On the open TS/ST correspondence

TL;DR

This work extends the TS/ST correspondence to open strings by constructing entire off-shell eigenfunctions for the quantized mirror curve of local $F_0$ from open topological-string partition functions.A matrix-model framework is developed, yielding explicit off-shell eigenfunctions in both matrix-model and outer topological-string coordinates, and a two-term symmetric structure that underpins their analytic properties.Two crucial four-dimensional limits are analyzed: the standard limit connects to the Fourier-transformed modified Mathieu operator, while the dual limit leads to the McCoy-Tracy-Wu operator; in both cases the open eigenfunctions become tractable and related by simple functional relations.The spectra in the two 4d limits are connected through NS/GV gauge-theory data, with precise formulas linking the eigenvalues to NS free energies and the arccosh-based relations between the two operators, reinforcing the deep bridge between topological strings and quantum integrable systems.

Abstract

The topological string/spectral theory correspondence establishes a precise, non-perturbative duality between topological strings on local Calabi-Yau threefolds and the spectral theory of quantized mirror curves. While this duality has been rigorously formulated for the closed topological string sector, the open string sector remains less understood. Building on the results of [1-3], we make further progress in this direction by constructing entire, off-shell eigenfunctions for the quantized mirror curve from open topological string partition functions. We focus on local $\mathbb{F}_0$, whose mirror curve corresponds to the Baxter equation of the two-particle, relativistic Toda lattice. We then study the standard and dual four-dimensional limits, where the quantum mirror curve for local $\mathbb{F}_0$ degenerates into the modified Mathieu and McCoy-Tracy-Wu operators, respectively. In these limits, our framework provides a way to construct entire, off-shell eigenfunctions for the difference equations associated with these operators. Furthermore, we find a simple relation between the on-shell eigenfunctions of the modified Mathieu and McCoy-Tracy-Wu operators, leading to a functional relation between the operators themselves.

Figures (3)

• Figure 1: From top left to bottom: the first term $\omega_{0,+}$ on the right-hand side of \ref{['eq:Psi0x_RationalB_Result']}, the second term $\omega_{0,-}$, and their sum $\psi_{0}$ for $\xi = - 7 / 4$ and $\hbar = 4 \pi$. The solid and dashed lines correspond to the real and imaginary parts, respectively.
• Figure 2: The on-shell eigenfunction from topological strings for $\xi = - 16 / 53$ and $\hbar = 8 \pi / 3$ at energy $E = E_4 \approx 7.69$. Top left: the $\sigma = 1$ term in \ref{['eq:sigmas']}. Top right: the $\sigma = 2$ term in \ref{['eq:sigmas']}. Bottom left: the full eigenfunction $\psi$ as given in \ref{['tsstopenf']}, after normalization. Bottom right: the absolute difference between the analytical and numerical eigenfunctions, including 0 (green), 2 (blue), and 4 (violet) instantons in the small $\exp\mathopen{}\mathclose{\left( -t_B \right)$ expansion of $\psi$. The plot on the bottom right uses a logarithmic scale with base 10. The solid and dashed lines correspond to the real and imaginary parts, respectively.
• Figure 3: The off-shell eigenfunction from topological strings for $\xi = - 16 / 53$ and $\hbar = 8 \pi / 3$ at energy $E = 225/31$, which is between $E_3$ and $E_4$. From top left to bottom: the $\sigma = 1$ term in \ref{['eq:sigmas']}, the $\sigma = 2$ term in \ref{['eq:sigmas']}, and the complete eigenfunction $\psi$ in \ref{['tsstopenf']}, after normalization. The solid and dashed lines correspond to the real and imaginary parts, respectively.