K3 Elliptic Genus and an Umbral Moonshine Module

Abstract

Umbral moonshine connects the symmetry groups of the 23 Niemeier lattices with 23 sets of distinguished mock modular forms. The 23 cases of umbral moonshine have a uniform relation to symmetries of K3 string theories. Moreover, a supersymmetric vertex operator algebra with Conway sporadic symmetry also enjoys a close relation to the K3 elliptic genus. Inspired by the above two relations between moonshine and K3 string theory, we construct a chiral CFT by orbifolding the free theory of 24 chiral fermions and two pairs of fermionic and bosonic ghosts.In this paper we mainly focus on the case of umbral moonshine corresponding to the Niemeier lattice with root system given by 6 copies of D 4 root system. This CFT then leads to the construction of an infinite-dimensional graded module for the umbral group G D ⊕6 4 whose graded characters coincide with the umbral moonshine functions. We also comment on how one can recover all umbral moonshine functions corresponding to the Niemeier root systems A ⊕4 5 D 4 , A ⊕2 7 D ⊕2 5 , A 11 D 7 E 6 , A 17 E 7 , and D 10 E ⊕2

Generalised Umbral Moonshine

“…g,1 for r = ±1 mod 4 and H (2) g,r = 0 for r = 0 mod 2, and the above conjecture recovers that of Mathieu moonshine.…”

“…The above conjecture was proven in [32,44], in the sense that the existence of the module K ( ) has been established using properties of (mock) modular forms. However, among the 23 cases of umbral moonshine, modules have only been constructed for the eight simpler cases [2,16,33,34]. A uniform construction of the umbral moonshine modules is to the best of our knowledge not yet in sight and is expected to be the key to a true understanding of this new moonshine phenomenon.…”

On Weierstrass mock modular forms and a dimension formula for certain vertex operator algebras

“…A first special case of this, now called Baby Monster moonshine was established earlier by Höhn [40]. Other instances of moonshine related to VOAs include Conway moonshine [25] and various instances of umbral moonshine [3,15,28,24].…”

TASI Lectures on Moonshine

“…However, the construction, or even an understanding of the exact nature of K X , is not yet obtained in general. Construction of K X has so far only been achieved for certain particularly simple cases of umbral moonshine, corresponding to Niemeier root systems 3E 8 [94], 4A 6 and 2A 12 [95], 4D 6 , 3D 8 , 2D 12 and D 24 [96], as well as 6D 4 [97]. The construction in [94] relies on special identities satisfied by the mock modular forms H 3E 8 g,r relating it to a latticetype sum, while in [95,96] the modules are constructed using the interpretation of the meromorphic PoS(TASI2017)010…”

“…Miranda C. N. Cheng Jacobi forms associated to Ψ X g as the twined partition function of certain vertex operator algebras (or chiral CFTs). In [97] the module for the 6D 4 case of umbral moonshine is constructed by exploiting the relation between the (twined) K3 elliptic genus, umbral and Conway moonshine, which we will explain in §8.2. Note that, so far, this is the only constructed module for which the corresponding umbral group (when embedded in Co 0 ) does not fix a 4-plane in the 24-dimensional representation of Co 0 .…”