*This is a commentary of the recent article by Creutzig, Hikida and Ronne, which I was asked to review for the Journal of High-Energy Physics. I am grateful to the authors for helpful correspondence.*

It has been known for a long time that $W$-algebras can be obtained from affine Lie algebras by Drinfeld-Sokolov reduction. The reduction eliminates a number of generators of the affine Lie algebra, leaving a $W$-algebra with fewer generators (but more complicated relations). The reductions of algebras are most useful when they can be promoted into relations between correlation functions of conformal field theories. For example, the reduction from the $\widehat{\mathfrak{sl}}_2$ affine Lie algebra, to the Virasoro algebra, can be promoted into a relation between correlation functions of the $H_3^+$ model and Liouville theory, two CFTs whose symmetry algebras are $\widehat{\mathfrak{sl}}_2$ and Virasoro respectively.

Generalizing the $H_3^+$-Liouville relation to models with larger symmetry algebras could be helpful for understanding, or even solving, such models. In order to find such generalizations, there are two approaches:

- Studying differential equations that correlation functions obey, as in the original derivation of the $H_3^+$-Liouville relation. That derivation indeed used a relation between the Knizhnik-Zamolodchikov equations associated with the $\widehat{\mathfrak{sl}}_2$ algebra, and the Belavin-Polyakov-Zamolodchikov equations associated with the Virasoro algebra.
- Working with a functional integral representation of correlation functions, as was done by Hikida and Schomerus for the $H_3^+$-Liouville relation.

When unable to directly solve a given difficult problem, one can instead look for the widest possible generalization of known results. This is what the recent article is doing, by demonstrating a vast generalization of the $H_3^+$-Liouville relation using the functional integral approach. This results in the very simple rule that the approach works whenever the relevant $W$-algebra has generators with spins at most two. This criterion excludes the $W_{N\geq 3}$ algebras, but includes large classes of other $W$-algebras, which I am now going to sketch.

**A zoo of $W$-algebras**

The relevant reductions of the affine Lie algebra $\widehat{\mathfrak{sl}}_N$ correspond to embeddings of $\mathfrak{sl}_2$ into $\mathfrak{sl}_N$. There are several inequivalent such embeddings, which can be parametrized by the corresponding decomposition of the fundamental representation $F_N$ of $\mathfrak{sl}_N$, into representations of $\mathfrak{sl}_2$.

In the case $N=3$, there are three inequivalent decompositions of the three-dimensional fundamental representation,

\[

F_3 = \underline{3} \quad , \quad F_3 = \underline{2} + \underline{1} \quad , \quad F_3 = \underline{1} + \underline{1} + \underline{1}

\]

Here $\underline{n}$ is the $n$-dimensional representation of $\mathfrak{sl}_2$, whose spin is $j=\frac{n-1}{2}$.

In order to find the three corresponding $W$-algebras, the rule is to derive the decomposition of the adjoint representation of $\mathfrak{sl}_N$,

\[

A_N = F_N \times F_N - \underline{1}

\]

In our example,

\[

A_3 = \underline{5} + \underline{3} \quad , \quad A_3 = \underline{3} + 2\cdot \underline{2} + \underline{1} \quad , \quad A_3 = 8 \cdot \underline{1}

\]

Then, for any representation $\underline{n}$ of $\mathfrak{sl}_2$ that appears in $A_N$, there is a $W$-algebra generator $W^s$ with spin $s=\frac{n+1}{2}$, and we find the $W$ algebras

\[

W_3 = (W^3, W^2) \quad , \quad W_3^{(2)} = (W^2, 2\cdot W^{\frac32}, W^1) \quad , \quad \widehat{\mathfrak{sl}}_3 = (8\cdot W^1)

\]

So we find not only the algebra $W_3$ with its two generators, but also the Bershadsky-Polyakov algebra $W_3^{(2)}$ with its four generators, and finally the affine Lie algebra $\widehat{\mathfrak{sl}}_3$ itself, in the case when the reduction does actually nothing. In particular, $W_3^{(2)}$ is the simplest example of a large class of $W$-algebras that are intermediates between $W_N$ and $\widehat{\mathfrak{sl}}_N$. The numbers of generators of these algebras are intermediates between $N-1$ and $N^2-1$, and the spins of these generators are intermediates between the spin $N$ of the highest spin generator of $W_N$, and the spin $1$ of the generators of $\widehat{\mathfrak{sl}}_N$. In particular, the presence of generators of spin $1$ corresponds to the survival of part of the global subalgebra $\mathfrak{sl}_N\subset \widehat{\mathfrak{sl}}_N$.

