The refined counting of BPS states in four-dimensional string theories with N=4 supersymmetry has brought interesting connections with modular forms, mock modular forms, Lie algebras and the surprising appearance of the sporadic simple group, M_24, is called Mathieu moonshine. The (inverse of the) generating function of dyon degeneracies is a Siegel modular form.
In this talk, we focus on the appearance of families of Lie Algebras that are obtained by interpreting the square-root of the generating function of dyons degeneracies as the denominator formulae of the Lie algebras. In most of the examples, the Lie algebras are generalized Kac-Moody Lie superalgebras or Borcherds-Kac-Moody Lie superalgebras. There are examples that are not BKM Lie superalgebras. In these cases the square-root provides us a tool to look for an extension of BKM Lie superalgebras. With this in mind, we discuss the decomposition of this square-root in terms of an affine Lie algebra. This leads to the appearance of vector valued modular forms that encode the multiplicities of simple roots of the (potential) Lie superalgebra.
Based on arXiv:2106.01605
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