SO(32) heterotic String theory

[Master replies:]

I have been following along the problem on SO(32) st. in Zwiebach's "A first course in String theory" and my question concerns this problem. I have no problem with the mathematics of SO(32), at least not at the simple level Zwiebach discusses it, rather the "why" behind a particular concept of SO(32).
If one bosonic string (D-1=25) and one super string (D=10-1) are combined, only 10 spatatial dimensions match and 16 remain of the bosonic string. So one wrapes the extra 16 dimensions into 32 two dimensional planes*.
Why, if one is trying to get rid of the extra 16, would one create 32 wraped up dimensions?If one were to wrap the dimensions up wouldn't the result be 8 two dimensional planes?

*I doubt this word is suiting, as with some others( e.g. "wraped")

Thankyou for all help

 

[fzero]

I'm not sure what part of the book you're referring to, as the copy I have doesn't discuss any details about the heterotic string and I couldn't find an end of chapter problem on them. As you say, in the heterotic string, the, say, right-movers consist of 26 free bosons and we are to think of 16 of them as compact, forming a 16-torus, $T^{16}$. When there is enough symmetry we can think of this as a product of 8 copies of a 2-torus, which is what I think you are referring to. One connection with $SO(32)$ we can immediately see is that $T^{16} = U(1)^{16}$ is the maximal abelian subgroup (aka maximal torus) of $SO(32)$, which has rank 16. It turns out that if we want to see the maximal $SO(32)$ symmetry, we should really replace the 16 free bosons (central charge $c=16$) with 32 free fermions (also $c=16$). Then $SO(32)$ corresponds to an internal rotational symmetry acting on these 32 free fermions.

 

[Master replies:]

Thanks. Very helpfull indeed.
I have the (I think) second edition of Zwiebach and the problem should be found at the end of chapter 14 part 1 Basics.
Anyway that helped.

 

[Master replies:]

To explain my naive questioning, I am unexperienced on the field (simply some of T-duality, Superstring, bosonic string, Ads CFT corespondence) so here is yet another question:

Is there are unorientated version of SO (32) similar to type 1 string theory?

 

[fzero]

To explain my naive questioning, I am unexperienced on the field (simply some of T-duality, Superstring, bosonic string, Ads CFT corespondence) so here is yet another question:

Is there are unorientated version of SO (32) similar to type 1 string theory?

There's no unoriented version of the heterotic string because the left-moving and right-moving sectors are so different. Specifically, we can't make a consistent world-sheet parity projection on the extra right-moving bosons and left-moving fermions. The heterotic nature of the worldsheet theory also prevents Dirichlet boundary conditions so there are no D-branes for the heterotic theories.

The IIB theory has the same chirality of fermions in the right and left sectors, so you can use the worldsheet parity to truncate the theory in such a way that you recover an unoriented closed string theory. To get a consistent theory you actually have to couple this unoriented closed string to an unoriented open string with SO(32) Chan-Paton factors. The SO(32) is selected by anomaly cancellation between the open and closed sectors. So this is actually the Type I string theory.

When D-branes were discovered, it was realized that this procedure on the IIB string could be put into a stronger context. The partity operation could be represented by adding an orientifold 9-plane in 10D. In order to balance the Ramond-Ramond charge on the O9, we need to also add 16 D9-branes. The open strings between the D9-branes now furnish precisely the open string sector that we needed to cancel anomalies. This correspondence between the Type I string and the IIB orientifold was one of the earliest recognized dualities between what were previously thought to be different string theories.

 

[Master replies:]

I see. Is that mathematically justified ?

 

[fzero]

I see. Is that mathematically justified ?

Yes, so long as you admit the physical techniques that allow you to compute Hilbert space of states, charges, scattering amplitudes, etc. into your mathematical techniques. A reference for the IIB orientifold - Type I duality is http://arxiv.org/abs/hep-th/9510017 (look around eq (17)), which has references to earlier work, not on the arxiv, where the equivalence was essentially established. Some more complicated models in lower dimensions are discussed in http://arxiv.org/abs/hep-th/9604129 and http://arxiv.org/abs/hep-th/9604178. The equivalence is also discussed in sects. 10.6 and 13.2 of Polchinski's String Theory text (vol 2). A set of lectures on orientifolds is http://arxiv.org/abs/hep-th/9804208, but as far as I can tell, it primarily discusses the Type II - Heterotic part of the triangle in lower dimensions, but there is a bit of Type I mentioned toward the end.