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- W3199957730 abstract "In 1992 Wirthm{u}ller 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 $E_4^{[t/5]}Delta^{[5t/6]}phi_t$ 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 $tleq 13$ (resp. $tleq 11$)." @default.
- W3199957730 created "2021-09-27" @default.
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- W3199957730 date "2021-09-22" @default.
- W3199957730 modified "2023-09-27" @default.
- W3199957730 title "Weyl invariant $E_8$ Jacobi forms and $E$-strings" @default.
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- W3199957730 doi "https://doi.org/10.48550/arxiv.2109.10578" @default.
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