Operators from mirror curves and the quantum dilogarithm

TL;DR

This paper explores the quantization of mirror curves associated with toric Calabi-Yau threefolds, demonstrating that the resulting operators are trace class and connecting their spectral properties to enumerative invariants.

Contribution

It introduces a method to compute the integral kernels of quantum operators from mirror curves using quantum dilogarithm and verifies a conjecture relating their determinants to geometric invariants.

Findings

Operators are trace class for many local del Pezzo Calabi-Yau threefolds.

Explicit integral kernels are computed for simple geometries like local P2.

Spectral traces are expressed as multi-dimensional integrals, matching topological state-integrals.

Abstract

Mirror manifolds to toric Calabi-Yau threefolds are encoded in algebraic curves. The quantization of these curves leads naturally to quantum-mechanical operators on the real line. We show that, for a large number of local del Pezzo Calabi-Yau threefolds, these operators are of trace class. In some simple geometries, like local P2, we calculate the integral kernel of the corresponding operators in terms of Faddeev's quantum dilogarithm. Their spectral traces are expressed in terms of multi-dimensional integrals, similar to the state-integrals appearing in three-manifold topology, and we show that they can be evaluated explicitly in some cases. Our results provide further verifications of a recent conjecture which gives an explicit expression for the Fredholm determinant of these operators, in terms of enumerative invariants of the underlying Calabi-Yau threefolds.