BKM Lie superalgebras from counting twisted CHL dyons

Abstract

Following Sen, we study the counting of ('twisted') BPS states that contribute to twisted helicity trace indices in four-dimensional CHL models with N = 4 supersymmetry. The generating functions of half-BPS states, twisted as well as untwisted, are given in terms of multiplicative eta products with the Mathieu group, M 24 , playing an important role. These multiplicative eta products enable us to construct Siegel modular forms that count twisted quarter-BPS states.The square-roots of these Siegel modular forms turn out be precisely a special class of Siegel modular forms, the dd-modular forms, that have been classified by Clery and Gritsenko. We show that each one of these dd-modular forms arise as the Weyl-Kac-Borcherds denominator formula of a rank-three Borcherds-Kac-Moody Lie superalgebra. The walls of the Weyl chamber are in one-to-one correspondence with the walls of marginal stability in the corresponding CHL model for twisted dyons as well as untwisted ones. This leads to a periodic table of BKM Lie superalgebras with properties that are consistent with physical expectations.

Generalized Mathieu Moonshine

“…It is therefore natural to wonder about the corresponding physical interpretation of the twisted twining genera φ g,h . In the special case of a trivial twist g = e, it has been shown that for some elements h the twining genera φ e,h correspond to BPS-indices in CHL-orbifolds [6,[51][52][53]. We may therefore expect that the twisted twining genera φ g,h should be related to the counting of 'twisted dyons' in CHL-models.…”

“…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.…”

Fricke S-duality in CHL models

“…These models have been a remarkable arena for testing string dualities and analyzing black hole microstates (see [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] for a sample of references). In the non-orbifold case the N = 4 theory has duality group SL(2, Z) × SO (6, 22; Z), where the first factor is the S-duality group which acts as a strong-weak duality on the coupling S, while the second factor is the T-duality group.…”

“…The role of the McKay-Thompson series is here played by the so called twining genera φ g (τ, z), which are weak Jacobi forms with respect to subgroups of SL(2, Z). In our previous work [53] (generalising the earlier results [18,54]), we defined a class of infinite-products, labelled by commuting pairs g, h of elements in M 24 : 19) where the first product runs over…”

Calabi-Yau manifolds and sporadic groups

“…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].…”