# Elliptic deformation of $\mathcal{W}_N$-algebras

### Submission summary

 As Contributors: Luc FRAPPAT · Eric Ragoucy Arxiv Link: https://arxiv.org/abs/1810.11410v2 Date accepted: 2019-04-29 Date submitted: 2019-04-17 Submitted by: FRAPPAT, Luc Submitted to: SciPost Physics Domain(s): Theoretical Subject area: Mathematical Physics

### Abstract

We construct $q$-deformations of quantum $\mathcal{W}_N$ algebras with elliptic structure functions. Their spin $k+1$ generators are built from $2k$ products of the Lax matrix generators of ${\mathcal{A}_{q,p}(\widehat{gl}(N)_c)}$). The closure of the algebras is insured by a critical surface condition relating the parameters $p,q$ and the central charge $c$. Further abelianity conditions are determined, either as $c=-N$ or as a second condition on $p,q,c$. When abelianity is achieved, a Poisson bracket can be defined, that we determine explicitly. One connects these structures with previously built classical $q$-deformed $\mathcal{W}_N$ algebras and quantum $\mathcal{W}_{q,p}(\mathfrak{sl}_N)$.

### Ontology / Topics

See full Ontology or Topics database.

### List of changes

Dear Editor,

We thank the referees for their valuable comments and suggestions.
We have done some modifications in the papier according to the recommendations.
More precisely:
- the notation has been fixed after eq. (2.3) (but the standard notation for P_12 R_12 is with the check, not with the hat).
- the algebras that we construct are not much wider than the W_N algebra. Indeed, the generators t_{mn}^{(k)}(z) are defined for a given pair of integers m and n defining the surface (hence the relation between p and q for a given central charge c). So to be clear, we have added a sentence just before corollary 3.2.
- the first sentence of section 3.2 could be ambiguous. We made the statement more explicit and rephrased the beginning of this section. We also added a sentence before Remark 3.3. We hope that this clarification answers to the requested changes of referee 2.
- we added a paragraph at the end of the conclusion ("A structural issue ... algebra [24]"). This paragraph is related to the point raised by the second referee in the weaknesses of the paper. Accordingly, we added two references (23 and 24 in the present version). We added also a sentence at the end of the second paragraph of the conclusion regarding possible physical interpretations of this algebra.
- we modified formula (5.22) (there was a mistake in this formula).
- finally, it does not seem appropriate to us to split the section 5 with the details moved to another appendix as suggested by the first referee, since the proof is not only technical and should be read as a whole.
- some references were not properly ordered, now it is fixed.

Sincerely yours,
The authors

### Submission & Refereeing History

Resubmission 1810.11410v2 on 17 April 2019
Submission 1810.11410v1 on 30 October 2018