The [formula omitted]-tower of weak Jacobi forms and applications

Abstract

We construct the tower of arithmetic generators of the bigraded polynomial ring J∗,∗w,O(Dn) of the weak Jacobi modular forms invariant with respect to the full orthogonal group O(Dn) of the root lattice Dn for 2≤n≤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.