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The geometry of Casimir Walgebras
by Raphaël Belliard, Bertrand Eynard, Sylvain Ribault
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Raphaël Belliard · Sylvain Ribault 
Submission information  

Preprint Link:  https://arxiv.org/abs/1707.05120v4 (pdf) 
Date accepted:  20181023 
Date submitted:  20180921 02:00 
Submitted by:  Belliard, Raphaël 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

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 Walgebra. 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 Walgebra, equivalently to moduli of the Fuchsian system.
Author comments upon resubmission
List of changes
Reply to Report 2 on revised version:
1. We have tried to improve the explanations, with a more detailed and explicit elaboration of our two examples. Basically, one should not confuse coefficients, variables and parameters: a basis of a space of functions of one variable is a oneparameter family of functions; a function in that space is a linear combination of basis functions, with infinitely many coefficients.
2. We now state that the diagram is indeed commutative, provided an embedding is appropriately chosen.
3. The reviewer rightly points out that the regularization in (3.3) differs from the regularization of amplitudes at coinciding points. Strictly speaking, this does not affect our proof of the relation between correlation functions and amplitudes in Section 3.3. Nevertheless, the third bullet item in that proof implicitly suggested a relation between amplitudes of Casimir elements, and correlation functions of generators of the Casimir algebra: this was wrong, and unnecessary for the proof. In order to clarify this point, we have added a paragraph at the end of Section 2.3, where we give an alternative definition of amplitudes of Casimir elements, using normal ordering rather than our earlier regularization. This leads to the new eq. (3.11), which provides an extra check of the relation between correlation functions and amplitudes.
4a. We agree that cocyles are hidden in the $\propto$ sign in (3.5) and in the $O(1)$ factor in (3.8). We prefer leaving the cocycles hidden as they are not essential to our argument. The reference [8] that we cite gives the correct cocycles.
4b. Our statement that twisted modules are affine highestweight representations only if $A_j=0$ was indeed too strong. We have corrected it. We have however refrained from discussing the precise conditions for twisted modules to be affine highestweight representations, as we do not need it.
4c. We have added the restriction of [9]'s free fermion construction to the A and Dseries.
Published as SciPost Phys. 5, 051 (2018)
Reports on this Submission
Anonymous Report 1 on 20181012 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1707.05120v4, delivered 20181012, doi: 10.21468/SciPost.Report.609
Report
As this is my fourth review, I will only say that I think the authors are not likely to agree on precisely the meaning of the number of parameters parametrising a space of functions. Given the fact that there is a continuous map from $\mathbb{R}^2$ to $\mathbb{R}$, I think this is a rather unclear concept without appropriate extra conditions of smoothness, but I think this is not sufficient to stop me recommending the paper. the authors have a clear point of view which I think should be heard. I personally also question the statement, or requirement, that the function f($\theta$) at the bottom of page 10 is analytic, since the coefficients can be freely chosen. When one of the representations has a null vector this seems likely to be the case, but in general I think it would have to be an extra assumption. Still, this is a very minor point which I hope will not confuse the reader. I am happy to recommend publication of the article as it stands. I am very grateful to the authors for all the changes they have made in response to my comments which has helped me understand the paper.