# Level-rank duality in WZW models and CS theories

Cross-posting from Physics Overflow: https://www.physicsoverflow.org/41281/level-rank-duality-in-wzw-models-and-cs-theories

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}

Any ideas?

• Good question, let's see what the experts here have to say. It seems to me that the sign is merely conventional (cf. "bare" vs. "renormalised" level). After all, you can flip the sign by means of a time-reversal transformation. – AccidentalFourierTransform Jun 1 '18 at 22:16
• I know that some dual pairs (e.g. Seiberg-like dual theories in $d=3,\mathcal{N}=2$) flow in the low-energy limit to $\mathsf{SU}(N)_l\leftrightarrow\mathsf{U}(l)_{-N}$, after an appropriate shift of the levels. I'd like to understand why a negative level on the right side seems more "natural" in $d=3$. – green.onion Jun 3 '18 at 14:45