Second-quantized Mathieu moonshine

Abstract

We study the second quantized version of the twisted twining genera of generalized Mathieu moonshine, and prove that they give rise to Siegel modular forms with infinite product representations. Most of these forms are expected to have an interpretation as twisted partition functions counting 1/4 BPS dyons in type II superstring theory on K3×T 2 or in heterotic CHL-models. We show that all these Siegel modular forms, independently of their possible physical interpretation, satisfy an "S-duality" transformation and a "wall-crossing formula". The latter reproduces all the eta-products of an older version of generalized Mathieu moonshine proposed by Mason in the '90s. Surprisingly, some of the Siegel modular forms we find coincide with the multiplicative (Borcherds) lifts of Jacobi forms in umbral moonshine. arXiv:1312.0622v3 [hep-th] 11 Nov 2014Contents * The terminology "second-quantized" originates in the work of Dijkgraaf, Moore, Verlinde, Verlinde [23]. * * We denote by the same symbolρ g,r also the projective representation of C M24 (g) on H g r withρ g,r (h) := ρ g r (h)/f g,r (h)

On symmetries of N $$ \mathcal{N} $$ = (4, 4) sigma models on T 4

“…In the context of the Mathieu moonshine phenomenon, the analysis of the present paper can lead to a generalization of the results of [39]. In this work, some Siegel modular forms were constructed as the multiplicative lifts of the twisted-twining genera of generalized Mathieu moonshine [11].…”

“…Many of these modular forms admit an interpretation as partition functions for 1/4 BPS states in four dimensional CHL models with 16 space-time supersymmetries. In [39], it was observed that there exists a particular modular transformation, induced by electric-magnetic duality in the four dimensional CHL model, that exchanges the multiplicative lifts of twining genera in two distinct K3 models, related by an orbifold. An obvious generalization of the Siegel modular forms of [39] can be obtained by taking the multiplicative lifts of twining genera in torus models, such as the ones considered in section 5 of the present paper.…”

“…In [39], it was observed that there exists a particular modular transformation, induced by electric-magnetic duality in the four dimensional CHL model, that exchanges the multiplicative lifts of twining genera in two distinct K3 models, related by an orbifold. An obvious generalization of the Siegel modular forms of [39] can be obtained by taking the multiplicative lifts of twining genera in torus models, such as the ones considered in section 5 of the present paper. In this case, one expects the 'electric-magnetic duality' to relate these forms to the multiplicative lifts of the twining genera in exceptional K3 models, i.e.…”

Fricke S-duality in CHL models

“…In recent years a new moonshine phenomenon has been discovered, dubbed Mathieu moonshine, which relates the representation theory of the Mathieu group M 24 with weak Jacobi forms and superstring theory on K3-surfaces [18,22,[35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53]. 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).…”

“…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…”