SciPost Submission Page
Elliptic deformation of $\mathcal{W}_N$-algebras
by J. Avan, L. Frappat, E. Ragoucy
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Luc Frappat · Eric Ragoucy |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/1810.11410v1 (pdf) |
| Date submitted: | Oct. 30, 2018, 1 a.m. |
| Submitted by: | Luc Frappat |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approach: | Theoretical |
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 $\mathcal{W}_{q,p}(\mathfrak{sl}_N)$.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2019-4-11 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1810.11410v1, delivered 2019-04-11, doi: 10.21468/SciPost.Report.906
Strengths
Weaknesses
The algebra looks very nice, but its practical use for physical applications is elusive at that point. Are the t^k_mn generators useful for some physical application ?. Could they be useful in connection with the spectrum of the Ruisjenaars-Schneider chains ?.
The algebra is much wider than the W_N (or Virasoro) algebra and contains a lot more generators, but no interpretation of these generators is given. Do they have a geometric interpretation in the classical limit? Are they connected in some ways to the bigger algebras such as Ding Iohara Miki?
Report
Requested changes
It would be nice if the authors clarify the relation between their paper and ref [2]. I can guess that t^j_2,-1 is related to F.F. T_j but the authors should make it clear. At the top of page 5, the t^k_mn are claimed to be defined in [2], could the authors be more precise.
Report #1 by Anonymous (Referee 1) on 2019-3-15 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1810.11410v1, delivered 2019-03-15, doi: 10.21468/SciPost.Report.869
Strengths
1.This paper develops an approach to a construction of q-deformations of quantum $W_N$ algebras based on the Lax operator of $A_{q,p}({\hat{gl}}(N)_c)$. 2. The main result is a derivation for exchange relations for generators satisfied on a special surface in the space of parameters. 3. Authors compare their construction with a previously known approach to construct quantum $W$-algebra generators. 4. They also analyze a critical level case in details.
Weaknesses
- Section 5 contains a lot of technical details which could be moved to another appendix.
- A notation $\hat R_{12}(z)$ in (2.3) is not the best one. It can be easily confused with the standard notation $\hat R_{12}(z)=P_{12}R_{12}(z)$. At least, formulas (B.14-B.15) should be quoted immediately after (2.3).
Report
The main result is the Theorem 3.1 where exchange relations for quantum generators are derived provided that parameters $q,p,c$ lie on a special surface. The generators are constructed in terms L-operators for the $A_{q,p}({\hat{gl}}(N)_c)$ algebra. Then the authors match their construction to the construction derived by Feigin and Frenkel and discuss their differences.
In Chapter 4 they introduce Abelian subalgebras with the required restrictions on parameters and derive Poisson structures on such subalgebras. Section 5 is dedicated to the case of critical level where they present in Proposition 5.2 structural constants of a closed Poisson algebra for the algebra generators.
Section 5 exlpains a reduction to $U_q(\hat{gl}_N)$ at ctirical level.
Overall, the paper is well written and opens some interesting new directions for a future research.
As noted in Conclusion, explicit realizations of these algebras are required similar to the case of
$W_{q,p}(A_N)$ algebras. A comparison with other approaches to elliptis $W$-algebras is also would be desirable.
Requested changes
Overall the paper is well written and I recommend it for publication in the current form.
I notices a couple of points above which could improve the paper but they are not compulsory.