Advances in Theoretical and Mathematical Physics

Volume 27 (2023)

Number 8

$K_2$ and quantum curves

Pages: 2261 – 2318

DOI: https://dx.doi.org/10.4310/ATMP.2023.v27.n8.a1

Authors

Charles F. Doran (Departmemt of Mathematics, University of Alberta, Edmonton, AB, Canada; and Center for Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts, U.S.A.)

Matt Kerr (Department of Mathematics, Washington University, St. Louis, Missouri, U.S.A.)

Soumya Sinha Babu (Department of Mathematics, University of Georgia, Athens, Ga., U.S.A.)

Abstract

A 2015 conjecture of Codesido-Grassi-Mariño 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.

Published 14 August 2024