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

Abstract

We show that the generating function of electrically charged 1 2 -BPS states in N = 4 supersymmetric CHL 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 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 Φ 3 (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 1 4 -BPS states, i.e., Φ 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. * suresh@physics.iitm.ac.in † gkrishna@imsc.res.in

Four-derivative couplings and BPS dyons in heterotic CHL orbifolds

“…, corresponding to left-moving excitations in the twisted sectors of the perturbative heterotic string, it is known that the helicity supertrace Ω 4 (Q, 0) is given by [22,17,24,25,23] 14) where q = 2 2πiτ and ∆ k (τ) is the same cusp form (2.4) which enters in the exact R 2 coupling.…”

Generalized Mathieu Moonshine

“…For some conjugacy classes [g] ∈ M 24 the associated second-quantized twining genera Φ g also give rise to denominator formulas of GKMs [56], which should presumably be identified with the wall-crossing algebras found in CHL-models [51,[57][58][59][60]. Although these observations are very suggestive, the precise role of the GKMs for Mathieu Moonshine remains to be understood.…”

Calabi-Yau manifolds and sporadic groups

“…In particular the product of any two CY 3-folds has χ 0 = χ 1 = 0. 7 The other interesting Jacobi forms in [21] have weight zero and index 4, 6 and 12. Let us look first at the case of d = 8 that leads to a Jacobi form with index 4.…”

“…1 Since the K3 surface plays a central role in mathematics and physics, this new observation 1 Relatedly, M24 has at the same time been connected to the counting functions of half BPS states in four dimensional N = 4 theories [2,7,8].…”