# What's the supersymmetric analogue of the Monster group?

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.

• Some remark: in the monster story, the even unimodularity condition guarantees the modularity of the holomorphic conformal field theory obtained by compactifying holomorphic bosons over the torus. To have a modular holomorphic conformal field theory, the central charge has to be divisible by 8 (for exemple holomorphic bosons over the E_8 torus define an holomorphic conformal field theory: the affine E_8 current algebra at level 1). Conformal field theory on a supertorus of dimension 8 has central charge: 8+4=12 and so cannot give rise to a modular holomorphic conformal field theory. – user25309 May 20 '16 at 20:27
• The theorem you cite was not Borcherds, but Frenkel-Lepowsky-Meurman. Borcherds's role was (a) defining vertex algebras and (b) showing that the moonshine module constructed by Frenkel-Lepowsky-Meurman satisfied the Conway-Norton Monstrous Moonshine conjectures. – S. Carnahan May 21 '16 at 6:25
• @user25309 You'll never get full $SL_2(\mathbb{Z})$-invariance from a supersymmetric theory, since odd states restrict translation symmetry to $\tau \mapsto \tau + 2$. However, even the free fermions of central charge $1/2$ satisfy a property analogous to holomorphicity, namely that the representation category is equivalent to vector spaces. – S. Carnahan May 21 '16 at 15:38
• I fixed the reference to Borcherds - thanks! – John Baez May 23 '16 at 5:07

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:

• Nora Ganter's work on categorifying Co_0 is surely relevant to this somehow. – David Roberts May 21 '16 at 1:53

The paper Beauty and the Beast (open access) shows that the Moonshine module contains a copy of the super-Virasoro algebra, and so in some sense is already supersymmetric. I don't know how to interpret it in terms of a susy string, however.

• The super-Virasoro algebra is not contained in the Moonshine module, but in a larger structure whose $\mathbb{Z}/2$-fixed points form the Moonshine module. That object only has symmetries given by an extraspecial extension of Conway, rather than the (substantially larger) monster. – S. Carnahan May 21 '16 at 6:21
• @S.Carnahan Thank you for the correction! I only briefly skimmed that paper. – Theo Johnson-Freyd May 23 '16 at 14:52

This may be of interest:

Monstrous BPS-Algebras and the Superstring Origin of Moonshine
by Natalie M. Paquette, Daniel Persson, Roberto Volpato

http://arxiv.org/abs/1601.05412

• Also the followup: "3D String Theory and Umbral Moonshine" arxiv.org/abs/1603.07330 – Urs Schreiber May 23 '16 at 15:52
• I see the Paquette et al paper mentions heterotic string theory. Indeed my next question was going to be about "heterotic Moonshine" where we treat the bosonic left-movers using the Leech lattice and the supersymmetric right-movers using the E8 lattice, and then (I guess) do a Z/2 orbifold. What symmetry group does this theory have? I guess the answer should at least include the symmetries of the two theories separately, which seem to be the Monster group and Conway's $\mathrm{Co}_0$ group. – John Baez May 23 '16 at 21:04
• This is correct except you should do a $Z/2 \times Z/2$ orbifold with the first factor acting on the left to give the Monster VOA and the second factor acting on the right to give the Conway Super VOA. – Jeff Harvey May 24 '16 at 1:25
• Jeff Harvey - is there a place to read about that construction? – John Baez May 24 '16 at 5:13
• JohnBaez-it is discussed in the above paper, but the construction is pretty obvious. The Monster VOA is constructed as a $Z/2$ asymmetric orbifold of the bosonic string on the Leech lattice and the Conway super VOA as an asymmetric $Z/2$ orbifold of the $E_8$ theory and given the known cancellation of modular anomalies in the heterotic string it is obvious you can take the tensor product to construct a consistent heterotic background with Monster x Conway symmetry. – Jeff Harvey May 24 '16 at 11:27