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FIELD Phys:StringTheory
DATE November 16 (Tue), 2021
TIME 16:00-18:00
PLACE 1423
SPEAKER Wang, Haowu
HOST Ghim, Dongwook
TITLE 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 Wirthm?ller 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 Wirthm?ller'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.
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