Topological Strings from Quantum Mechanics

TL;DR

The paper establishes a non-perturbative bridge between quantum spectral problems arising from quantized toric Calabi–Yau mirror curves and topological string theory. It conjectures that the spectral determinant of the associated positive, trace-class operator equals e^{J_X(μ,ħ)} times a generalized theta function Θ_X(μ,ħ), with J_X built from perturbative, M2, and worldsheet NS contributions, thereby encoding the full spectrum via zeros of Θ_X. The framework yields exact quantization conditions, reproduces known spectra for local P^2, F_1, and P^1×P^1, and provides a background-independent, M-theoretic reformulation of topological strings through a Fermi-gas picture. It also demonstrates how the genus expansion emerges in a ’t Hooft limit and connects spectral data to refined BPS invariants, GV invariants, and NS limit data. Extensive checks on multiple geometries (including orbifold and maximally supersymmetric cases) support the conjectures and open avenues for higher-genus generalizations, rigorous proofs, and links to resurgence and holography.

Abstract

We propose a general correspondence which associates a non-perturbative quantum-mechanical operator to a toric Calabi-Yau manifold, and we conjecture an explicit formula for its spectral determinant in terms of an M-theoretic version of the topological string free energy. As a consequence, we derive an exact quantization condition for the operator spectrum, in terms of the vanishing of a generalized theta function. The perturbative part of this quantization condition is given by the Nekrasov-Shatashvili limit of the refined topological string, but there are non-perturbative corrections determined by the conventional topological string. We analyze in detail the cases of local P2, local P1xP1 and local F1. In all these cases, the predictions for the spectrum agree with the existing numerical results. We also show explicitly that our conjectured spectral determinant leads to the correct spectral traces of the corresponding operators. Physically, our results provide a non-perturbative formulation of topological strings on toric Calabi-Yau manifolds, in which the genus expansion emerges as a 't Hooft limit of the spectral traces. Since the spectral determinant is an entire function on moduli space, it leads to a background independent formulation of the theory. Mathematically, our results lead to precise, surprising conjectures relating the spectral theory of functional difference operators to enumerative geometry

Figures (6)

• Figure 1: The figure on the left shows the region (\ref{['reg-E']}) in phase space for the quantum operator associated to local ${\cal B}_2$, for $E=35$ and $m_1=m_2=1$. The figure on the right is the polyhedron representing toric ${\cal B}_2$.
• Figure 2: The figure on the left shows the region (\ref{['reg-E']}) in phase space for the quantum operator associated to local ${\cal B}_3$, for $E=35$ and $m_1=m_2=m_3=1$. The figure on the right is the polyhedron representing toric ${\cal B}_3$.
• Figure 3: The contour ${\cal C}$ in the complex plane of the chemical potential, which can be used to calculate the canonical partition function from the modified grand potential.
• Figure 4: The difference $\Delta(4\pi,m)$, defined in (\ref{['diff']}), for the local $\mathbb{P}^2$ geometry.
• Figure 5: The smooth line gives the free energy $F(N, 2 \pi)$ of local ${\mathbb P}^2$ as a function of $N$, computed from (\ref{['zn-airy']}), while the points give the values of of the free energy as computed from the spectral traces (\ref{['p2-traces']}).

Theorems & Definitions (2)

• Example 2.1
• Example 3.1