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Rationality of the Exceptional \(\mathcal {W}\)-Algebras \(\mathcal {W}_k(\mathfrak {sp}_{4},f_{subreg})\) Associated with Subregular Nilpotent Elements of \(\mathfrak {sp}_{4}\)

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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

  1. Ai and Lin [2] recently gave a counterexample to this conjecture in the context of vertex superalgebras.

  2. 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 \(\mathfrak {g}\).

  3. The notion of exceptional pair coincides with [21] if f is of Levi type, and with [35] if moreover \((q,r^\vee )=1\), where \(r^\vee \) is the lacety of \(\mathfrak {g}\).

  4. 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 \(p\geqslant 8\).

  5. It means that the Dynkin grading associated with f is even.

  6. 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.

  7. 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.

  8. The matrix \(I_n\) denotes the n-size square identity matrix.

References

  1. Abe, T., Buhl, G., Donc, C.: Rationality, regularity, and \(C_2\)-cofiniteness. Trans. Am. Math. Soc. 356(8), 3391–3402 (2004)

    Article  Google Scholar 

  2. Ai, C., Lin, X.: Module category and \(C_2\)-cofiniteness of affine vertex operator superalgebras (2021). arXiv:2101.10567, [math.QA]

  3. Arakawa, T.: Representation theory of superconformal algebras and the Kac–Roan–Wakimoto conjecture. Duke Math. J. 130(3), 435–478 (2005)

    Article  MathSciNet  Google Scholar 

  4. Arakawa, T.: Representation theory of \({\cal{W}}\)-algebras. II. Adv. Stud. Pure Math. 61, 51–90 (2011)

    Article  MathSciNet  Google Scholar 

  5. Arakawa, T.: A remark on the \(C_2\) cofiniteness condition on vertex algebras. Math. Z. 270(1), 559–575 (2012)

    Article  MathSciNet  Google Scholar 

  6. Arakawa, T.: Rationality of Bershadsky–Polyakov vertex algebras. Commun. Math. Phys. 323(2), 627–633 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  7. Arakawa, T.: Associated varieties of modules over Kac–Moody algebras and \(C_2\)-cofiniteness of \({\cal{W}}\)-algebras. Int. Math. Res. Not. 2015, 11605–11666 (2015)

    MATH  Google Scholar 

  8. Arakawa, T.: Rationality of \({\cal{W}}\)-algebras: principal nilpotent cases. Ann. Math. 182(2), 565–694 (2015)

    Article  MathSciNet  Google Scholar 

  9. Arakawa, T.: Chiral algebras of class \(\cal{S}\) and Moore–Tachikawa symplectic varieties (2018). arXiv:1811.01577, [math.RT]

  10. Arakawa, T., van Ekeren, J., Rationality and fusion rules of exceptional \({\cal{W}}\)-algebras. J. Eur. Math. Soc. (2019) (to appear). arXiv:1905.11473 [math.RT]

  11. Arakawa, T., Frenkel, E.: Quantum Langlands duality of representations of W-algebras. Compos. Math. 155(12), 2235–2262 (2019)

    Article  MathSciNet  Google Scholar 

  12. Arakawa, T., Moreau, A.: Arc Spaces and Vertex Algebras. CEMPI, Lectures Series. Springer (in preparation)

  13. Beem, C., Lemos, M., Liendo, P., Peelaers, W., Rastelli, L., van Rees, B.: Infinite chiral symmetry in four dimensions. Commun. Math. Phys. 336(3), 1359–1433 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  14. Beem, C., Peelaers, W., Rastelli, L., van Rees, B.C.: Chiral algebras of class \(\cal{S}\). J. High Energy Phys. 5, 2015 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  15. Creutzig, T., Linshaw, A.: Trialities of orthosymplectic \({\cal{W}}\)-algebras (2021). arXiv:2102.10224, [math.RT]

  16. Collingwood, D., McGovern, W.: Nilpotent Orbits in Semisimple Lie Algebras. Routledge, New York (2017)

    Book  Google Scholar 

  17. De Sole, A., Kac, V.: Finite vs affine \({\cal{W}}\)-algebras. Jpn. J. Math. 1, 137–261 (2006)

    Article  MathSciNet  Google Scholar 

  18. Dong, C., Lin, X.: Unitary vertex operator algebras. J. Algebra 397, 252–277 (2014)

    Article  MathSciNet  Google Scholar 

  19. Dong, C., Li, H., Mason, G.: Twisted representations of vertex operator algebras. Math. Ann. 310(3), 571–600 (1998)

    Article  MathSciNet  Google Scholar 

  20. Elashvili, A., Kac, V.: Classification of good gradings of simple Lie algebras. Am. Math. Soc. Transl. Ser. 2(213), 85–104 (2005)

