Weyl invariant $E_8$ Jacobi forms and $E$-strings

In 1992 Wirthmüller showed that for any irreducible root system not of type $E_8$ the ring of weak Jacobi forms invariant under Weyl group is a polynomial algebra. However, it has recently been proved that for $E_8$ the ring is not a polynomial algebra. Weyl invariant $E_8$ Jacobi forms have many applications in string theory and it is an open problem to describe such forms. The scaled refined free energies of $E$-strings with certain $\eta$-function factors are conjectured to be Weyl invariant $E_8$ quasi-holomorphic Jacobi forms. It is further observed that the scaled refined free energies up to some powers of $E_4$ can be written as polynomials in nine Sakai’s $E_8$ Jacobi forms and Eisenstein series $E_2, E_4, E_6$. Motivated by the physical conjectures, we prove that for any Weyl invariant $E_8$ Jacobi form $\phi_t$ of index $t$ the function $E^{[t/5]}_4 \Delta^{[5t/6]} \phi_t$ can be expressed uniquely as a polynomial in $E_4$, $E_6$ and Sakai’s forms, where $[x]$ is the integer part of $x$. This means that a Weyl invariant $E_8$ Jacobi form is completely determined by a solution of some linear equations. By solving the linear systems, we determine the generators of the free module of Weyl invariant $E_8$ weak (resp. holomorphic) Jacobi forms of given index $t$ when $t \leq 13$ (resp. $t \leq 11$).

Keywords

Jacobi forms, $E_8$ root system, Weyl groups, $E$-strings

2010 Mathematics Subject Classification

Primary 11F50, 17B22. Secondary 81T30.

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This work began in July 2020, when both authors were postdoctoral fellows at the Max Planck Institute for Mathematics in Bonn. The authors thank the institute for its stimulating environment and financial support. The authors would like to thank Valery Gritsenko, Min-xin Huang, Albrecht Klemm, Kimyeong Lee, Kazuhiro Sakai, XinWang for useful discussions. KS is also supported by Korea Institute for Advanced Study Grant QP081001. H. Wang also thanks the Institute for Basic Science (IBS-R003-D1) for its hospitality and financial support.

Received 18 August 2022

Accepted 18 April 2023

Published 7 November 2023