$\widehat{sl(2)}$ decomposition of denominator formulae of some BKM Lie superalgebras

Abstract

We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a sl(2) subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of sl(2) characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients of these vector-valued modular forms are related to multiplicities of roots appearing on the sum side of the Weyl-Kac-Borcherds denominator formulae.

$\widehat{sl(2)}$ decomposition of denominator formulae of some BKM Lie superalgebras -- II

“…This is why we expect that B CHL N (A (N ) ) are Borcherds extensions. Unlike the examples considered in our previous work [1], we are unaware of a proof that this is indeed the case for N ≤ 4.…”

“…The Jacobi forms Ψ (N ) 0,m (τ, z) will be the main object of our study. They are Jacobi forms of the congruence subgroup Γ 0 (N) with weight zero and index m. We obtain explicit formulae for these Jacobi forms in terms of standard modular forms for m ≤ N. The analogous expansion in our previous work [1] had nonvanishing terms only for indices that were multiples of N.…”

“…T is the Borcherds correction term incorporates the inclusion of imaginary simple roots. (See appendix B of [1] and references therein for a detailed description.) A key aspect of the Borcherds extension is that ∆ is a suitable automorphic form that admits a product formula.…”

“…In our previous work [1], we studied a family of Siegel modular forms that are associated with Umbral moonshine [2]. Here we consider Siegel modular forms that are associated with L 2 (11)-moonshine [3].…”

“…The dyon counting generating function is provided us with Siegel modular forms that transform covariantly under the Weyl group of g(A (N ) ). We have three examples of this variety, one of which was considered in [1]. We restrict to the case of N = 5 for simplicity in this work and the N = 6 example should work similarly.…”