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Resurgence and Riemann-Hilbert problems for elliptic Calabi-Yau threefolds

Tom Bridgeland, Iván Tulli

Abstract

Let $X$ be a Calabi-Yau threefold with an elliptic fibration. We investigate the non-linear Riemann-Hilbert problems associated to the Donaldson-Thomas theory of fibre classes, and relate them to the Borel sum of the $A$-model topological string free energy for such classes.

Resurgence and Riemann-Hilbert problems for elliptic Calabi-Yau threefolds

Abstract

Let be a Calabi-Yau threefold with an elliptic fibration. We investigate the non-linear Riemann-Hilbert problems associated to the Donaldson-Thomas theory of fibre classes, and relate them to the Borel sum of the -model topological string free energy for such classes.
Paper Structure (27 sections, 20 theorems, 150 equations, 1 figure)

This paper contains 27 sections, 20 theorems, 150 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Borel sum of the free energy
  3. DT invariants and the RH problem
  4. Further remarks
  5. Structure of the paper
  6. Acknowledgements
  7. Summary of the relevant complex analysis
  8. Elliptic functions
  9. Borel transforms
  10. Irrationality measure
  11. Borel sums and integrals along detour paths
  12. Free energy, DT invariants and the RH problem
  13. Free energy and its Borel sums
  14. Stability conditions and DT invariants
  15. Riemann-Hilbert problem
  16. ...and 12 more sections

Key Result

Theorem 1.1

Fix $\omega_1,\omega_2\in \mathbb{C}^{*}$ with $\operatorname{Im}(\omega_2/\omega_1)>0$.

Figures (1)

  • Figure 1: The vectors denote the generators $\omega_1$ and $\omega_2$ of the lattice $\Lambda(\omega_1,\omega_2)$. The discs are centered at the points in $\Lambda(\omega_1,\omega_2)^*$ and have radius $\delta>0$ small enough that they do not intersect. The bold path $r(\delta)$ is determined by the direction of the non-Stokes ray $r$, and takes a detour along the boundary of any disc intersected by $r$. These detours traverse arcs of the boundary of angle $<\pi$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Proposition 2
  • Proposition 3
  • Theorem 3.1
  • Proposition 4
  • ...and 22 more