# From Quantum Curves to Topological String Partition Functions II

## TL;DR

This paper explores how quantum curves can be used to characterize topological string partition functions in certain supersymmetric field theories.

## Contribution

The paper introduces a geometric characterization of partition functions using quantized Calabi–Yau manifolds and isomonodromic tau-functions.

## Key findings

- Partition functions correspond to preferred coordinates on moduli spaces of quantum curves.
- Changes in normalization of tau-functions define a line bundle crucial for the geometric characterization.
- The Exact WKB method is used to define coordinates on moduli spaces.

## Abstract

We propose a geometric characterisation of the topological string partition functions associated with the local Calabi–Yau (CY) manifolds used in the geometric engineering of \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {S}}$$\end{document}S. A quantisation of these CY manifolds defines differential operators called quantum curves. The partition functions are extracted from the isomonodromic tau-functions associated with 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 normalization of the tau-functions associated with these jumps define a natural line bundle playing a key role in the geometric characterisation of the topological string partition functions proposed here.