Bosonic string theory lives in 26 dimensions, and it gives a conformal field theory where the field is a map from a Riemann surface to $\mathbb{R}^{24}$. The Leech lattice $L$ is an even unimodular lattice in $\mathbb{R}^{24}$. We can form a conformal field theory where the field is a map from a Riemann surface to the torus $T = \mathbb{R}^{24}/L$, and this theory almost has the Monster group as its symmetry group. In fact we need to go one step further and replace $T$ by the orbifold where we mod out by the involution of $T$ coming from the transformation $x \mapsto -x$ of $\mathbb{R}^{24}$. In this case Frenkel, Lepowsky and Meurman showed we get a conformal field theory, or more technically a vertex operator algebra, whose symmmetry group includes the Monster group.
There could be a supersymmetric analogue of this, and it's probably been studied. What group does that give?
More precisely:
Superstring theory lives in 10 dimensions, and it should give a conformal field theory where the field is a map from a Riemann surface to $\mathbb{R}^{8}$, or actually a super-vector space $V$ with $\mathbb{R}^8$ as its even part. The $\mathrm{E}_8$ lattice is an even unimodular lattice in $\mathbb{R}^8$. I suspect we should be able to form a form a conformal field theory where the field is a map from a Riemann surface to the 'supertorus' $T_\mathrm{super} = V/\mathbb{E}_8$. Is the symmetry group of the corresponding vertex operator algebra known? We may have to replace $T_\mathrm{super}$ by a super-orbifold, e.g. mod out by an involution, to get a really interesting group.
Best Answer
There is a super analog constructed just as you describe with the Conway group $Co_0$ replacing the Monster and commuting with the superconformal algebra. The construction is described in detail in:
and in John Duncan's paper: