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The geometry of Casimir W-algebras
by Raphaël Belliard, Bertrand Eynard, Sylvain Ribault
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Raphaël Belliard · Sylvain Ribault |
| Submission information | |
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| Preprint Link: | http://arxiv.org/abs/1707.05120v3 (pdf) |
| Date submitted: | April 26, 2018, 2 a.m. |
| Submitted by: | Raphaël Belliard |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Let $\mathfrak{g}$ be a simply laced Lie algebra, $\widehat{\mathfrak{g}}_1$ the corresponding affine Lie algebra at level one, and $\mathcal{W}(\mathfrak{g})$ the corresponding Casimir W-algebra. We consider $\mathcal{W}(\mathfrak{g})$-symmetric conformal field theory on the Riemann sphere. To a number of $\mathcal{W}(\mathfrak{g})$-primary fields, we associate a Fuchsian differential system. We compute correlation functions of $\widehat{\mathfrak{g}}_1$-currents in terms of solutions of that system, and construct the bundle where these objects live. We argue that cycles on that bundle correspond to parameters of the conformal blocks of the W-algebra, equivalently to moduli of the Fuchsian system.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2018-7-13 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1707.05120v3, delivered 2018-07-13, doi: 10.21468/SciPost.Report.534
Strengths
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The paper is an interesting attempt to provide a geometric framework for new ways to consider W-algebra conformal blocks and correlation functions
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It is likely to generate further work to understand this area
Weaknesses
- There are several places where the arguments are not clear or explained well. See the report for details
Report
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
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
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
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.
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.
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 thattwisted 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 atranscendental'' free fermion construction for $E_8$ but the methods of [9] would not appear to be easy to apply.Requested changes
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.
Report #1 by Anonymous (Referee 1) on 2018-6-28 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1707.05120v3, delivered 2018-06-28, doi: 10.21468/SciPost.Report.518
Report
I notice that they have also made some corrections in response to the other referee.