# E_n Jacobi forms and Seiberg-Witten curves

## TL;DR

This paper constructs explicit generators of E_n weak Jacobi forms for n=6,7, and uses them to derive Seiberg-Witten curves for E-string theory, linking algebraic forms to physical models.

## Contribution

It provides explicit constructions of E_n Jacobi forms for n=6,7 and derives corresponding Seiberg-Witten curves, advancing the understanding of their algebraic and physical connections.

## Key findings

- Explicit generators of E_6 and E_7 weak Jacobi forms constructed.
- Seiberg-Witten curves for E_6 and E_7 derived using these forms.
- Coefficients of the curves are E_n weak Jacobi forms of specific weights and indices.

## Abstract

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 E_6 and 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 Wirthm\"uller.