# The geometry of Casimir W-algebras

### Submission summary

 As Contributors: Raphaël Belliard · Sylvain Ribault Arxiv Link: https://arxiv.org/abs/1707.05120v4 (pdf) Date accepted: 2018-10-23 Date submitted: 2018-09-21 02:00 Submitted by: Belliard, Raphaël Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Mathematical Physics 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.

### Ontology / Topics

See full Ontology or Topics database.

Published as SciPost Phys. 5, 051 (2018)

We are grateful to the reviewer for this report, which allowed us to correct several technical mistakes and imprecisions. (This does not affect our conclusions.)

### 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 one-parameter 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 highest-weight 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 highest-weight representations, as we do not need it.

4c. We have added the restriction of [9]'s free fermion construction to the A- and D-series.

### Submission & Refereeing History

Resubmission 1707.05120v4 on 21 September 2018
Resubmission 1707.05120v3 on 26 April 2018
Submission 1707.05120v2 on 18 January 2018

## Reports on this Submission

### Anonymous Report 1 on 2018-10-12 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1707.05120v4, delivered 2018-10-12, 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.

• validity: high
• significance: good
• originality: high
• clarity: good
• formatting: excellent
• grammar: excellent