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. See the report for details

1.
I find it hard to agree with the counting of parameters in W-algebra conformal blocks.
The authors give two examples where the counting ${\cal N}_{N,g}=1$, which are for $N=4$, $g=sl(2)$ and for $N=3$, $g=sl(3)$.
For the first case, I agree that this counts the number of parameters needed to fix the conformal block.
The four point block on primary fields is given by
\[
\langle h | \phi_{h'}(1) \phi_{h''}(z) | h'''\rangle
= z^{H - h''-h'''}(1 + z \frac{(H +h-h')(H + h''-h''')}{2H} + \ldots)
\]
which is fixed, for given $z$, by determining H, and the four-point block of any Virasoro descendants is also fixed uniquely once $H$ is fixed.

This is very different to the case of the W3 algebra 3-point block where the three point functions of any W3 descendants are fixed up to the infinite set of parameters
\[\langle h,w| \phi_{h',w'}(z) (W_{-1})^n |h'',w''\rangle
= c_n z^{h-h'-h''-n}\]
It is true that this is equivalent to a single undetermined function defined e.g. as $F(u) = \sum c_n u^n$, but I really think that this is not the same as the block being determined up to one parameter. One parameter means one real number, not an infinite set of real numbers.

It is also a little odd to say that there are $1/2(dim(g) - rank(g))$ ''undetermined descendants'' in the OPEs ${\cal W}^i V_j$ : for $sl(2)$ this is just 1 and the ''undetermined descendant'' which is the pole of order 1 in the OPE
\[
T(z) V_j(w) = \frac{ h_j V_j(w)}{(z-w)^2} + \frac{ \partial V_j(w)}{(z-w)} + O(1)
\]
is exactly the derivative field.
Saying that this is undetermined is the same as saying that one does not understand the complex structure of the conformal block. Perhaps this is related to what ${\cal N}_{N,g}$ counts - but it is not explained.

The example of the 4 point block of the Virasoro algebra being determined up to one parameter makes it clear that the block (as a function of $z$) is fully determined once this one single real number is known. This is not the same for the 3-pt block of the W3 algebra.
\[\]
I wonder if the authors really mean that the Virasoro four point function is undetermined up to a FUNCTION of $H$, e.g. $c(H)$ as might appear below
\[
\langle h | \phi_{h'}(1) \phi_{h''}(z) | h'''\rangle
= \int {\mathrm d} H \; c(H)\;F(h,h',h'',h''';H;z)
\]
where $F(h,h',h'',h''';H;z)$ is the conformal block depending on the one parameter $H$.\[\]

The counting ${\cal N}_{N,g}$ clearly counts something, but it is not the number of parameters appearing in a block if one only uses W-algebra Ward-identities.

If one can also use identities resulting from the construction in terms of the affine algebra at level 1, then this may be the case, I do not know, but that is not what is stated here.
\[\]
I think this point is sufficiently important that the paper should not be published as it is without a clearly elucidation. If one real number (the weight of the intermediate channel in a Virasoro 4-pt block) is "one parameter", then an arbitrary function of one variable is not "one parameter". It could be that ${\cal N}_{N,g}$ counts the number of undetermined functions appearing in the correlator [as opposed to the block]. It could just be that the "explanation" added after (3.11) is wrong. I really would like to see a clear explanation.
\[\]
2. The authors seem to have misunderstood my previous comment on the diagram (3.6). What I was saying was that this diagram should actually commute, for a suitable definition of the maps; it was the previous statement that it only commuted up to a scalar factor that was wrong. To include this diagram and then to say that ``This diagram is in general not commutative'' is not a helpful statement to make.
\[\]
3. The normal ordering implicitly defined through the regularised kernels (2.11) does not seem to be the same as that defined in (3.3). I am not sure that the Casimir algebras as described in this paper can be constructed through the correlation functions defined by (2.12), (2.13). To reproduce (3.3), one would need to add derivative fields at coincident points which is not discussed here.
\[\]
4. Having re-read section 3.2, I have started to worry whether the requirement of cocycles for the free-field construction of $g$ at level 1 will play a role. They are not mentioned explicitly here, being hidden in the $\sim$ in equation (3.5), but they do complicate the construction of the affine fields and I do not see why they are also not needed in the construction of the fields $V_j(z_j)$ just after (3.8).
I also think the statement that ``twisted modules differ ... except in the case $A_j=0$'' is wrong. The ``twisted modules'' will be the same as standard highest weight modules whenever $A_j$ is in the weight lattice of $g$.
For example, for $g=sl(2)$, with the simple root $\alpha=(\sqrt 2)$, then
the field $exp( \int J(z.(1/\sqrt 2)))$ is just the field correpsonding to the highest weight of the non-trivial representations of spin $s=1/2$, and the field
$exp( \int J(z.(\sqrt 2)))$ is just the field $J^+(z)$, which is in the vacuum representation.
I would also point out that reference [9] only deals with $A_n$ and $D_n$ (of the simply-laced algebras), not $E_n$, and so it is possibly an open question whether one can construct a twisted module for all simply-laced affine algebras using free fermions. There is a ``transcendental'' free fermion construction for $E_8$ but the methods of [9] would not appear to be easy to apply.

1- to explain clearly how ${\cal N}_{N,g}$ does, or does not, count parameters or unknown functions or whatever.

2- to clear up when the diagram (3.6) is commutative

3- to comment on the normal ordering implied by (2.12), (2.13)

4- to clear up when the "twisted" fields they define are actually twisted and when not.

The authors have made a change I suggested (additional motivation for their work) but have not adjusted the referencing. Instead, they have provided a sentence of justification for their policy of referring only to arxiv preprints. If their justification is acceptable to SciPost (despite the fact it appears to be at variance with the SciPost instructions with regard to references), then I will not insist on any changes. Though I have to say I agree with the SciPost policy in this regard - I would prefer to see the published journal reference together with the arxiv post and then be able to decide for myself, if necessary, whether changes to a published version are significant - moreover, in their parenthetic remark, the authors do not object to it either provided SciPost does the job of supplying the missing data!

I notice that they have also made some corrections in response to the other referee.