Strong generators of the subregular \(\mathcal {W}\)-algebra \(\mathcal {W}^{K-N}(\mathfrak {sl}_N, f_{sub})\) and combinatorial description at critical level

  • Naoki Genra
  • Toshiro KuwabaraEmail author


We construct explicitly strong generators of the affine \(\mathcal {W}\)-algebra \(\mathcal {W}^{K_0-N}(\mathfrak {sl}_N, f_{sub})\) of subregular type A. Moreover, we are able to describe the OPEs between them at critical level. We also give a description the affine \(\mathcal {W}\)-algebra \(\mathcal {W}^{-N}(\mathfrak {sl}_N, f_{sub})\) in terms of certain fermionic fields, which was conjectured by Adamović.


Vertex algebra Affine W-algebra Critical level Feigin–Frenkel center 

Mathematics Subject Classification

Primary 17B69 Secondary 81R10 



One of the main results of this paper, Theorem 4.5, was first conjectured by Dražen Adamović. The authors would like to express their gratitude to him for showing us his private notes [1]. The authors are deeply grateful to Tomoyuki Arakawa for valuable comments. The authors also thank Boris Feigin and Alexei Semikhatov for fruitful discussion on their construction of the vertex algebra \(W^{(2)}_N\). The second author thanks Yoshihiro Takeyama for discussion on the proof of Lemma 3.6. The first author was supported by Grant-in-Aid for JSPS Fellows (No.17J07495) and is supported by JSPS Overseas Research Fellowships. The second author was supported by JSPS KAKENHI Grant Number JP17K14151.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  2. 2.Department of Mathematics, Faculty of Pure and Applied SciencesUniversity of TsukubaTsukubaJapan

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