Generalized Kac-Moody algebras from CHL dyons

Abstract

We provide evidence for the existence of a family of generalized Kac-Moody(GKM) superalgebras, G N , whose Weyl-Kac-Borcherds denominator formula gives rise to a genus-two modular form at level N , ∆ k/2 (Z), for (N, k) = (1, 10), (2, 6), (3, 4), and possibly (5, 2). The square of the automorphic form is the modular transform of the generating function of the degeneracy of CHL dyons in asymmetric Z N -orbifolds of the heterotic string compactified on T 6 . The new generalized Kac-Moody superalgebras all arise as different 'automorphic corrections' of the same Lie algebra and are closely related to a generalized Kac-Moody superalgebra constructed by Gritsenko and Nikulin. The automorphic forms, ∆ k/2 (Z), arise as additive lifts of Jacobi forms of (integral) weight k/2 and index 1/2. We note that the orbifolding acts on the imaginary simple roots of the unorbifolded GKM superalgebra, G 1 leaving the real simple roots untouched. We anticipate that these superalgebras will play a role in understanding the 'algebra of BPS states' in CHL compactifications. * suresh@physics.iitm.ac.in † gkrishna@imsc.res.in 12 2 also happens to be the size of a 1/4 BPS multiplet of states.

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.…”

“…Forĝ = (g, δ), with g ∈ O(Γ 6,22 ) of order N and δ ∈ Γ 6,22 ⊗ R fixed by g, this means 14) where ∆E g := E g,L − E g,R is the level mismatch for the g-twisted ground state in the heterotic model andN is the least common multiple of the orders of g and δ. In general,…”

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.…”

Twisted Elliptic Genus for K3 and Borcherds Product

“…The twisted elliptic genera of K3 for conjugacy classes 2A, 3A, 5A, 7A are closely related to the Z N orbifold partition function of CHL models [9,25,29,10]. This paper is organized as follows: in section 2 we recall the twisted elliptic genera.…”