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)!
 A: interesting questions. I am currently looking into understanding some of these questions about the connections between heterotic compactifications to 2D and moonshine. In particular from free fermionic CFT approach used in https://arxiv.org/abs/1610.04898 and https://arxiv.org/pdf/0901.3055.pdf, for example. In the former my supervisor and former student show how to write various Niemeier lattices in more familiar language of free fermions on torus CFT. In the latter paper 2D models with massive SUSY are found where you get the usual Klein modular function in antiholomorphic part of heterotic PF, suggestive of moonshine connection. Seems very possible that this kind of approach can probe some of moonshine connections, by looking at various different kind of heterotic 2D models- different orbifoldings, gauge groups...
I'm interested in what you've found out since posting this question. Sorry I haven't got onto understanding broader links of moonshine to other constructions in string theory really yet but am working through so would be happy to keep in touch.
