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Geometric Kac–Moody modularity
Affiliation:1. Indiana University South Bend, 1700 Mishawaka Av., South Bend, IN 46634, USA;2. Kennesaw State University, 1000 Chastain Road, Kennesaw, GA 30144, USA;1. Physics Department, Stony Brook University, Stony Brook, NY 11794-3840, USA;2. C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794-3840, USA;1. Department of Mathematics, University of Sussex, UK;2. School of Mathematics & Statistics, University of St Andrews, UK;3. ibidi GmbH Am Klopferspitz 19, 82152 Martinsried, Germany;1. School of Mathematical Sciences, Capital Normal University, Beijing 100048, China;2. Department of Mathematics, China University of Mining and Technology, Beijing 100083, China;1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA;2. Institute for Information Transmission Problems of Russian Academy of Sciences, Moscow, Russia
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.
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