# 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?

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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.

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Do they satisfy the required specific "nice properties" to construct a heterotic string? –  Yuji Oct 5 '11 at 14:20
I'm not sure what these "nice properties" are, the only thing I can tell you in general is that their character will give you the Witten genus, this is related to your question on MO –  Reimundo Heluani Oct 5 '11 at 14:30
Perhaps I do not have enough privileges to comment on Lubos' answer, but definitely c=24k is not necessary: any C2-cofinite vertex algebra gives rise to modular invariance of the character (cf. Zhu) As for Lubos claim that those are the only modular-invariant CFTs of central charge 8 that is also not true, any rank 8 Dynkin lattice will give rise to modular invariant characters and Chiral differential operators as I mentioned give rise to the Witten genus which also has modular properties. –  Reimundo Heluani Oct 5 '11 at 22:25
Edit: I think I might have missunderstood Lubos' comment. If you want a modular function instead of a modular form then you want a vertex algebra with only one representation. In that case I think there is such a classification. –  Reimundo Heluani Oct 5 '11 at 23:10
The reference Yuji's asking in Lubos' answer is. "Holomorphic vertex operator algebras of small central charge" Dong and Mason. Pacific Journal of Mathematics. Vol 213 (2) 2004. –  Reimundo Heluani Oct 6 '11 at 10:52

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.

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Don't you need c=24k for modular invariance? –  Yuji Oct 5 '11 at 18:11
And could you suggest me where it is shown that these two solutions are unique? That was my question. –  Yuji Oct 6 '11 at 9:16
Hi @Yuji: Let me just copy Reimundo's reference. The paper is fully available here: pjm.berkeley.edu/pjm/2004/213-2/pjm-v213-n2-p05-s.pdf –  Luboš Motl Oct 9 '11 at 7:01
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