# Understanding Moonshine via Heterotic $\rm E_8\times E_8$ (resubmitted, originally asked on mathematics stack exchange)

Recently I have become familiar with the conjectured relationship of monstrous moonshine and pure (2+1)-dimensional quantum gravity in AdS with maximally negative cosmological constant and, it’s being dual to c=24 (if I recall) homomorphic conformal field theory w/ partition function proportional to the graded character of the module (Frenkel-Lepowsky-Meurman, Li-Song-Strominger, Witten, Etc.). Now, I assume the entirety of THIS work was built on leech, or at least a general niemeier (I’ve seen reference to the work of Eguchi, Ooguri, Tachikawa on Mathieu Moonshine, regarding elliptic genus on K3 and it’s split into the character of either N=4,4 or N=2 super conformal algebras if I recall correctly). I assume the statement this would make is that there exists a cft on K3 target with m24? In both these cases (leech or general niemeier) I would appreciate reference to literature on the topic w/ regards to Umbral Moonshine as an unrelated reading.

Now, as someone who is less acquainted with the mathematics than I am the physics, I was curious if E8xE8 heterotic (or general heterotic) could be used to study monstrous moonshine specifically (likely with respect to K3 target?).

Assuming the strongest condition is that the worldsheet CFT(?) must be built on an even, self-dual lattice of dimension 8k (in order to satisfy a modular invariant partition function), k=2 has of particular interest E8xE8, and in the case k=3 there is leech! Assuming one could split the theta function of leech into three k=1 E8 roots (w/o damaging winding in the lattice), and the same for k=2 does there not exist a direct method of studying the monster in the context of worksheet CFTs on heterotic strings?

To finalize the question, are there properties or heterotic E8xE8, either with worldsheet conformal field theories or heterotic branes on K3 that are appealing in the context of monster or even the smaller sporadic simple groups? (I have seen reference to heterotic five branes on K3 w/ massive U(1) and was pondering the topic)

I appreciate any responses!

Edit: To refine this question, would a worldsheet c24 homomorphic conformal field theory satisfy Strominger for E8E8 heterotic? If so, what is the virasoro primary fields interpretation in heterotic? I assume if I was to calculate, in the large mass limit, the logarithm of the virasoro primary of the module it would agree well with radiative terms in heterotic (supergravity action w/ gauss bonnet, assuming curvature terms exist at tree-level heterotic)!