Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 036, 23 pages      arXiv:2411.08406      https://doi.org/10.3842/SIGMA.2025.036
Contribution to the Special Issue on Recent Advances in Vertex Operator Algebras in honor of James Lepowsky

On Kazama-Suzuki Duality between $\mathcal W_k(\mathfrak{sl}_4, f_{\rm sub})$ and $N=2$ Superconformal Vertex Algebra

Dražen Adamović a and Ana Kontrec b
a) Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, Zagreb, Croatia
b) Research Institute for Mathematical Sciences (RIMS), Kyoto University, Kyoto 606-8502, Japan

Received December 31, 2024, in final form May 08, 2025; Published online May 17, 2025

Abstract
We classify all possible occurrences of Kazama-Suzuki duality between the ${N=2}$ superconformal algebra $L^{N=2}_c$ and the subregular $\mathcal{W}$-algebra $\mathcal{W}_{k}(\mathfrak{sl}_4, f_{\rm sub})$. We establish a new Kazama-Suzuki duality between the subregular $\mathcal{W}$-algebra $\mathcal{W}_k(\mathfrak{sl}_4, f_{\rm sub})$ and the $N = 2$ superconformal algebra $L^{N=2}_{c}$ for $c=-15$. As a consequence of the duality, we classify the irreducible $\mathcal{W}_{k=-1}(\mathfrak{sl}_4, f_{\rm sub})$-modules.

Key words: vertex algebra; subregular $W$-algebras; $N=2$ superconformal vertex algebra; Kazama-Suzuki duality.

