Matthew Mahowald | Searcharxiv

arXiv2016

Open Gromov-Witten Invariants from the Augmentation Polynomial

A conjecture of Aganagic and Vafa relates the open Gromov-Witten theory of $X=\mathcal{O}_{\mathbb{P}^{1}}(-1,-1)$ 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 $L_{K}\subset X$ obtained from the conormal bundles of knots $K\subset S^{3}$. 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.

hep-th↗

arXiv2016

A Gamma Class Formula for Open Gromov-Witten Calculations

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.

hep-th↗