The geometry of Casimir W-algebras

1. The paper is an interesting attempt to provide a geometric framework for new ways to consider W-algebra conformal blocks and correlation functions

2. It is likely to generate further work to understand this area

1. There are several places where the arguments are not clear or explained well.

This looks an interesting paper, I think it is likely to stimulate further work in this active area, but there are several places where I could not follow the arguments or where they were unexpected and not explained. I think that if these were addressed it would be considerably improved, from my point of view. I'll make a list of places where I thought it was not clear in the detailed changes, but two points, one possibly important, one just historical, I would like to mention here.

It is from one point of view very surprising that, as in reference [3], the conformal blocks depend on so few parameters. From simple W-algebra identities one would expect a conformal block to have an infinite number of parameters which could be reduced if the fields involved were suitably degenerate, but this does not appear to happen here. I would have expected a comment on this fact - if it is well understood by now, then a reference would have been good.

I was a little surprised that the authors decided to choose the name "Casimir W-algebra" for the specialisation of what is already known as a Casimir W-algebra to the value of $c$=rank$\mathfrak g$ - perhaps they were unaware that this term had already been used, e.g. in arXiv:hep-th/9404113

These are in order of occurrence in the paper. Some requests are just that the authors consider changing their presentation, others are really requests for more clarity that I think are needed (from my point of view)

1. Section 2.3, there is no discussion of the choice of $C_i$, whether a basis is chosen such that the fields $W_i$ are Virasoro primary or not. I could not easily see if this was needed or not in what follows - a comment on this would be helpful.

2. Section 3.1 - the name "Casimir W-algebra" is already used in the literature to describe the algebra ${\cal W}_c(\mathfrak g)$, I can see that the authors would like to give their particular restriction to $c=rank(\mathfrak g)$ a name, but perhaps they could choose one that is not already used, or at least refer to this so that readers who are familiar with the older literature do not get confused?

3. On page 7, it is not clear which representations of $\mathfrak g$ are allowed if $V_j$ are to be primary fields for $\mathfrak g$. If they are the unitary or integrable representations of $\mathfrak g$, then clearly this is too few; if they are more general, then they will not be representations of the vertex algebra corresponding to $\mathfrak g$ and so possibly some assumptions from CFT would fail. I know that the authors say that this is not what they plan to do, but it is still not clear what they are saying they are not going to do.

4. On page 7, I am not sure what the authors mean by "This diagram commutes only up to multiplication with scalar factors". If they mean that restricting the fields from $\mathfrak g$ to $\mathfrak h$, it is not the case that for higher $W_i$ fields the relation is just a scalar factor - the restriction of the tensor $C_i$ to the fields in $\mathfrak h$ which typically not be equal to the corresponding field when summed over all the fields in $\mathfrak g$, up to a factor. This can be easily seen since the requirement of primality is that the tensor is traceless, and the requirement that an invariant tensor of rank 4 for $sl(n)$ is traceless when summed over all indices does not lead to it being traceless when summed over only the Cartan indices. If they actually mean using the maps then the relations in $\cal I$ should mean that (since the the energy-momentum tensor is unique) the two expressions given are in fact the same (modulo $\cal I$), for example for $sl(2)$, with the relations in $\cal I$ being $J^\pm = \exp(\pm i\sqrt 2 \varphi)$, we get $ (J^+ J^- + J^- J^+) = 2 (i \partial\varphi)^2$, and so we end up with equality of the two expressions (modulo $\cal I$).

