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Quantum Curve for strip geometries, Topological Recursion and open GW/DT invariants

Sibasish Banerjee, Alexander Hock

Abstract

Open topological string partition function gives rise to open Gromov-Witten invariants, open Donaldson-Thomas invariants and 3D-5D BPS indices. Utilizing the remodelling conjecture which connects topological recursion and topological string theory, in this paper we study open topological string theory for the subclass of toric Calabi-Yau threefold known as strip geometries. For this purpose, certain new developments in the theory of topological recursion are applied as its extension to Logarithmic Topological Recursion (Log-TR) and the universal $x$--$y$ duality. Through this we derive the open topological string partition function and also the associated quantum curve. We also explain how this is related to the open Donaldson-Thomas partition function associated with certain symmetric quivers, exponential networks and $q$-Barnes type integrals. In the process, we also connect how 3D-5D wall crossing affects these partition functions as one varies $x$, in examples.

Quantum Curve for strip geometries, Topological Recursion and open GW/DT invariants

Abstract

Open topological string partition function gives rise to open Gromov-Witten invariants, open Donaldson-Thomas invariants and 3D-5D BPS indices. Utilizing the remodelling conjecture which connects topological recursion and topological string theory, in this paper we study open topological string theory for the subclass of toric Calabi-Yau threefold known as strip geometries. For this purpose, certain new developments in the theory of topological recursion are applied as its extension to Logarithmic Topological Recursion (Log-TR) and the universal -- duality. Through this we derive the open topological string partition function and also the associated quantum curve. We also explain how this is related to the open Donaldson-Thomas partition function associated with certain symmetric quivers, exponential networks and -Barnes type integrals. In the process, we also connect how 3D-5D wall crossing affects these partition functions as one varies , in examples.

Paper Structure

This paper contains 14 sections, 75 equations.

Table of Contents

  1. Introduction and summary
  2. Quantum Curve from Topological Recursion for Strip Geometry
  3. Background on Topological Recursion
  4. The dual Wave Function and the dual Quantum Spectral Curve for Strip Geometries
  5. The Wave Function and the Quantum Spectral Curve for Strip Geometry
  6. Quantum Spectral Curve for Strip Geometry with base point $z_0=\infty$
  7. Quantum Spectral Curve for Strip Geometry with base point $z_0=1$
  8. Relations to open DT invariants and quiver generating series
  9. Connection to symmetric quivers
  10. Open BPS states and exponential networks
  11. $q$-Barnes type integral representation of wave functions
  12. Example : $\mathbb{C}^3$
  13. Framing $f=-2$
  14. Example : Resolved conifold

Theorems & Definitions (6)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 3.1
  • Remark 3.2