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Non-perturbative topological strings from resurgence

Murad Alim

Abstract

The partition function of topological string theory on any family of Calabi-Yau threefolds is defined perturbatively as an asymptotic series in the topological string coupling and encodes, in a holomorphic limit, higher genus Gromov-Witten as well as Gopakumar-Vafa invariants. We prove that the partition function of topological strings of any CY in this limit can be written as a product, where each factor is given by the partition function of the resolved conifold with shifted arguments, raised to the power of certain sheaf invariants. We use this result to put forward an expression for the non-perturbative topological string partition function in this limit, as a product over analytic functions in the topological string coupling which correspond to the Borel sums for the resolved conifold found previously. The non-perturbative corrections to the partition function are identified with Stokes jumps of a Borel summation. They depend only on genus zero GV invariants and can be expressed entirely in terms of a single function which is introduced as a deformation of the prepotential.

Non-perturbative topological strings from resurgence

Abstract

The partition function of topological string theory on any family of Calabi-Yau threefolds is defined perturbatively as an asymptotic series in the topological string coupling and encodes, in a holomorphic limit, higher genus Gromov-Witten as well as Gopakumar-Vafa invariants. We prove that the partition function of topological strings of any CY in this limit can be written as a product, where each factor is given by the partition function of the resolved conifold with shifted arguments, raised to the power of certain sheaf invariants. We use this result to put forward an expression for the non-perturbative topological string partition function in this limit, as a product over analytic functions in the topological string coupling which correspond to the Borel sums for the resolved conifold found previously. The non-perturbative corrections to the partition function are identified with Stokes jumps of a Borel summation. They depend only on genus zero GV invariants and can be expressed entirely in terms of a single function which is introduced as a deformation of the prepotential.

Paper Structure

This paper contains 23 sections, 7 theorems, 126 equations, 1 figure.

Table of Contents

  1. Introduction and summary
  2. Introduction
  3. Summary
  4. Topological strings and enumerative geometry
  5. The Gromov-Witten potential
  6. The Gopakumar-Vafa resummation
  7. Finiteness of GV invariants
  8. GW potential of the resolved conifold
  9. Building blocks of non-perturbative topological strings
  10. The resolved conifold
  11. Constant maps contributions
  12. Non-perturbative topological strings
  13. Sheaf invariants
  14. Free energy in terms of the resolved conifold
  15. Non-perturbative free energy
  16. ...and 8 more sections

Key Result

Proposition 3.1

Product form of the partition functions. For the partition functions we have the following product forms: where $Q=\exp(2\pi i t),\quad q=e^{\textnormal{i}\lambda}\,, \quad Q'=\exp(2\pi i t/\check{\lambda})\,, \quad q'=\exp(2\pi i/\check{\lambda})$ .

Figures (1)

  • Figure 1: Illustration of some of the singularity rays in the Borel plane $l_{d,k}=\mathbb{R}_{<0}\cdot 2\pi {\mathrm i} (d t+k)$ in the Borel plane, plotted for $t= \frac{1}{\pi}\left(\frac{1}{2}+ {\mathrm i}\right)$ and $(d,k)=(1,-10),\dots,(1,10)$ as well as $(d,\pm1)= (2,\pm1)\dots (10,\pm1)$

Theorems & Definitions (15)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • ...and 5 more