# 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?

• You mean rank 16. – user320 Oct 5 '11 at 23:23
• Yes I'm sorry. I updated it. – Yuji Oct 6 '11 at 9:12

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.

I think that the two solutions are the only modular-invariant chiral CFTs with the right central charge. They have the right transformation law under $\tau\to\tau+1$ and especially (and less trivially) $\tau\to-1/\tau$ where $\tau$ is the complex structure of the world sheet torus. That's needed for a consistent path integral interpretation of the histories and for unitarity when used as a portion of string theory.
• Concerning $c=24k$, yes, it's needed, but one must be careful what $c$ is. It's the $c$ of the total theory which is really 0 in the full world sheet theory, once the ghosts are added. Note that $bc$ ghosts have $c=-26$ and the $bc$ plus $\beta\gamma$ superconformal ghosts have $c=-15$, cancelling 10 bosons plus 10 fermions (equivalent to 15 bosons' $c$). In the light cone gauge, the counting is different: 24 bosons in the bosonic string is OK but 8 bosons plus 8 fermions in the superstring is also OK - $c=12$ sort of. – Luboš Motl Oct 9 '11 at 7:04