Disk enumeration on the quintic 3-fold

2006/10/31 by R. Pandharipande, J. Solomon, J. Walcher · 4 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Boundary (topology) #Cover (algebra) #Enumeration #Extension (predicate logic) #Function (biology) #Generating function #Geometric and Algebraic Topology #Intersection (aeronautics) #Quintic function #hep-th #math.AG #math.SG #msc:14J32 #msc:14N35 #msc:53D45

paper · pdf · doi:10.1090/s0894-0347-08-00597-3

published as J. Amer. Math. Soc. 21 (2008), 1169-1209. · 52 pages, 5 figures. Added background on open mirror symmetry in Section 0.4, and added details especially in Lemma 7 and Lemma 18

arxiv created 2007/05/28 · openalex publication_date 2008/02/12 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06

Abstract

Holomorphic disk invariants with boundary in the real Lagrangian of a quintic 3-fold are calculated by localization and proven mirror transforms. A careful discussion of the underlying virtual intersection theory is included. The generating function for the disk invariants is shown to satisfy an extension of the Picard-Fuchs differential equations associated to the mirror quintic. The Ooguri-Vafa multiple cover formula is used to define virtually enumerative disk invariants. The results may also be viewed as providing a virtual enumeration of real rational curves on the quintic.

Citations

• A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory
• A general Fredholm theory I: a splicing-based differential geometry
• Mirror principle I

Cited by

• Open Gromov-Witten invariants in genus zero and Lagrangian cobordisms
• Symmetries of Lagrangian fibrations
• Point-like bounding chains in open Gromov–Witten theory
• Real loci in (log) Calabi–Yau manifolds via Kato–Nakayama spaces of toric degenerations