BKM Lie superalgebras from counting twisted CHL dyons – II

Govindarajan, Suresh; Samanta, Sutapa

doi:10.1016/j.nuclphysb.2019.114770

Cited by 10 publications

(12 citation statements)

References 42 publications

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“…Thus, the simple CHL Z N orbifolds occur for N ≤ 8. In our forthcoming paper [29], we revisit these considerations and provide evidence for a new type of BKM Lie algebras that arise for the CHL Z 5 and Z 6 orbifolds. The cycle shapes associated with the conjugacy classes of L 2 (11) A with orders 2, 3 and 6 appear in considering cases involving generalized moonshine associated with commuting pairs of elements.…”

Section: Discussionmentioning

confidence: 99%

Two moonshines for L2(11) but none for M12

Govindarajan

Samanta

2019

Nuclear Physics B

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In this paper, we revisit an earlier conjecture by one of us that related conjugacy classes of M 12 to Jacobi forms of weight one and index zero. We construct Jacobi forms for all conjugacy classes of M 12 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 M 12 : 2 ⊂ M 24 . We show that there exists no solution where the Jacobi forms (for order 4/8 elements of M 12 ) transform with phases under the appropriate level. In the absence of a moonshine for M 12 , we show that there exist moonshines for two distinct L 2 (11) sub-groups of the M 12 . We construct Siegel modular forms for all L 2 (11) conjugacy classes and show that each of them arises as the denominator formula for a distinct Borcherds-Kac-Moody Lie superalgebra.3. We address the non-existence of a moonshine for M 12 by providing two moonshines for L 2 (11) that arise as two distinct sub-groups of M 12 . Using this, we construct Siegel modular forms for all conjugacy classes using a product formula that arises naturally as a consequence of L 2 (11) moonshine. The modularity of this product is proven in two ways: (i) as an additive lift determined by the eta product, ηρ(τ ) -this works for all but three conjugacy classes (1 1 11 1 , 3 4 and 6 2 ), and (ii) as a product of rescaled Borcherds products -this works for all conjugacy classes.4. For all conjugacy classes of L 2 (11), we show the existence of Borcherds-Kac-Moody (BKM) Lie superalgebras for conjugacy classes of both the L 2 (11) by showing that the Siegel modular forms, ∆ρ k (Z), arise as their Weyl-Kac-Borcherds deominator identity. Figure 1 pictorially summarises the moonshines for L 2 (11).The plan of the manuscript is as follows. Following the introductory section, in section 2, we describe the conjecture 2.1 for M 12 moonshine. We find multiple solutions that satisfy all the constraints that are imposed. In proposition 2.4, we show that a stronger form of M 12 moonshine does not hold. We then show that there are unique solutions for two distinct L 2 (11) subgroups of M 12 . In section 3, we construct genus-two Siegel modular forms for all conjugacy classes for both

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Section: Discussionmentioning

confidence: 99%

Two moonshines for L2(11) but none for M12

Govindarajan

Samanta

2019

Nuclear Physics B

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“…These properties imply that the Siegel modular forms transform covariantly under the extended Weyl group. The proof of these properties is given, for instance, in [4,5,13].…”

Section: Properties Of the Siegel Modular Formsmentioning

confidence: 99%

“…It is known that for N = 1, 2, 3, 4, the genus two Siegel modular form ∆ k(N ) (Z) arises as the WKB denominator formula for an extension of the Kac-Moody Lie algebra g A (N ) by the addition of imaginary simple roots. For N = 1, 2, 3, 4, 6, it is known that the modular forms admit the following expansion [4,5]:…”

Section: Charactersmentioning

confidence: 99%

“…For N = 4, the Cartan matrix has to be obtained as a limit N → 4 leading to a nm = 2 − 4(n − m) 2 . The square of these modular forms appear as the generating function of the refined counting of quarter-BPS states in certain CHL Z N orbifolds of the heterotic string compactified on T 6 [4,5]. Further, the root lattices associated with these Cartan matrices are closely related to the walls of marginal stability where quarter BPS states decay [6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

“…For N = 6, let B(A (6) ) denote an as yet unknown Lie superalgebra whose WKB superdenominator formula is given by ∆ k(6) (Z). All five Siegel modular forms admit a product formula of the form [5] (see Eq. (2.16))…”

Section: Introductionmentioning

confidence: 99%

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$\widehat{sl(2)}$ decomposition of denominator formulae of some BKM Lie superalgebras

Govindarajan,

Shabbir,

Viswanath

2021

Preprint

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We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a sl(2) subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of sl(2) characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients of these vector-valued modular forms are related to multiplicities of roots appearing on the sum side of the Weyl-Kac-Borcherds denominator formulae.

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Mathieu moonshine and Siegel Modular Forms

Govindarajan

Samanta

2021

J. High Energ. Phys.

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A second-quantized version of Mathieu moonshine leads to product formulae for functions that are potentially genus-two Siegel Modular Forms analogous to the Igusa Cusp Form. The modularity of these functions do not follow in an obvious manner. For some conjugacy classes, but not all, they match known modular forms. In this paper, we express the product formulae for all conjugacy classes of M24 in terms of products of standard modular forms. This provides a new proof of their modularity.

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BKM Lie superalgebras from counting twisted CHL dyons – II

Cited by 10 publications

References 42 publications

Two moonshines for L2(11) but none for M12

Two moonshines for L2(11) but none for M12

$\widehat{sl(2)}$ decomposition of denominator formulae of some BKM Lie superalgebras

Mathieu moonshine and Siegel Modular Forms

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