Two moonshines for L2(11) but none for M12

Abstract

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

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

“…The connection with Mathieu and L 2 (11) moonshine leads to a product formula given in Eq. (2.12), for the Siegel modular forms [3,8,9]. For the prime cases, it is consistent with the product formulae given by David et al [10] in the context of dyon counting.…”

“…In our previous work [1], we studied a family of Siegel modular forms that are associated with Umbral moonshine [2]. Here we consider Siegel modular forms that are associated with L 2 (11)-moonshine [3]. The squares of these Siegel modular forms are the generating function of quarter BPS states in CHL Z N orbifolds (for N = 1, 2, .…”

BKM Lie superalgebras from counting twisted CHL dyons – II

“…In [33], we have constructed several Lorentzian Kac-Moody Lie superalgebras -one for every conjugacy class of two inequivalent L 2 (11) subgroups of M 12 . The real simple roots for all of them have the same Cartan matrix, i.e., A (1) .…”

“…Since, we do not have an additive lift, we do not have an independent formula for the sum side. That does not preclude the existence of a BKM Lie superalgebra as was the case in some examples studied in [33].…”

“…These appear as rows 5 and 6 of Table 3. (1) It was shown by us in [33] that there exist inequivalent automorphic corrections to g(A (1) ) (associated with all conjugacy classes of two distinct L 2 (11) sub-groups of M 12 that we denote by L 2 (11) A and L 2 (11) B . In this paper, we focus only on the L 2 (11) B cases and these appear in row 1 of Table 3.…”