# Is there a WZW string theory?

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This is a naive question. Is there a way to couple the WZW model to gravity to obtain a perturbatively consistent string theory?

So I thought about it some. I guess to have a string theory, we need a CFT with central charge $c=0$. For the WZW model

$c = \frac{k dim G}{k + n_G}$,

where $n_G$ is the Coxeter number of $G$. This $c$ is always positive, so the $WZW$ model must be coupled to gravity in some non-trivial way. Conformal field theories with negative central charges include bosonic and $N=1$ supersymmetric sigma models below the critical dimension. Thus, by judicious choice of target space $X$, we get a composite theory with target space $X \times G$. So far, the two theories are uncoupled. Maybe something interesting happens if I make the target space some non-trivial $G$ bundle over $X$?

retagged Apr 19, 2014

Look up coset models, you can make string theory on group manifolds, but the more interesting spaces for this purpose are cosets of groups, which are smaller manifolds, the dimensions of Lie groups go up fast. You get currents and stress tensors by Sugawara construction like in WZ models, it's an 80s method of finding vacua. (just a comment, proper answer later).

You maybe interested in Urs Schreiber's article on WZW SFT.

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Here is something quick, as I don't have time. I didn't reply because it wasn't clear to me what the original question really is. But here are some quick comments.

Of course the WZW model famously exists and describes strings propagating on a spacetime which is a group manifold. Via the FRS theorem and related facts the WZW model is one of the mathematically best understood string models. As such it has received a great deal of attention.

Of course it is not critical and needs to be combined with more stuff to make a critical string background. I guess that's what the question is wondering about, and the answer is: sure! That's what one considers all the time.

The only trouble is that KK-compactification on group manifolds is not quite realistic.

On the other hand, the chiral half of the WZW model, also known as "current algebra", appears all over the place in realistic models: namely as such the WZW model appears notably in the heterotic string -- and hence all over the place in string theory. See also at parameterized WZW model for a geometric interpretation of the Green-Schwarz anomaly cancellation of the heterotic string in terms of "bundles of WZW models". The discussion there is the most general realistic answer to how WZW models and quantum gravity appear jointly. (I should expand on that, but not right now.)

Another story is if we allow ourselves to pass to WZW models for "higher Lie groups", hence for the global objects which Lie integrate Lie n-algebras . Doing so one finds that all the Green-Schwarz action functionals for superstrings and super-p-branes are WZW models --- on super-spacetime. One can classify all these super-Lie-n algebra WZW models and finds... the full brane scan structure of M-theory:

On the other hand WZW-type superstring field theory is something different. This is not about strings propagating on group manifolds (well it is about strings generally, so it will also include strings on group manifolds) but is instead about a way of formulating "string field theory" (second quantized strings) in a form that exhibits a kind of "second quantized WZW term".

Actually string field theory maybe more naturall involves a "second quantized Chern-Simons term", namely a Chern-Simons term for a Lie n-algebra as $n\to \infty$. (Everything is higher Chern-Simons, that flows right out of the foundations of mathematics...)

answered Apr 15, 2014 by (6,095 points)
edited Apr 15, 2014

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Urs Schreiber has written a (currently stub) article on the $n$CatLab about "WZW-type superstring field theory" here.

The article links to this and this article for closed and open superstrings respectively. I haven't read the papers yet, but they seem interesting. I may expand this answer once I do read the papers, if I understand enough out of it to write an answer about : )

I hope @UrsSchreiber, if he finds time, may want to write an answer, since he's started to write an nLab article on it.

answered Apr 15, 2014 by (1,975 points)

Urs Schreiber explained why this is unrelated.

@RonMaimon Yes you're right, I'm not sure if I should hide this answer?

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