through examples.

Application to disk enumeration on the quintic 3-fold demonstrated.

Unexpected applicability to Lagrangian cycles from torus knots.

Abstract

For toric Calabi-Yau threefolds, open Gromov-Witten invariants associated to Riemann surfaces with one boundary component can be written as the product of a disk factor and a closed invariant. Using the Brini-Cavalieri-Ross formalism, these disk factors can often be expressed in terms of gamma classes. When the Lagrangian boundary cycle is preserved by the torus action and can be locally described as the fixed locus of an anti-holomorphic involution, we prove a formula that expresses the disk factor in terms of a gamma class and combinatorial data about the image of the Lagrangian cycle in the moment polytope. We verify that this formula encodes the expected invariants obtained from localization by comparing with several examples. We then examine a novel application of this formula to disk enumeration on the quintic 3-fold. Finally, motivated by large $N$ duality, we show that this formula also unexpectedly applies to Lagrangian cycles on $\mathcal{O}_{\mathbb{P}^1}(-1,-1)$ constructed from torus knots.