Topological Strings from Quantum Mechanics

TL;DR

This paper establishes a non-perturbative link between quantum mechanics and topological string theory on toric Calabi-Yau manifolds, providing exact quantization conditions and spectral determinants that match numerical results.

Contribution

It introduces a conjectured explicit formula for spectral determinants in terms of topological string free energy, unifying non-perturbative quantum spectra with enumerative geometry.

Findings

Spectral determinants match numerical spectra for local P2, P1xP1, and F1 geometries.

The quantization condition involves a generalized theta function with non-perturbative corrections.

Spectral traces derived from the spectral determinant agree with theoretical predictions.

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