The D_8-tower of weak Jacobi forms and applications

TL;DR

This paper constructs a tower of weak Jacobi modular forms invariant under orthogonal groups related to D_n lattices, revealing their algebraic structure and differential equations with applications to Kac-Moody algebras and Enriques surfaces.

Contribution

It introduces a new tower of generators for the ring of weak Jacobi forms invariant under O(D_n), linking modular forms, differential equations, and geometric applications.

Findings

Constructed a tower of generators for J_{*,*}^{w, O}(D_n).

Derived modular differential equations satisfied by key generators.

Connected modular forms to automorphic discriminants of Enriques surfaces.

Abstract

We construct a tower of arithmetic generators of the bigraded polynomial ring J_{*,*}^{w, O}(D_n) of weak Jacobi modular forms invariant with respect to the full orthogonal group O(D_n) of the root lattice D_n for 2\le n\le 8. This tower corresponds to the tower of strongly reflective modular forms on the orthogonal groups of signature (2,n) which determine the Lorentzian Kac-Moody algebras related to the BCOV (Bershadsky-Cecotti-Ooguri-Vafa)-analytic torsions. We prove that the main three generators of index one of the graded ring satisfy a special system of modular differential equations. We found also a general modular differential equation of the generator of weight 0 and index 1 which generates the automorphic discriminant of the moduli space of Enriques surfaces.