On the open TS/ST correspondence

Technical Summary: On the open TS/ST correspondence

Problem Statement
The Topological String/Spectral Theory (TS/ST) correspondence establishes a non-perturbative duality between topological strings on local Calabi-Yau (CY) threefolds and the spectral theory of quantized mirror curves. While the closed string sector is well-understood—specifically, the relation between the spectral determinant of the quantized mirror curve and the sum of topological string grand potentials is rigorously formulated—the open string sector remains less understood. A primary challenge is constructing entire, off-shell eigenfunctions for the quantized mirror curve. Previous attempts using open topological string partition functions yielded functions that were not entire in the open string modulus $x$, failing to provide a background-independent, non-perturbative formulation. The goal of this work is to extend the TS/ST correspondence to the open string sector by constructing such entire eigenfunctions for the specific case of local $F_0$.

Methodology
The authors focus on local $F_0$, whose mirror curve corresponds to the Baxter equation of the two-particle, relativistic Toda lattice. The methodology proceeds through several interconnected steps:

1. Matrix Model Formulation: The spectral problem is first analyzed in "matrix model coordinates" ($q, p$). The authors utilize the canonical transformation relating these coordinates to the "outer topological string coordinates" ($x, y$). They construct off-shell eigenfunctions in the matrix model frame, denoted as $\Xi(q, \kappa)$, as a sum of two contributions involving $O(2)$ matrix model expectation values.
2. Analytic Continuation and Resummation: To address convergence issues in the canonical transformation for certain values of $\hbar$ (specifically $\hbar \leq 2\pi$), the authors employ a technique involving the integration of quasi-periodic functions using Faddeev's non-compact quantum dilogarithm ($\Phi_b$). This allows for the explicit computation of eigenfunctions even when the original integral transform is ill-defined.
3. Symmetric Structure Identification: By analyzing the $N=0$ and $N=1$ cases in the matrix model and taking the 't Hooft limit, the authors identify a symmetric structure in the eigenfunctions. They propose that the eigenfunction in topological string coordinates, $\psi(x, \kappa)$, is a sum of two terms related by a specific shift in the complex plane and a phase factor.
4. Topological String Reconstruction: The authors map the matrix model results back to the topological string frame. They define an open string grand potential $J(x, \mu, \xi, \hbar)$ incorporating perturbative and non-perturbative (NS and GV) contributions. They then construct the full eigenfunction as a sum over integer shifts of the closed string modulus $\mu$ and a specific combination of the open string grand potential evaluated at shifted arguments.
5. Four-Dimensional Limits: The framework is tested by taking two distinct four-dimensional limits: the standard limit (leading to the modified Mathieu operator) and the dual limit (leading to the McCoy-Tracy-Wu operator).

Key Contributions and Results

• Construction of Entire Eigenfunctions: The paper provides an explicit formula for the eigenfunctions of the quantized mirror curve for local $F_0$. The proposed eigenfunction is:
  $$\psi(x, \kappa) = \sum_{k \in \mathbb{Z}} \left( e^{J(x, \mu+i2\pi k, \xi, \hbar)} + e^{\frac{i}{\hbar}\frac{\pi^2}{2} + \frac{\pi x}{\hbar} + J(-x-i\pi, \mu+i\pi+i2\pi k, \xi, \hbar)} \right)$$
  where $J$ is the full grand potential (closed plus open). The authors demonstrate that this combination is an entire function of $x$ for all $\kappa \in \mathbb{C}$, resolving the issue of non-entireness found in previous formulations.
• Verification of Properties: The constructed function is shown to:
  1. Be a well-defined solution to the functional difference equation (Baxter equation) associated with the quantized mirror curve.
  2. Reduce to proper, square-integrable eigenfunctions when $\kappa$ coincides with a root of the spectral determinant (on-shell).
  3. Exhibit the correct symmetry properties under parity and shifts.
• Matrix Model Connection: The eigenfunctions are expressed in terms of $O(2)$ matrix models, providing a concrete computational framework. The authors explicitly compute the $N=1$ term in the matrix model expansion to verify the analytic properties of the proposed structure.
• Four-Dimensional Limits:
  • In the standard 4D limit, the difference equation reduces to the modified Mathieu equation. The authors show their construction yields entire off-shell eigenfunctions that reproduce known on-shell results related to 2d/4d surface defects in the Nekrasov-Shatashvili (NS) phase.
  • In the dual 4D limit, the equation reduces to the McCoy-Tracy-Wu operator. The construction yields eigenfunctions related to surface defects in the Gopakumar-Vafa (GV) phase.
• Operator Relation: A significant result is the discovery of a simple functional relation between the on-shell eigenfunctions of the modified Mathieu and McCoy-Tracy-Wu operators. This leads to a direct functional relation between the two operators themselves, bridging the standard and dual 4D limits.

Significance and Claims
The paper claims to make "further progress" in understanding the open string sector of the TS/ST correspondence. The primary significance lies in providing a concrete, non-perturbative definition of off-shell eigenfunctions that are entire in the open string modulus. This addresses a gap in the rigorous formulation of the duality, where previous open string partition functions failed to be entire.

The authors are modest regarding the mathematical rigor of their main proposal. They state that while they do not have a "complete, mathematical proof" that the proposed combination is the unique entire solution, they have performed "many analytic and numerical tests" that support the claim. The work relies on the insights of previous studies [1–3] to generalize the TS/ST correspondence, specifically by identifying the precise summation over "saddles" (or shifts in the modulus) required to smooth out singularities in the moduli space.

The results suggest that the TS/ST correspondence naturally encodes the necessary non-perturbative completions (via the summation over $k$ and the specific combination of $J$ terms) to define well-behaved spectral objects, extending the duality from the closed string spectrum to the open string eigenfunctions.