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References

  1. Adamović D., Representations of the $N=2$ superconformal vertex algebra, Int. Math. Res. Not. 1999 (1999), 61-79, arXiv:math.QA/9809141.
  2. Adamović D., Vertex algebra approach to fusion rules for ${N=2}$ superconformal minimal models, J. Algebra 239 (2001), 549-572.
  3. Adamović D., A construction of admissible $A^{(1)}_1$-modules of level $-\frac 43$, J. Pure Appl. Algebra 196 (2005), 119-134, arXiv:math.QA/0401023.
  4. Adamović D., Realizations of simple affine vertex algebras and their modules: the cases $\widehat{\mathfrak{sl}(2)}$ and $\widehat{\mathfrak{osp}(1,2)}$, Comm. Math. Phys. 366 (2019), 1025-1067, arXiv:1711.11342.
  5. Adamović D., Kontrec A., Bershadsky-Polyakov vertex algebras at positive integer levels and duality, Transform. Groups 28 (2023), 1325-1355, arXiv:2011.10021.
  6. Adamović D., Milas A., Wang Q., On parafermion vertex algebras of $\mathfrak{sl}(2)$ and $\mathfrak{sl}(3)$ at level $-\frac32$, Commun. Contemp. Math. 24 (2022), 2050086, 23 paages, arXiv:2005.02631.
  7. Adamović D., Möseneder Frajria P., Papi P., New approaches for studying conformal embeddings and collapsing levels for $W$-algebras, Int. Math. Res. Not. 2023 (2023), 19431-19475, arXiv:2203.08497.
  8. Adamović D., Möseneder Frajria P., Papi P., Perše O., Conformal embeddings in affine vertex superalgebras, Adv. Math. 360 (2020), 106918, 50 pages, arXiv:1903.03794.
  9. Adamović D., Perše O., Vukorepa I., On the representation theory of the vertex algebra $L_{-5/2}(\mathfrak{sl}(4))$, Commun. Contemp. Math. 25 (2023), 2150104, 42 pages, arXiv:2103.02985.
  10. Blumenhagen R., Eholzer W., Honecker A., Hübel R., Hornfeck K., Coset realization of unifying $\mathcal W$ algebras, Internat. J. Modern Phys. A 10 (1995), 2367-2430, arXiv:hep-th/9406203.
  11. Creutzig T., Fasquel J., Linshaw A.R., Nakatsuka S., On the structure of $\mathcal W$-algebras in type $A$, Jpn. J. Math. 20 (2025), 1-111, arXiv:2403.08212.
  12. Creutzig T., Genra N., Nakatsuka S., Duality of subregular $\mathcal W$-algebras and principal $\mathcal W$-superalgebras, Adv. Math. 383 (2021), 107685, 52 pages, arXiv:2005.10713.
  13. Creutzig T., Linshaw A.R., Cosets of the $\mathcal{W}^k(\mathfrak{sl}_4, f_{\rm subreg})$-algebra, in Vertex Algebras and Geometry, Contemp. Math., Vol. 711, American Mathematical Society, Providence, RI, 2018, 105-117, arXiv:1711.11109.
  14. Creutzig T., Linshaw A.R., Trialities of $\mathcal W$-algebras, Camb. J. Math. 10 (2022), 69-194, arXiv:2005.10234.
  15. Dong C., Lam C.H., Yamada H., $\mathcal W$-algebras related to parafermion algebras, J. Algebra 322 (2009), 2366-2403, arXiv:0809.3630.
  16. Fehily Z., Subregular $\mathcal W$-algebras of type $A$, Commun. Contemp. Math. 25 (2023), 2250049, 44 pages, arXiv:2111.05536.
  17. Feigin B.L., Semikhatov A.M., $\mathcal W^{(2)}_n$ algebras, Nuclear Phys. B 698 (2004), 409-449, arXiv:math.QA/0401164.
  18. Feigin B.L., Semikhatov A.M., Tipunin I.Y., Equivalence between chain categories of representations of affine $\mathfrak{sl}(2)$ and $N=2$ superconformal algebras, J. Math. Phys. 39 (1998), 3865-3905, arXiv:hep-th/9701043.
  19. Genra N., Screening operators for $\mathcal{W}$-algebras, Selecta Math. (N.S.) 23 (2017), 2157-2202, arXiv:1606.0096.
  20. Goddard P., Kent A., Olive D., Virasoro algebras and coset space models, Phys. Lett. B 152 (1985), 88-92.
  21. Gorelik M., Kac V., On simplicity of vacuum modules, Adv. Math. 211 (2007), 621-677, arXiv:math-ph/0606002.
  22. Kac V., Vertex algebras for beginners, 2nd ed., Univ. Lecture Ser., Vol. 10, American Mathematical Society, Providence, RI, 1998.
  23. Kac V., Roan S.S., Wakimoto M., Quantum reduction for affine superalgebras, Comm. Math. Phys. 241 (2003), 307-342, arXiv:math-ph/0302015.
  24. Kazama Y., Suzuki H., New $N=2$ superconformal field theories and superstring compactification, Nuclear Phys. B 321 (1989), 232-268.
  25. Lepowsky J., Li H., Introduction to vertex operator algebras and their representations, Progr. Math., Vol. 227, Birkhäuser, Boston, MA, 2004.
  26. Li H., Certain extensions of vertex operator algebras of affine type, Comm. Math. Phys. 217 (2001), 653-696, arXiv:math.QA/0003038.
  27. Linshaw A.R., Universal two-parameter $\mathcal W_\infty$-algebra and vertex algebras of type $\mathcal W(2, 3, \ldots, N)$, Compos. Math. 157 (2021), 12-82, arXiv:1710.02275.
  28. Ridout D., $\widehat{\mathfrak{sl}}(2)_{-1/2}$ and the triplet model, Nuclear Phys. B 835 (2010), 314-342, arXiv:1001.3960.
  29. Wang W., $\mathcal W_{1+\infty}$ algebra, $\mathcal W_3$ algebra, and Friedan-Martinec-Shenker bosonization, Comm. Math. Phys. 195 (1998), 95-111, arXiv:q-alg/9708008.
  30. Zhu Y., Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), 237-302.

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