Finite groups and automorphic products

Abstract: In 1992 Borcherds proved Monstrous Moonshine Conjecture by lifting Mckay--Thomson series to twisted denominator identities of the monster Lie algebra. Borcherds' proof inspires an intriguing relation between vertex algebras, Borcherds--Kac--Moody algebras and automorphic products. In this talk we introduce several new results related to this relation. We classify nice BKM algebras whose denominators are automorphic products. We prove twists of denominator identities of the fake monster Lie algebra by the Conway group define automorphic products on orthogonal groups. To prove the results, we establish a new expression of automorphic products in terms of Jacobi forms. Using this representation, we also show that some infinite products involving twisted elliptic genera of K3 surfaces (appearing in Mathieu Moonshine) are meromorphic Siegel modular forms on congruence subgroups, which proves a conjecture of Miranda Cheng. This talk is based on joint work with Brandon Williams.

HEP - theory

Audience: researchers in the topic

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