Given the constraint that the spins of $W$-algebra generators do not exceed $2$, the cases that are considered in the recent article therefore correspond to decompositions of the type

\[

F_N = p\cdot \underline{2} + (N-2p) \cdot \underline{1} \qquad (2p\leq N)

\]

This leads to a $W$-algebra with $2p(N-p)$ fewer generators than the original affine Lie algebra. At best, the number of generators is therefore about halved. This works similarly for the affine superalgebras $\widehat{\mathfrak{sl}}_{N|P}$.

**So which CFTs are you now inventing?**

Now that we have an idea of the relevant algebras, let us see which theories appear in the generalizations of the $H_3^+$-Liouville relation. To begin with, which generalization of $H_3^+$ has an $\widehat{\mathfrak{sl}}_N$ symmetry algebra? Obviously not a Wess-Zumino-Witten model: such models have complicated spectrums, involving not only continuous but also discrete and/or spectrally flowed representations. On the other hand, the $H_3^+$ models, and its generalizations in the recent article, have purely continuous spectrums made of affine highest-weight representations. We should not be misled by the use of the WZW action in the functional integral formalism: this action is evaluated not on a group element $g\in SL_N(\mathbb{R})$ as in the $SL_N(\mathbb{R})$ WZW model, but rather on the Hermitian combination $gg^\dagger$. This means that we are dealing with Gawedzki's $SL_N(\mathbb{C})/SU_N$ coset model, which reduces to the $H_3^+$ model in the case $N=2$, and which is sometimes confusingly called a WZW or gauged WZW model.

On the $W$-algebra side of the relation, the zoo of possible $W$-algebras must correspond to a zoo of possible theories, that are intermediates between the $SL_N(\mathbb{C})/SU_N$ coset model, and the conformal $\mathfrak{sl}_N$ Toda theory. The Toda theory in question is the generalization of Liouville theory that would correspond to the decomposition $F_N= \underline{N}$. The intermediate theories seem to be new, what do we know about them beyond their symmetry algebras? The recent article's calculations lead to Lagrangian descriptions, that are rather more complicated than either WZW models' or Toda theories' Lagrangian descriptions. Moreover, it seems reasonable to conjecture that the spectrums of the intermediate theories are purely continuous, and made of highest-weight representations of the relevant $W$-algebras. Moreover, these spectrums are likely to be non-unitary, as removing the non-unitarity of the $SL_N(\mathbb{C})/SU_N$ coset model presumably requires the full reduction to $\mathfrak{sl}_N$ Toda theory.

**What about the differential equations?**

The affine $\widehat{\mathfrak{sl}}_N$ symmetry implies that an $n$-point correlation functions obeys $n$ Knizhnik-Zamolodchikov differential equations. If this $n$-point function is rewritten in terms of a correlation function in another theory, the corresponding differential equations must have an interpretation in terms of the $W$-algebra symmetry of that other theory. Investigating this issue is not the aim of the recent article, which used the alternative approach of computing functional integrals. Still, it should not be complicated to see whether we have $n$ differential equations, even if computing them explicitly would be tedious. The point is that in a theory with a $W$-algebra symmetry algebra, a generic correlation function does not necessarily obey any differential equation, apart from the three equations of global conformal symmetry. In order to have more differential equations, we need some fields to be degenerate. Not surprisingly, the correlation functions that are found in the recent article involve some fields with very specific parameters, that therefore have very good chances of being degenerate. It should be easy to check that they are indeed degenerate, and provide the expected number of differential equations.

**Conclusion**

Along with the authors of the recent article, one may hope that the $\widehat{\mathfrak{sl}}_3 \to W_3^{(2)}$ reduction is an important step towards an $\widehat{\mathfrak{sl}}_3\to W_3$ reduction at the level of correlation functions. After all, the remaining task is the $W_3^{(2)}\to W_3$ reduction, which involves eliminating only two generators. However, I fear that this is more difficult, if not impossible. Consider the spins of the relevant $W$-algebra generators in the $\widehat{\mathfrak{sl}}_N$ case: we now know how to replace some fields of spin $1$ with other fields of spin $2$ or less, but the $W_N$ algebra has generators with spins up to $N$. Actually, the main problem might be to go beyond spin $2$, as the corresponding $W$-algebras are no longer Lie algebras (with linear commutators), but associative algebras (with quadratic commutators).

So the main interest of these results may not be how close they come to the full reduction, but rather how widely applicable they are. They provide many non-trivial relations between large families of CFTs, and most of these CFTs were not known before.