Open topological strings and integrable hierarchies: Remodeling the A-model

Abstract

We set up, purely in A-model terms, a novel formalism for the global solution of the open and closed topological A-model on toric Calabi-Yau threefolds. The starting point is to build on recent progress in the mathematical theory of open Gromov-Witten invariants of orbifolds; we interpret the localization formulae as relating D-brane amplitudes to closed string amplitudes perturbed with twisted masses through an analogue of the "loop insertion operator" of matrix models. We first generalize this form of open/closed string duality to general toric backgrounds in all chambers of the stringy Kaehler moduli space; secondly, we display a neat connection of the (gauged) closed string side to tau functions of 1+1 Hamiltonian integrable hierarchies, and exploit it to provide an effective computation of open string amplitudes. In doing so, we also provide a systematic treatment of the change of flat open moduli induced by a phase transition in the closed moduli space. We test our proposal in detail by providing an extensive number of checks. We also use our formalism to give a localization-based derivation of the Hori-Vafa spectral curves as coming from a resummation of A-model disc instantons.

49 pages, 10 figures; v2: typos fixed, updated references, version to appear on Comm. Math. Phys

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• Open Gromov-Witten Invariants of Toric Calabi-Yau 3-Folds
• Crepant resolutions and open strings
• Chern-Simons Theory in SIM(1) Superspace
• Supersymmetric Chern-Simons Theory in Presence of a Boundary in the Light-Like Direction
• On integrable conservation laws
• Flat Connections in Open String Mirror Symmetry
• Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs
• The Hodge-FVH Correspondence
• Symplectic cuts and open/closed strings I
• Symplectic cuts and open/closed strings II
• A Gamma Class Formula for Open Gromov-Witten Calculations
• Enumerative geometry of surfaces and topological strings