Recent results on Jacobi forms of lattice index

Abstract: In 1985 Eichler and Zagier introduced the theory of Jacobi forms. Such forms are holomorphic functions which are modular under SL(2,Z) in the first variable and quasi-periodic in the second variable. Later, the Jacobi form of lattice index was defined by replacing the second variable with many variables associated with a positive-definite lattice. Jacobi forms are an elegant intermediate between different types of modular forms and have many applications in mathematics and physics. In this talk, I will give a brief introduction to this theory and present some recent results. (1) In 1992 Wirthmüller proved that for any irreducible root system not of type E_8 the algebra of weak Jacobi forms invariant under the Weyl group is a polynomial algebra. I will define the Jacobian of Jacobi forms and present an automorphic proof of Wirthmüller's theorem. (2) Weyl invariant weak Jacobi forms for the exceptional root system E_8 appear in E-string theory. I will prove several conjectures proposed by some physicists. This gives a clear picture of the (non-free) algebra of such Jacobi forms. (3) I will talk about the algebra of weak Jacobi forms for lattices of rank 2. This talk is based on joint works with Brandon Williams and with Kaiwen Sun.

HEP - theory

Audience: researchers in the topic

KIAS Strings Seminar

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