Orthosymplectic Feigin–Semikhatov duality

• Justine Fasquel ^[1] ; Shigenori Nakatsuka ^[2]
  1. [1] University of Melbourne

     University of Melbourne

     Australia

     
  2. [2] Department Mathematik, FAU Erlangen–Nürnberg, Germany
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 4, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01070-5
• Enlaces
  • Texto completo
• Resumen

  • We study the representation theory of the subregular W-algebra Wk (so2n+1, fsub) of type B and the principal W-superalgebra W(osp2|2n), which are related by an orthosymplectic analogue of Feigin–Semikhatov duality in type A. We establish a block-wise equivalence of weight modules over the W-superalgebras by using the relative semi-infinite cohomology functor and spectral flow twists, which generalizes the result of Feigin–Semikhatov–Tipunin for the N = 2 superconformal algebra. In particular, the correspondence of Wakimoto type free field representations is obtained. When the level of the subregular W-algebra is exceptional, we classify the simple modules over the simple quotients Wk (so2n+1, fsub) and W(osp2|2n) and derive the character formulae

• Referencias bibliográficas
  • Abe, T.: Fusion rules for the charge conjugation orbifold. J. Algebra 242(2), 624–655 (2001). https:// doi.org/10.1006/jabr.2001.8838
  • Abe, T., Buhl, G., Dong, C.: Rationality, regularity, and C2-cofiniteness. Trans. Amer. Math. Soc. 356(8), 3391–3402 (2004). https://doi.org/10.1090/S0002-9947-03-03413-5
  • Adamovi´c, D.: Rationality of Neveu-Schwarz vertex operator superalgebras. Int. Math. Res. Not. 17, 865–874 (1997). https://doi.org/10.1155/S107379289700055X
  • Adamovi´c, D.: Representations of the N = 2 superconformal vertex algebra. Int. Math. Res. Not. 1999(2), 61–79 (1999). https://doi.org/10.1155/S1073792899000033
  • Adamovi´c, D.: Vertex Algebra Approach to Fusion Rules forN = 2 SuperconformalMinimalModels. J. Algebra 239(2), 549–572 (2001). https://doi.org/10.1006/jabr.2000.8728
  • Adamovi´c, D.: Realizations of simple affine vertex algebras and their modules: the cases sl(2) and osp(1, 2). Comm. Math. Phys. 366, 1025–1067...
  • Adamovi´c, D., Creutzig, T., Genra, N.: Relaxed and logarithmic modules of sl(3). Math. Ann. 389, 281–324 (2024). https://doi.org/10.1007/s00208-023-02634-6
  • Adamovi´c, D., Kawasetsu, K., Ridout, D.: A realisation of the Bershadsky-Polyakov algebras and their relaxed modules. Lett. Math. Phys....
  • Adamovi´c, D., Möseneder Frajria, P., Papi, P.: New approaches for studying conformal embeddings and collapsing levels for W-algebras. Int....
  • Ai, C., Lin, X.: On the unitary structures of vertex operator superalgebras. J. Algebra 487, 217–243 (2017). https://doi.org/10.1016/j.jalgebra.2017.05.030
  • Arakawa, T.: Representation theory of W -algebras. Invent. Math. 169(2), 219–320 (2007). https:// doi.org/10.1007/s00222-007-0046-1
  • Arakawa, T.: Rationality of Bershadsky-Polyakov vertex algebras. Comm. in Math. Phys. 323(2), 627–633 (2013). https://doi.org/10.1007/s00220-013-1780-4
  • Arakawa, T.: Two-sided BGG resolutions of admissible representations. Represent. Theory 18, 183– 222 (2014). https://doi.org/10.1090/S1088-4165-2014-00454-0
  • Arakawa, T.: Rationality of W-algebras: principal nilpotent cases. Ann. of Math. 182(2), 565–604 (2015). https://doi.org/10.4007/annals.2015.182.2.4
