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

Suresh Govindarajan, K. Gopala Krishna

TL;DR

<3-5 sentence high-level summary>The paper addresses the microscopic counting of dyons in four-dimensional N=4 CHL orbifolds with composite N by constructing genus-two Siegel modular forms that generate dyon degeneracies. It develops additive lifts from Jacobi seeds determined by twisted elliptic genera and eta-products, yielding explicit Siegel forms for N=4,6,8; notably, Φ_3(Z) and ˜Φ_3(Z) factor as squares of products of even genus-two theta constants. The authors demonstrate that the square roots Δ_{3/2}(Z) and ˜Δ_{3/2}(Z) serve as denominator formulae for two distinct Borcherds-Kac-Moody Lie superalgebras, whose Weyl data and walls of marginal stability map to the dyon stability structure, including a parabolic-type algebra for the ˜Δ_{3/2}(Z) case. They further discuss integrality, physical interpretation, and generalizations to product groups and Type II models, hinting at a broad unifying BKM framework for dyons in diverse N=4 compactifications.

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, Φ_3(Z) and \widetildeΦ_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., \widetildeΦ_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.

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

TL;DR

<3-5 sentence high-level summary>The paper addresses the microscopic counting of dyons in four-dimensional N=4 CHL orbifolds with composite N by constructing genus-two Siegel modular forms that generate dyon degeneracies. It develops additive lifts from Jacobi seeds determined by twisted elliptic genera and eta-products, yielding explicit Siegel forms for N=4,6,8; notably, Φ_3(Z) and ˜Φ_3(Z) factor as squares of products of even genus-two theta constants. The authors demonstrate that the square roots Δ_{3/2}(Z) and ˜Δ_{3/2}(Z) serve as denominator formulae for two distinct Borcherds-Kac-Moody Lie superalgebras, whose Weyl data and walls of marginal stability map to the dyon stability structure, including a parabolic-type algebra for the ˜Δ_{3/2}(Z) case. They further discuss integrality, physical interpretation, and generalizations to product groups and Type II models, hinting at a broad unifying BKM framework for dyons in diverse N=4 compactifications.

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, Φ_3(Z) and \widetildeΦ_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., \widetildeΦ_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.

Paper Structure

This paper contains 41 sections, 123 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The model
  3. Microscopic counting of dyonic degeneracies
  4. The additive lift
  5. Counting $\frac{1}{2}$-BPS states
  6. Heterotic string on $T^6$
  7. The CHL orbifold of the heterotic string on $T^6$
  8. Symplectic automorphisms of $K3$ and the Mathieu group $\mathbf{M_{24}}$
  9. The additive lift
  10. $N=4$
  11. $N=6$
  12. $N=8$
  13. Product formulae
  14. Product formula for $\Phi_3(\mathbf{Z})$
  15. Product formula for $\widetilde{\Phi}_3(\mathbf{Z})$
  16. ...and 26 more sections

Figures (2)

  • Figure 1: Fundamental domains/Weyl chamber for $N=1,2,3$
  • Figure 2: The fundamental domain/Weyl chamber for $N=4$ is bounded by an infinite number of semi-circles as the BKM Lie superalgebra has infinite number of real simple roots. Each of the semi-circles indicated represent real simple roots that appear with multiplicity one in the sum side of the denominator formula. Note that the diameter of the semi-circles are reducing as one gets closer to $\tfrac{1}{2}$. The point $\tfrac{1}{2}$ is approached as a limit point of the infinite sequence of semi-circles.