# Level-rank duality in WZW models and CS theories

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I know that the classical level-rank duality in the $\widehat{\mathfrak{sl}}(r)_l$ WZW model states that the space of conformal blocks of $\widehat{\mathfrak{sl}}(r)_l$ is isomorphic to that of $\widehat{\mathfrak{sl}}(l)_r$, with $r,l>0$. This has been shown from a physical point of view here: https://www.sciencedirect.com/science/article/pii/055032139090380V and also proved by mathematicians in this article: https://projecteuclid.org/euclid.cmp/1104249321. It has been also shown that this level-rank follows from a "strange duality" (the Beauville-Donagi-Tu conjecture, no longer a conjecture, by the way).

By the WZW/CS connection, the corresponding $d=3$ topological CS theory ($\mathsf{SU}(N)$ at level $k$) also enjoys this duality, which I expect to be: $\mathsf{SU}(r)_l\leftrightarrow \mathsf{SU}(l)_r$ (or $\mathsf{SU}(r)_l\leftrightarrow \mathsf{U}(l)_r$).

Now, the duality in the WZW model follows from the conformal embedding

\begin{equation}\widehat{\mathfrak{sl}}(r)_l\oplus \widehat{\mathfrak{sl}}(l)_r\oplus\widehat{\mathfrak{u}}(1)\subset \widehat{\mathfrak{gl}}(lr)_1\end{equation}

which means that the central charge of $\widehat{\mathfrak{sl}}(r)_l\oplus \widehat{\mathfrak{sl}}(l)_r$ is equal to that of $\widehat{\mathfrak{sl}}(lr)_1$.

In the last few years physicists became interested with level-rank dualities in connection with CS theories with matter, for example here: https://arxiv.org/abs/1607.07457

What I don't understand is why they write the above duality for topological CS theories with a level $-r$ on the right-hand side, namely

\begin{equation} \mathsf{SU}(r)_l\leftrightarrow \mathsf{U}(l)_{-r}\end{equation}

where $\mathsf{U}(r)_l=\frac{\mathsf{U}(1)_{lr}\times\mathsf{SU}(r)_l}{\mathbb{Z}_r}$.

My questions are:

1. What's the meaning of that minus sign (apart from putting a minus in the Lagrangian, of course) and where does it come from (since there's no sign of it in the WZW level-rank)?
2. How is the CS theory with a negative level related with the corresponding WZW model (for example if we just substitute $-r$ in the central charge of $\widehat{\mathfrak{sl}}(r)_{-l}$, then the above embedding is no more a conformal embedding)?
3. Why $\mathsf{SU}(r)_l\leftrightarrow \mathsf{U}(l)_r$ is not a good CS level-rank duality?

Moreover, from the CFT point of view the $\widehat{\mathfrak{u}}(1)_k$ is not really a WZW model, in fact there is no unambiguous notion of level, since by rescaling the generators of its current algebra we can change the level of the "would-be" model at our will. Thus, the level-rank $\mathsf{U}(1)_2\leftrightarrow\mathsf{U}(1)_{-2}$ seems something very "formal" to me, since all the $\widehat{\mathfrak{u}}(1)$ Heisenberg algebras are isomorphic, independently of the value of $Z$ in

.\begin{equation}[J_m,J_n]=Zm\delta_{m+n,0}.\end{equation}

Thanks

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