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BKM Lie superalgebras from dyon spectra in Z N CHL orbifolds for composite N

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Abstract

We show that the generating function of electrically charged \( \frac{1}{2} \) -BPS states in \( \mathcal{N} = 4 \) supersymmetric CHL \( {\mathbb{Z}_N} \)-orbifolds of the heterotic string on T 6 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 \( {\mathbb{Z}_N} \)-orbifolds when N is non-prime. We study the \( {\mathbb{Z}_4} \) CHL orbifold in detail and show that the associated Siegel modular forms, \( {\Phi_3}\left( \mathbb{Z} \right) \) and \( {\widetilde\Phi_3}\left( \mathbb{Z} \right) \), 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 \( \frac{1}{4} \) -BPS states, i.e., \( {\widetilde\Phi_3}\left( \mathbb{Z} \right) \) 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 \( \frac{1}{4} \) -BPS states.

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Authors and Affiliations

  1. Department of Physics, Indian Institute of Technology Madras, Chennai, 600036, India

    Suresh Govindarajan

  2. The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai, 600113, India

    K. Gopala Krishna

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  1. Suresh Govindarajan
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  2. K. Gopala Krishna
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Correspondence to Suresh Govindarajan.

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ArXiv ePrint: 0907.1410

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Govindarajan, S., Gopala Krishna, K. BKM Lie superalgebras from dyon spectra in Z N CHL orbifolds for composite N . J. High Energ. Phys. 2010, 14 (2010). https://doi.org/10.1007/JHEP05(2010)014

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