Geometric Kac-Moody Modularity

TL;DR

This paper explores the connection between the arithmetic properties of algebraic curves, specifically the Hasse-Weil L-function, and affine Kac-Moody algebras, revealing a link to exactly solvable theories in mathematical physics.

Contribution

It establishes a novel relationship between the Hasse-Weil L-function of algebraic curves and affine Kac-Moody algebra structures, especially for Calabi-Yau varieties and Fermat curves.

Findings

Hasse-Weil L-function related to affine Kac-Moody algebra characters

Identified Mellin transform of modular form with L-function for genus three Fermat curve

Connected twist characters to quantum dimensions in conformal field theory

Abstract

It is shown how the arithmetic structure of algebraic curves encoded in the Hasse-Weil L-function can be related to affine Kac-Moody algebras. This result is useful in relating the arithmetic geometry of Calabi-Yau varieties to the underlying exactly solvable theory. In the case of the genus three Fermat curve we identify the Hasse-Weil L-function with the Mellin transform of the twist of a number theoretic modular form derived from the string function of a non-twisted affine Lie algebra. The twist character is associated to the number field of quantum dimensions of the conformal field theory.