From quantum curves to topological string partition functions

TL;DR

This work presents a non-perturbative reconstruction of the topological string partition function $Z_{ m top}$ for certain local Calabi–Yau manifolds by quantising their defining curves into quantum curves and solving the accompanying Riemann–Hilbert problem. The central device is the isomonodromic tau-function, whose chamber-dependent normalisations encode different phases in the extended Kähler moduli space; these tau-functions admit generalised theta-series expansions that reproduce chamber-specific topological-string data. By relating quantum curves to ${ m D}$-modules, opers with apparent singularities, and isomonodromic deformations, the authors connect non-perturbative string data to free fermion partition functions and conformal blocks, with abelianisation providing a canonical coordinate framework. The results are checked against topological vertex calculations in the main example (the SU$(2)$, $N_f=4$ theory) and lead to a coherent picture in which integrable structures govern quantum corrections and chamber structure, offering a path to generalisations via spectral networks and beyond.

Abstract

This paper describes the reconstruction of the topological string partition function for certain local Calabi-Yau (CY) manifolds from the quantum curve, an ordinary differential equation obtained by quantising their defining equations. Quantum curves are characterised as solutions to a Riemann-Hilbert problem. The isomonodromic tau-functions associated to these Riemann-Hilbert problems admit a family of natural normalisations labelled by the chambers in the extended Kähler moduli space of the local CY under consideration. The corresponding isomonodromic tau-functions admit a series expansion of generalised theta series type from which one can extract the topological string partition functions for each chamber.

Figures (7)

• Figure 1: The toric graph of the mirror $X_{R;U,z}$ to the local CY $Y_{R;U,z}$.
• Figure 2: Representation of the flop transition on a subgraph of a toric graph.
• Figure 3: Two toric diagrams related via flopping both engineering $SU(2)$ superconformal QCD with $N_f=4$ fundamental hypermultiplets.
• Figure 4: FN-networks on $C_{0,3}$, with punctures depicted at positions $z=0,1$. These isotopies occur in different regions of the parameter space and correspond to the possible choices of chamber $\mathfrak{C}_{\alpha}$ in Section \ref{['ExtKaehler']}. Network $\mathcal{W}_1$ corresponds to chamber $\mathfrak{C}^{\rm in}_{\mathfrak 1}$, while $\mathcal{W}_3$ corresponds to $\mathfrak{C}^{\rm in}_{\mathfrak 3}$ and $\mathcal{W}_s$ to $\mathfrak{C}^{\rm in}_{\mathfrak 2}$. The triplets of parameters $\{a_1,a_2,a\}$ in equation \ref{['C03curve']} take here the values: $\mathcal{W}_1$) $\{$0.51, 0.32, 0.18$\}$, $\mathcal{W}_3$) $\{$0.49, 0,48, 1$\}$$\mathcal{W}_2$) $\{$0.32, 0.51, 0.18$\}$ and $\mathcal{W}_s$) $\{$0.51, 0.5, 1$\}$.
• Figure 5: Fenchel-Nielsen network on the four-punctured sphere.

Theorems & Definitions (2)

• Remark 1
• Remark 2