From quantum curves to topological string partition functions II

TL;DR

This work develops a non-perturbative, geometry-driven definition of topological string partition functions for local Calabi–Yau geometries arising in class S theories. By quantising the defining curves to obtain quantum curves and using isomonodromic tau-functions, the authors express partition functions as generalized theta-series tied to distinguished coordinate systems produced by exact WKB (Fenchel–Nielsen and Fock–Goncharov types). They show that normalization changes across monodromy loci encode a natural line bundle, unifying perturbative and non-perturbative data and linking to BPS spectra, spectral determinants, and topological recursion. The framework harmonises RH problems, free-fermion tau-functions, and abelianisation to produce a coherent, coordinate-dependent description of non-perturbative topological string partition functions, with broad connections to DT theory, quantum Teichmüller theory, and hypermultiplet moduli spaces.

Abstract

We propose a geometric characterisation of the topological string partition functions associated to the local Calabi-Yau (CY) manifolds used in the geometric engineering of $d=4$, $\mathcal{N}=2$ supersymmetric field theories of class $\mathcal{S}$. A quantisation of these CY manifolds defines differential operators called quantum curves. The partition functions are extracted from the isomonodromic tau-functions associated to the quantum curves by expansions of generalised theta series type. It turns out that the partition functions are in one-to-one correspondence with preferred coordinates on the moduli spaces of quantum curves defined using the Exact WKB method. The coordinates defined in this way jump across certain loci in the moduli space. The changes of normalisation of the tau-functions associated to these jumps define a natural line bundle playing a key role in the geometric characterisation of the topological string partition functions proposed here.

Figures (11)

• Figure 1: Triangulation of the annulus of the Painlevé III system.
• Figure 2: Fat graph on $C_{0,4}$ with six basic paths.
• Figure 3: Stokes graphs for the differential (\ref{['eq:SU2-quad-diff']}) with $\Lambda=1$ and $u=2$, varying $\arg(\hbar)$ until a non-degenerate ring domain appears.
• Figure 4: Top row: Stokes graph examples from AT. Bottom row: Corresponding Fenchel-Nielsen networks, in this case identical with Anti-Stokes graphs.
• Figure 5: The flip of a triangulation on the annulus.