BKM Lie superalgebras from dyon spectra in ℤ<inf>n</inf> CHL orbifolds for composite N

We show that the generating function of electrically charged 12-BPS states in M = 4 supersymmetric CHL ℤn-orbifolds of the heterotic string on T6 are given by multiplicative η-products. The η-products are determined by the cycle shape of the corresponding symplectic involution in the dual type II picture. This enables us to complete the construction of the genus-two Siegel modular forms due to David, Jatkar and Sen [arXiv:hep-th/ 0609109] for ℤn-orbifolds when N is non-prime. We study the Z4 CHL orbifold in detail and show that the associated Siegel modular forms, φ3(ℤ) and φ3(ℤ), are given by the square of the product of three even genus-two theta constants. Extending work by us as well as Cheng and Dabholkar, we show that the 'square roots' of the two Siegel modular forms appear as the denominator formulae of two distinct Borcherds-Kac-Moody (BKM) Lie superalgebras. The BKM Lie superalgebra associated with the generating function of 14-BPS states, i.e., φ3 (ℤ) has a parabolic root system with a lightlike Weyl vector and the walls of its fundamental Weyl chamber are mapped to the walls of marginal stability of the 1/4BPS states. © SISSA 2010.