Integral Regulators on Mirror Curves with Mass Parameter

TL;DR

This paper extends the connection between integral regulators on mirror curves and dilogarithm values to include cases with a mass parameter, broadening the scope of the 2015 conjecture linking mirror curves and operator spectra.

Contribution

It demonstrates that the relation between integral regulators and dilogarithm values persists even when a mass parameter is introduced in mirror curves.

Findings

Integral regulators remain connected to dilogarithm values with a mass parameter.

The conjecture linking mirror curves and operator spectra is expanded.

The results apply to a broader class of mirror curves with mass parameters.

Abstract

In 2015, Codesido-Grassi-Marino laid the foundation of a connection between local CY 3-folds and the spectra of operators attached to their mirror curves in the context of Topological String Theory. In a 2024 paper with C. Doran and M. Kerr, the author previously deduced two consequences of this conjecture; one relating zeroes of higher normal function to the spectra of operators of genus one curves, the other connecting integral regulators of $K_2$-classes on mirror curves to dilogarithm values at algebraic arguments. We now show that the latter continues to hold in the presence of a mass parameter, thus expanding the range of the conjecture.