Nuclear Physics B (Dec 2021)

sl(2)ˆ decomposition of denominator formulae of some BKM Lie superalgebras

  • Suresh Govindarajan,
  • Mohammad Shabbir,
  • Sankaran Viswanath

Journal volume & issue
Vol. 973
p. 115614

Abstract

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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)ˆ 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)ˆ 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.