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

TL;DR

This paper links the generating functions of BPS states in Z_N-CHL orbifolds to eta-products and Siegel modular forms, revealing their connection to Borcherds-Kac-Moody Lie superalgebras and their root systems.

Contribution

It constructs new BKM Lie superalgebras from dyon spectra in non-prime N orbifolds, extending previous modular form and algebraic frameworks.

Findings

Generated eta-products from cycle shapes of symplectic involutions.

Constructed Siegel modular forms as squares of genus-two theta constants.

Identified BKM Lie superalgebras with specific root systems related to BPS spectra.

Abstract

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.