Abstract
We prove the rationality of the exceptional \(\mathcal {W}\)-algebras \(\mathcal {W}_k(\mathfrak {g},f)\) associated with the simple Lie algebra \(\mathfrak {g}=\mathfrak {sp}_{4}\) and a subregular nilpotent element \(f=f_{subreg}\) of \(\mathfrak {sp}_{4}\), proving a new particular case of a conjecture of Kac–Wakimoto. Moreover, we describe the simple \(\mathcal {W}_k(\mathfrak {g},f)\)-modules and compute their characters. We also work out the nontrivial action of the component group on the set of simple \(\mathcal {W}_k(\mathfrak {g},f)\)-modules.
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Notes
Ai and Lin [2] recently gave a counterexample to this conjecture in the context of vertex superalgebras.
The level k is admissible if \(L_k(\mathfrak {g})\) is admissible as a representation of \(\widehat{\mathfrak {g}}\), cf. Sect. 3 for a precise definition. It is non-degenerate if the associated variety of \(L_k(\mathfrak {g})\) is the whole nilpotent cone of \(\mathfrak {g}\).
In fact the results of Creutzig and Linshaw [15] cover all cases of the Main Theorem except for the levels \(-3+p/3\) where p is even and \(p\geqslant 8\).
It means that the Dynkin grading associated with f is even.
For instance we will show in a future work that \(\mathcal {W}_{-1}(\mathfrak {sp}_{4},f_{subreg})\simeq M(1)\) and \(\mathcal {W}_{-2}(\mathfrak {sp}_{4},f_{subreg})\simeq \mathrm{Vir}^{-2}\). We thanks Dražen Adamović for pointing out these results to us.
The functor \(M\mapsto H^{\text {Lie}}_0(M)\) defines a correspondence between a subcategory of the category \(\mathcal {O}\) of \(\mathfrak {g}\)-modules and the category of the finite dimensional representations of \(U(\mathfrak {g},f)\), see [4, Sect. 5] for a precise definition.
The matrix \(I_n\) denotes the n-size square identity matrix.
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Acknowledgements
The author is very grateful to her thesis advisors Anne Moreau and Tomoyuki Arakawa for suggesting the problem and for useful conversations and comments. She also thanks the members of the \(\mathcal {W}\)-algebra working group of the University of Lille. She is thankful to the referees for their valuable advice. The author acknowledges support from the Labex CEMPI (ANR-11-LABX-0007-01).
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The author receives support from the Labex CEMPI (ANR-11-LABX-0007-01).
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Fasquel, J. Rationality of the Exceptional \(\mathcal {W}\)-Algebras \(\mathcal {W}_k(\mathfrak {sp}_{4},f_{subreg})\) Associated with Subregular Nilpotent Elements of \(\mathfrak {sp}_{4}\). Commun. Math. Phys. 390, 33–65 (2022). https://doi.org/10.1007/s00220-021-04294-6
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DOI: https://doi.org/10.1007/s00220-021-04294-6