Correspondences among CFTs with different W-algebra symmetry

This is a review of the preprint by Creutzig, Genra, Hikida and Liu.

Context and general comments
This article is a follow-up of a 2015 article with practically the same title, which shares two authors with the present article. The 2015 article was the subject of a publicly available review, which is still relevant as an exposition of the motivations for investigating relations between correlation functions of two-dimensional CFTs with different symmetry algebras, since little progress in that direction has occured between 2015 and 2020.

The 2015 article found relations between CFTs with affine Lie algebra symmetries, and CFTs based on W-algebras whose generators had spins at most $$2$$. The upper bound on spins prevented making contact with CFTs such as the conformal Toda theory based on an algebra $$W_{N\geq 3}$$, which includes a generator of spin $$N$$. At the time, this seemed to be a fundamental limitation of the free field techniques. The 2020 article now overcomes this limitation by exploiting the existence of different free field realizations of the same CFT. These realizations were studied in previous work by one of the authors.

In the example of the relation between the affine Lie algebra $$\hat{\mathfrak{sl}}_3$$ and the W-algebra $$W_3$$, we need two different realizations of the Bershadsky-Polyakov algebra $$W_3^{(2)}$$, which can be viewed as an intermediate between $$\hat{\mathfrak{sl}}_3$$ and $$W_3$$:

\begin{align} \hat{\mathfrak{sl}}_3 \quad \overset{\text{2015 article}}{\longrightarrow} \quad & W_3^{(2)} \text{ (first realization)} \\ & W_3^{(2)} \text{ (second realization)} \quad \underset{\text{this article}}{\longrightarrow} \quad W_3 \end{align} $$ This article stops short of studying the combined relation from $$\hat{\mathfrak{sl}}_3 $$ to $$W_3$$, and of translating their free field results to a more general language. Rather, they study the generalization to $$\hat{\mathfrak{sl}}_4 $$ and the related W-algebras. In that case, the relation between $$\hat{\mathfrak{sl}}_4 $$ and $$W_4$$ takes three steps, with two intermediate W-algebras. By the same logic, the relation between $$\hat{\mathfrak{sl}}_N $$ and $$W_N$$ would take $$N-1$$ steps. Appendix B explores that relation for particular correlation functions, which are such that only $$3$$ steps are needed for any value of $$N$$.

Validity
The article relies exclusively on the free field techniques which were introduced by Hikida and Schomerus in the prototypical case of the $$H_3^+$$-Liouville relation. These techniques are reliable, they seem to be applied correctly, and they lead to reasonable results.

Standard disclaimers about the limitations of free-field techniques in non-rational CFTs nevertheless apply. Fundamental works on correlation functions in Liouville theory and conformal Toda theory complement free-field calculations with other independent approaches.

Interest
This article overcomes a non-trivial technical problem in the free field approach to relations between correlation functions of non-rational CFTs. Potentially, this could lead to relations between correlation functions of WZNW models based on $$\hat{\mathfrak{sl}}_N$$, and correlation functions in conformal Toda theory. Such relations are so far known only in the case $$N=2$$, and there are doubts about their very existence for $$N\geq 3$$.

However, this article is narrowly focussed on the free field approach to the question, and does not study the combined relation from $$\hat{\mathfrak{sl}}_3 $$ to $$W_3$$. That relation would surely be complicated, and involve an integral transformation. Nevertheless, it should be possible to at least write the correlation functions that appear on the $$W_3$$ side, and to determine the representation-theoretic nature of the degenerate fields that appear alongside the generic fields. This could go some way towards showing that the proposed relation is compatible with the Knizhnik-Zamolodchikov equations.

Clarity
The article's style is generally clear, and the logic of the calculations is relatively easy to follow.

However, there are notational clashes, where the same notation can be used for several different objects. This is because the different realizations of the same algebra are not distinguished by different notations, so that for example the same notations $$\mathcal{D}_{H},\mathcal{D}_+,\mathcal{D}_-$$ mean different things in Eqs. (2.19) and (2.24), same problem with $$|j,s;m\rangle$$ in Eqs. (2.25) and (2.28), and with $$\Phi_{j,s;m}(z)$$ in Eqs. (2.30) and (2.34).

Moreover, the calculations do not come with narratives of their salient features, with hints of what parts are straightforward or tricky, or with discussions of what we learn from the results. To give a few example:
 * 1) When checking that the Bershadsky-Polyakov algebra generators (2.11) obey the OPEs (2.1), it is not pointed out that the last term in $$G^-$$ is responsible for the leading divergence in $$G^+G^-$$. This is actually made obscure by using different notations for the coefficient of the term and of the divergence.
 * 2) It is pointed out that the conformal weight (2.18) is common to both free field realizations, but it is not said whether this is a consistency requirement or a happy coincidence.
 * 3) What was an insurmountable obstacle in the 2015 article is now an apparently straightforward calculation in Section 3.2. It would be interesting to say a bit more than "Fortunately, we have the second realization".
 * 4) The result of the calculation in Section 3.3 is not commented, in particular not compared with the more general result of Section 3.1.

Recommended major changes

 * 1) Formulate the results in a manner that depends as little as possible on the free field approach. In particular, characterize the $$\hat{\mathfrak{sl}}_N$$ vertex operators by their spins and conformal dimensions, and the W-algebra vertex operators by their charges. Indicate which operators have null vectors.
 * 2) Explain what can be learned from the results and from the calculations: which features are expected or surprising, which steps are difficult.

Recommended minor changes

 * 1) Number all formulas.
 * 2) A few commas are missing in the introduction, after "true matching of correlation functions" and "Mathematically".
 * 3) Clumsy sentence "One of the differential operator..."
 * 4) Clumsy paragraph "For the first realization, we found the expression..."
 * 5) "in a way slightly different from [30]": what is the difference and why? See also footnote 3.
 * 6) "we propose several ways to obtain direct relations": too vague statement that covers very different ways.
 * 7) State somewhere that the correlation functions are on the sphere.
 * 8) "We can see that the action describes...": not so clear, as the notations are not the same cf. $$\phi$$ vs $$\varphi$$.
 * 9) "As was done in the previous subsection, we can reduce...": do it!
 * 10) In the conclusion: "the structure constants for $$\mathfrak{sl}(N)$$ Toda field theory have been computed": The cited references only compute special cases of the structure constants.