$K_2$ and quantum curves

TL;DR

This paper provides evidence for a conjecture linking topological string invariants to operator spectra by proving two theorems that connect normal functions and regulator periods on mirror curves.

Contribution

It proves two theorems related to the CGM conjecture, connecting spectral properties and regulator periods of mirror curves in topological string theory.

Findings

Zeroes of higher normal functions relate to operator spectra for genus one curves.

Dilogarithm formulas describe limits of regulator periods at conifold points.

Abstract

A 2015 conjecture of Codesido-Grassi-Mari\~no in topological string theory relates the enumerative invariants of toric CY 3-folds to the spectra of operators attached to their mirror curves. We deduce two consequences of this conjecture for the integral regulators of $K_2$-classes on these curves, and then prove both of them; the results thus give evidence for the CGM conjecture. (While the conjecture and the deduction process both entail forms of local mirror symmetry, the consequences/theorems do not: they only involve the curves themselves.) Our first theorem relates zeroes of the higher normal function to the spectra of the operators for curves of genus one, and suggests a new link between analysis and arithmetic geometry. The second theorem provides dilogarithm formulas for limits of regulator periods at the maximal conifold point in moduli of the curves.