The D1-D5-KK-p system naturally provides an infinite-dimensional module graded by the dyonic charges whose dimensions are counted by the Igusa cusp form, Φ 10(Z). We show that the Mathieu group, M 24, acts on this module by recovering the Siegel modular forms that count twisted dyons as a trace over this module. This is done by recovering Borcherds product formulae for these modular forms using the M 24 action. This establishes the correspondence ('moonshine') proposed in arXiv:0907.1410 that relates conjugacy classes of M 24 to Siegel modular forms. This also, in a sense that we make precise, subsumes existing moonshines for M 24 that relates its conjugacy classes to eta-products and Jacobi forms. © 2012 Elsevier B.V.

Suresh Govindarajan

Department Of Physics

Fulltext

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IIT Madras

Nuclear Physics B

In this paper, we revisit an earlier conjecture by one of us that related conjugacy classes of M12 to Jacobi forms of weight zero and index one. We construct Jacobi forms for all conjugacy classes of M12 that are consistent with constraints from group theory as well as modularity. However, we obtain 1427 solutions that satisfy these constraints (to the order that we checked) and are unable to provide a unique Jacobi form. Nevertheless, as a consequence, we are able to provide a group theoretic proof of the evenness of the coefficients of all EOT Jacobi forms associated with conjugacy classes of M12:2⊂M24. We show that there exists no solution where the Jacobi forms (for order 4/8 elements of M12) transform with phases under the appropriate level. In the absence of a moonshine for M12, we show that there exist moonshines for two distinct L2(11) sub-groups of the M12. We construct Siegel modular forms for all L2(11) conjugacy classes and show that each of them arises as the denominator formula for a distinct Borcherds–Kac–Moody Lie superalgebra. © 2019

Two moonshines for L2(11) but none for M12

We revisit earlier work by one of us which lead to a periodic table of Borcherds-Kac-Moody algebras that appeared in the context of the refined generating function of quarter-BPS states (dyons) in N=4 supersymmetric four-dimensional string theory. We make new additions to the periodic table by making use of connections with generalized Mathieu moonshine as well as umbral moonshine. We show the modularity of some Siegel modular forms that appear in umbral moonshine associated with Niemeier lattices constructed from A-type root systems and further show that the same Siegel modular forms appear for generalized Mathieu moonshine in some cases. We argue for the existence of a new kind of BKM Lie superalgebras that arise from the dyon generating functions for the Z5 and Z6 CHL orbifolds. © 2019 The Author(s)

BKM Lie superalgebras from counting twisted CHL dyons – II

Unravelling Mathieu moonshine

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, M24, 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. © SISSA 2011.

Journal of High Energy Physics

BKM Lie superalgebras from counting twisted CHL dyons

We show that the generating function of electrically charged 12-BPS states in M = 4 supersymmetric CHL ℤn-orbifolds of the heterotic string on T6 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 ℤn-orbifolds when N is non-prime. We study the Z4 CHL orbifold in detail and show that the associated Siegel modular forms, φ3(ℤ) and φ3(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 14-BPS states, i.e., φ3 (ℤ) has a parabolic root system with a lightlike Weyl vector and the walls of its fundamental Weyl chamber are mapped to the walls of marginal stability of the 1/4BPS states. © SISSA 2010.

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

We provide evidence for the existence of a family of generalized Kac-Moody (GKM) superalgebras, N whose Weyl-Kac-Borcherds denominator formula gives rise to a genus-two modular form at level N, Δ(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 N-orbifolds of the heterotic string compactified on T6. 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, Δ2(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, 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. © 2009 SISSA.

Journal	Nuclear Physics B
ISSN	05503213
Open Access	No