5. On page 8, equation (3.8) only fixes the OPEs of the fields in $\mathfrak h_j$, yet the sum in (3.7) appears to be over all $\mathfrak g$. Is the sum in (3.7) over only the CSA? If so, which CSA? This is rather confusing - since the tensor $C_i$ is group invariant, it should not matter to which CSA one restricts, but the restriction should be done after the choice of CSA. It appears in the paper that once can use the full expression, summing over $\mathfrak g$ in (3.7) and then restrict to any particular $\mathfrak h_j$ and still use the same expression. One has $W_i = \sum_{a \in\mathfrak g} C_{a_1,\ldots a_i} (J^{a_1} ( \ldots ( J^{a_m})..)
= \sum_{\alpha \in \mathfrak h} C'_{\alpha_1\ldots \alpha_i} (J^{\alpha_1} ( \ldots ( J^{\alpha_m})..)$ but $C_{\alpha_1\ldots \alpha_i} \neq A C'_{a_1,\ldots a_i} $ for some number $A$, in general, and so the relation between (3.7), (3.8) and (3.9) needs to be clarified.

5. On page 8, I did not easily see why $V_{j'}(z')$ could not be expressed using the same free boson as $V_j(z_j)$ if the intention is only to replicate the W-algebra primary fields. They may not be expressed as simple exponential, but if $A_j$ and $A_{j'}$ define different Cartan subalgebra, then could one not find the conjugation required to take $A_{j'}$ to be in the $\mathfrak h_j$ using the generators $\mathfrak g_0$ expressed in terms of the bosons in $\mathfrak h_j$ in and similarly conjugate the primary field? The resulting expression will be in terms of the free bosons of $\mathfrak h_j$ - the expressions may be ugly, but wouldn't $V_{j'}$ will still be expressed in terms of those free bosons?

6. On page 8, two lines above section 3.3, it says "These representations" - I was not sure which representations were being refered to .

7. This is surely my ignorance, but in section 3.3, why is only the leading behaviour as $\pi(X_1)\to \pi(X_2)$ needed? If $r(A_j)$ in (3.9) is large and negative, are not all the singular terms important?

8. On page 9, why does each field $V_j$ come with only $1/2 (dim(\mathfrak g) - rank(\mathfrak g))$ undetermined descendents? I would have said that for a W-algebra, all the descendents in the OPE were undetermined from the W-algebra commutation relations, it is only for the Virasoro algebra that they are determined by the descent relations (from BPZ). If instead the authors mean there are $1/2 (dim(\mathfrak g) - rank(\mathfrak g))$ undetermined parameters, which appears to be what they mean given the parameter counting in (3.12), why is this? I thought this was one of the surprising results of [3] which I took to be an assumption in this paper. If it is explained elsewhere why the number of parameters is fixed, a reference would be most helpful.

9. On page 11, it says "We conjecture is that".

1. The authors succeed in their goal to develop a generalisation to an arbitrary (simply-laced) Lie algebra of their earlier work (in the context of $gl_2$) on the interesting relationship between Fuchsian systems and conformal field theory, to explain the mathematics behind it, then to use it to calculate correlation functions that involve currents within certain $W(g)$ symmetric conformal field theories and discuss their properties. It seems to me this is a positive mathematical development that is likely to be elaborated further, possibly to evaluate correlation functions of primary fields, or to extend its scope.

1. The paper fails to cite properly the work of others in the sense that the data provided with the references is incomplete compared to standard conventions. For example, Ref 8 is referred to as '1997 book', while most other papers are referred to only by their arxiv numbers (the exception being a PhD thesis that does not have an arxiv number). According to the instructions for authors, the DOI numbers should also be included for published papers yet none are provided in this article.

While the paper is inspired by ideas from physics (since conformal field theories have their origins there), and is a continuation of earlier work by the authors and others, the focus is on the mathematics, developing ideas and techniques without any clear indication (it seems to me) concerning where they might be useful in a physical context. For this reason it is likely to be of interest to a relatively small group of mathematical physicists rather than more broadly to the readership of a journal devoted to physics. Nevertheless, as this is quite common, the ideas, techniques and results reported should be published as interesting steps within a larger program of work.

Improve the data provided alongside the references (ie add the publication data, journal etc, and also DOIs for the purposes of creating external links to journals).

Consider adding some explanation as to why the investigation ought to be of interest to physicists.