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arXiv · 2106.01605

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

Suresh Govindarajan·Mohammad Shabbir·Sankaran Viswanath
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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 $\widehat{sl(2)}$ subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of $\widehat{sl(2)}$ 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.

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Suresh Govindarajan, Mohammad Shabbir, Sankaran Viswanath. 2021-06-03. $\widehat{sl(2)}$ decomposition of denominator formulae of some BKM Lie superalgebras. https://doi.org/10.1016/j.nuclphysb.2021.115614

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