    MathSciNet  MATH  Google Scholar 

  21. Elashvili, A., Kac, V., Vinberg, E.: On exceptional nilpotents in semisimple Lie algebras. J. Lie Theory 19(2), 371–390 (2009)

    MathSciNet  MATH  Google Scholar 

  22. Feigin, B., Frenkel, E.: Quantization of the Drinfel’d–Sokolov reduction. Phys. Lett. B 246, 75–81 (1990)

  23. Frenkel, E.: Langlands Correspondence for Loop Groups. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  24. Frenkel, E., Kac, V., Wakimoto, M.: Characters and fusion rules for \({\cal{W}}\)-algebras via quantized Drinfel’d–Sokolov reduction. Commun. Math. Phys. 147(2), 295–328 (1992)

  25. Frenkel, I., Zhu, Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66(1), 123–168 (1992)

    Article  MathSciNet  Google Scholar 

  26. Gaitsgory, D.: Quantum Langlands correspondence (2016). arXiv:1601.05279 [math.AG]

  27. Kac, V.: Introduction to vertex algebras, Poisson vertex algebras, and integrable Hamiltonian PDE. In: Caselli, F., De Concini, C., De Sole, A., Callegaro, F., Carnovale, G. (eds.) Perspectives in Lie Theory. Springer INdAM Series, vol. 9, pp. 3–72 (2017)

  28. Kac, V.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  29. Kanade, S., Linshaw, A.: Universal two-parameter even spin \({\cal{W}}_{\infty }\)-algebra. Adv. Math. 355, 106774 (2019)

    Article  MathSciNet  Google Scholar 

  30. Kawasetsu, K.: \(\cal{W}\)-algebras with non-admissible levels and the Deligne exceptional series. Int. Math. Res. Not. 2018(3), 641–676 (2018)

    MathSciNet  MATH  Google Scholar 

  31. Kac, V., Roan, S.-S., Wakimoto, M.: Quantum reduction for affine superalgebras. Commun. Math. Phys. 241, 307–342 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  32. Kac, V., Wakimoto, M.: Modular invariant representations of infinite-dimensional Lie algebras and superalgebras. Proc. Nat. Acad. Sci. USA 85(14), 4956–4960 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  33. Kac, V., Wakimoto, M.: Classification of modular invariant representations of affine algebras, infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988). Adv. Ser. Math. Phys. 7, 138–177 (1989)

    MATH  Google Scholar 

  34. Kac, V., Wakimoto, M.: Quantum reduction and representation theory of superconformal algebras. Adv. Math. 185(2), 400–458 (2004)

    Article  MathSciNet  Google Scholar 

  35. Kac, V., Wakimoto, M.: On rationality of \({\cal{W}}\)-algebras. Transform. Groups 13, 671–713 (2008)

    Article  MathSciNet  Google Scholar 

  36. Li, H.: The physics superselection principle in vertex operator algebra theory. J. Algebra 196, 436–457 (1997)

    Article  MathSciNet  Google Scholar 

  37. McRae, R.: On rationality for C2-cofinite vertex operator algebras (2021). arXiv:2108.01898, [math.QA]

  38. Moody, R., Pianzola, A.: Lie Algebras with Triangular Decompositions. Wiley, New York (1995)

    MATH  Google Scholar 

  39. Premet, A.: Special transverse slices and their enveloping algebras. Adv. Math. 170(1), 1–55 (2002)

    Article  MathSciNet  Google Scholar 

  40. Song, J., Xie, D., Yan, W.: Vertex operator algebras of Argyres-Douglas theories from M5-branes. J. High Energy Phys. 123, 1–36 (2017)

  41. Thielemans, K.: A mathematica package for computing operator product expansions. Int. J. Mod. Phys. C 2(3), 787–798 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  42. Wang, W.: Rationality of Virasoro vertex operator algebras. Duke Math. J. 7, 197–211 (1993)

    MathSciNet  MATH  Google Scholar 

  43. Zhu, Y.: Modular invariance of characters of vertex operator algebras. J. Am. Math. Soc. 9(1), 237–302 (1996)

    Article  MathSciNet  Google Scholar 

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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).

Funding

The author receives support from the Labex CEMPI (ANR-11-LABX-0007-01).

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  1. CNRS, UMR 8524, Laboratoire Paul Painlevé, University of Lille, 59000, Lille, France

    Justine Fasquel

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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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