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arXiv · 2110.08482

$K_2$ and quantum curves

Charles F. Doran·Matt Kerr·Soumya Sinha Babu
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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.

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Charles F. Doran, Matt Kerr, Soumya Sinha Babu. 2021-10-16. $K_2$ and quantum curves. https://arxiv.org/abs/2110.08482

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