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# Communications in Number Theory and Physics

## Volume 17 (2023)

### Number 3

### Resurgence, Stokes constants, and arithmetic functions in topological string theory

Pages: 709 – 820

DOI: https://dx.doi.org/10.4310/CNTP.2023.v17.n3.a4

#### Author

#### Abstract

The quantization of the mirror curve to a toric Calabi–Yau threefold gives rise to quantum-mechanical operators, whose fermionic spectral traces produce factorially divergent power series in the Planck constant. These asymptotic expansions can be promoted to resurgent trans-series. They show infinite towers of periodic singularities in their Borel plane and infinitely many rational Stokes constants, which are encoded in generating functions expressed in closed form in terms of $q$-series. We provide an exact solution to the resurgent structure of the first fermionic spectral trace of the local $\mathbb{P}^2$ geometry in the semiclassical limit of the spectral theory, corresponding to the strongly-coupled regime of topological string theory on the same background in the conjectural TS/ST correspondence. Our approach straightforwardly applies to the dual weakly-coupled limit of the topological string. We present and prove closed formulae for the Stokes constants as explicit arithmetic functions and for the perturbative coefficients as special values of known $L$-functions, while the duality between the two scaling regimes of strong and weak string coupling constant appears in number-theoretic form. A preliminary numerical investigation of the local $\mathbb{F}_0$ geometry unveils a more complicated resurgent structure with logarithmic sub-leading asymptotics. Finally, we obtain a new analytic prediction on the asymptotic behavior of the fermionic spectral traces in an appropriate WKB double-scaling regime, which is captured by the refined topological string in the Nekrasov–Shatashvili limit.

#### Keywords

Calabi–Yau threefolds, TS/ST correspondence, strong-weak duality, resurgence, non-perturbative effects, peacock patterns, enumerative invariants, conifold volume conjecture, divisor sum functions, Dirichlet $L$-functions, Riemann zeta function, WKB double-scaling regime

#### 2010 Mathematics Subject Classification

Primary 11M06, 14J32, 30E15, 32Q25, 58K65. Secondary 11A25, 30B50, 40G10, 81Q10, 81T30.

This work has been supported by the ERC-SyG project “Recursive and Exact New Quantum Theory” (ReNewQuantum), which received funding from the European Research Council (ERC) within the European Union’s Horizon 2020 research and innovation program under Grant No. 810573, and by the Swiss National Centre of Competence in Research SwissMAP (NCCR 51NF40-141869 The Mathematics of Physics).

Received 13 February 2023

Accepted 5 July 2023

Published 7 November 2023