Generalized Kac-Moody algebras from CHL dyons

Abstract

We provide evidence for the existence of a family of generalized Kac-Moody (GKM) superalgebras, N, whose Weyl-Kac-Borcherds denominator formula gives rise to a genus-two modular form at level N, Δk/2(Z), for (N, k) = (1, 10), (2, 6), (3, 4), and possibly (5, 2). The square of the automorphic form is the modular transform of the generating function of the degeneracy of CHL dyons in asymmetric N-orbifolds of the heterotic string compactified on T6. The new generalized Kac-Moody superalgebras all arise as different `automorphic corrections' of the same Lie algebra and are closely related to a generalized Kac-Moody superalgebra constructed by Gritsenko and Nikulin. The automorphic forms, Δk/2(Z), arise as additive lifts of Jacobi forms of (integral) weight k/2 and index 1/2. We note that the orbifolding acts on the imaginary simple roots of the unorbifolded GKM superalgebra, 1, leaving the real simple roots untouched. We anticipate that these superalgebras will play a role in understanding the `algebra of BPS states' in CHL compactifications.