$K_2$ and quantum curves
Abstract
A 2015 conjecture of CodesidoGrassiMariño in topological string theory relates the enumerative invariants of toric CY 3folds 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.08482
 Bibcode:
 2021arXiv211008482D
 Keywords:

 Mathematics  Algebraic Geometry;
 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  KTheory and Homology;
 Mathematics  Number Theory;
 14D07;
 14J33;
 19E15;
 32G20;
 34K08
 EPrint:
 49 pages, 1 figure