research level - Uniqueness of supersymmetric heterotic string theory

Usually we say there are two types of heterotic strings, namely $E_8\times E_8$ and $Spin(32)/\mathbb{Z}_2$. (Let's forget about non-supersymmetric heterotic strings for now.)

The standard argument goes as follows.

1. 
   
   

   To have a supersymmetric heterotic string theory in 10d, you need to use a chiral CFT with central charge 16, such that its character $Z$ satisfies two conditions:

   
   
   
   
   1. $Z(-1/\tau)=Z(\tau)$
   
   
   2. $Z(\tau+1)=\exp(2\pi i/3) Z(\tau)$
   
   
   
   


2. 
   

   Such a chiral CFT, if we use the lattice construction, needs an even self-dual lattice of rank 16.

   
   


3. There are only two such lattices, corresponding to the two already mentioned above.

We can replace the lattice construction with free fermion construction, and we still get the same result. But mathematically speaking, there might still be a chiral CFT of central charge 16, with the correct property, right? Is it studied anywhere?

Answer

There are plenty of chiral CFTs with central charge 16 and nice properties studied in the mathematics literature. A nice example in this context would be chiral differential operators on a 8-manifold. If you want modularity of the character so that you want a holomorphic vertex algebra then the reference is

"Holomorphic vertex operator algebras of small central charge" Dong and Mason. Pacific Journal of Mathematics. Vol 213 (2) 2004.

as discussed in the comments and in Lubos' answer.