2016/01/02 by Matthew Mahowald, Mahowald, Matthew
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #FOS: Physical sciences #Geometric and Algebraic Topology #High Energy Physics - Theory (hep-th) #hep-th #math.AG
paper · pdf · doi:10.48550/arxiv.1601.00203
27 pages, 6 figures. v2: Added quintic 3-fold example, minor corrections
openalex publication_date 2016/01/02 · arxiv created 2016/01/26 · arxiv updated 2016/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Oℙ1(-1,-1) constructed from torus knots.