$E_n$ Jacobi forms and Seiberg–Witten curves

We discuss Jacobi forms that are invariant under the action of the Weyl group of type $E_n (n = 6, 7, 8)$. For $n = 6, 7$ we explicitly construct a full set of generators of the algebra of $E_n$ weak Jacobi forms. We first construct $n + 1$ independent $E_n$ Jacobi forms in terms of Jacobi theta functions and modular forms. By using them, we obtain Seiberg–Witten curves of type $\tilde{E}_6$ and $\tilde{E}_7$ for the $E$-string theory. The coefficients of each curve are $E_n$ weak Jacobi forms of particular weights and indices specified by the root system, realizing the generators whose existence was shown some time ago by Wirthmüller.

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This work was supported in part by JSPS KAKENHI Grant Number 26400257 and the JSPS Japan–Russia Research Cooperative Program.

Received 28 June 2017

Accepted 15 August 2018

Published 29 April 2019