## BKM Lie superalgebras from dyon spectra in Z_N-CHL orbifolds for composite N

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Govindarajan, Suresh

Krishna, K. Gopala

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We show that the generating function of electrically charged 1/2-BPS states
in N=4 supersymmetric Z_N-CHL orbifolds of the heterotic string on T^6 are
given by multiplicative eta-products. The eta-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 Z_N
orbifolds when N is non-prime. We study the Z_4 CHL orbifold in detail and show
that the associated Siegel modular forms, \Phi_3(Z) and \widetilde{\Phi}_3(Z),
are given by the square of the product of three even genus-two theta constants.
Extending work by us[arXiv:0807.4451] as well as Cheng and
Dabholkar[arXiv:0809.4258], we show that their `square roots' appear as the
denominator formulae of two distinct Borcherds-Kac-Moody (BKM) Lie
superalgebras. The BKM Lie superalgebra associated with the generating function
of 1/4-BPS states, i.e., \widetilde{\Phi}_3(Z) has a parabolic root system with
a light-like Weyl vector and the walls of its fundamental Weyl chamber are
mapped to the walls of marginal stability of the 1/4-BPS states.

Comment: LaTeX, 44 pages, 2 figures (v2)Physical interpretation for a family of BKM Lie superalgebras provided; typos corrected

Comment: LaTeX, 44 pages, 2 figures (v2)Physical interpretation for a family of BKM Lie superalgebras provided; typos corrected

##### Keywords

High Energy Physics - Theory, Mathematics - Algebraic Geometry