Open Gromov-Witten Invariants from the Augmentation Polynomial

2016/08/09 by Mahowald, Matthew
#Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th)

paper · doi:10.48550/arxiv.1608.02978

Abstract

A conjecture of Aganagic and Vafa relates the open Gromov-Witten theory of X=O_ℙ¹(-1,-1) to the augmentation polynomial of Legendrian contact homology. We describe how to use this conjecture to compute genus zero, one boundary component open Gromov-Witten invariants for Lagrangian submanifolds L_K⊂ X obtained from the conormal bundles of knots K⊂ S³. This computation is then performed for two non-toric examples (the figure-eight and three-twist knots). For (r,s) torus knots, the open Gromov-Witten invariants can also be computed using Atiyah-Bott localization. Using this result for the unknot and the (3,2) torus knot, we show that the augmentation polynomial can be derived from these open Gromov-Witten invariants.

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