  • Arakawa, T.: Associated varieties of modules over Kac-Moody algebras and C2-cofiniteness of Walgebras. Int. Math. Res. Notices 22, 11605–11666...
  • Arakawa, T.: Rationality of admissible affine vertex algebras in the category O. Duke Math. J. 165(1), 67–93 (2016). https://doi.org/10.1215/00127094-3165113
  • Arakawa, T.: Introduction to W-algebras and their representation theory, Perspectives in Lie Theory. Springer INdAM Ser., Vol. 19, Springer,...
  • Arakawa, T., Creutzig, T., Feigin, B.: Urod algebras and Translation of W-algebras. Forum of Mathematics, Sigma 10(1), E33 (2022). https://doi.org/10.1017/fms.2022.15
  • Arakawa, T., Creutzig, T., Kawasetsu, K., Linshaw, A.R.: Orbifolds and cosets of minimal W-algebras. Comm. Math. Phys. 355(1), 339–372 (2017)....
  • Arakawa, T., Creutzig, T., Linshaw, A.R.: W-algebras as coset vertex algebras. Invent. Math. 218, 145–195 (2019). https://doi.org/10.1007/s00222-019-00884-3
  • Arakawa, T., van Ekeren, J.: Rationality and fusion rules of exceptional W-algebras. J. Eur. Math. Soc. 25(7), 2763–2813 (2023). https://doi.org/10.4171/JEMS/1250
  • Arakawa, T., van Ekeren, J., Moreau, A.: Singularities of nilpotent Slodowy slices and collapsing levels of W-algebras, arXiv:2102.13462 [math.RT]
  • Arakawa, T., Frenkel, E.: Quantum Langlands duality of representations of W-algebras. Compos. Math. 155(12), 2235–2262 (2019). https://doi.org/10.1112/S0010437X19007553
  • Beem, C., Meneghelli, C.: A geometric free field realisation for the genus-two class S theory of type a1. Phys. Rev. D 104, 065015 (2021)....
  • Bernstein, I.N., Gelfand, I.M., Gelfand, S.I.: Differential operators on the base affine space and a study of g-modules, Lie groups and their...
  • Collingwood, D., McGovern, W.: Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold...
  • Creutzig, T.: Fusion categories for affine vertex algebras at admissible levels. Sel. Math. New Ser. 25, 27 (2019). https://doi.org/10.1007/s00029-019-0479-6
  • Creutzig, T., Kanade, S., McRae, R.: Tensor categories for vertex operator superalgebra extensions, Mem. Am. Math. Soc., 295(1472), (2024),...
  • Creutzig, T., Genra, N., Nakatsuka, S.: Duality of subregular W-algebras and principal Wsuperalgebras. Adv. Math. 383, 107685 (2021). https://doi.org/10.1016/j.aim.2021.107685
  • Creutzig, T., Genra, N., Nakatsuka, S., Sato, R.: Correspondences of categories for subregular Walgebras and W-superalgebras Commun. Math....
  • Creutzig, T., Huang, Y., Yang, J.: Braided tensor categories of admissible modules for affine Lie algebras. Commun. Math. Phys. 362, 827–854...
  • Creutzig, T., Kanade, S., Linshaw, A.R., Ridout, D.: Schur-Weyl duality for Heisenberg cosets. Transform. Groups 24, 301–354 (2019). https://doi.org/10.1007/s00031-018-9497-2
  • Creutzig, T., Linshaw, A.R.: Cosets of the Wk (sl4, fsub)-algebra. Contemp. Math. 711, 105–118 (2018). https://doi.org/10.1090/conm/711
  • Creutzig, T., Linshaw, A.R.: Cosets of affine vertex algebras inside larger structures. J. Algebra 517, 396–438 (2019). https://doi.org/10.1016/j.jalgebra.2018.10.007
  • Creutzig, T., Linshaw, A.R.: Trialities of W-algebras. Cambridge J. Math. 10(1), 69–194 (2022). https://doi.org/10.4310/CJM.2022.v10.n1.a2
  • Creutzig, T., Linshaw, A.R.: Trialities of orthosymplecticW-algebras. Adv.Math. 409, 108678 (2022). https://doi.org/10.1016/j.aim.2022.108678
  • Creutzig, T., Linshaw, A.R., Nakatsuka, S., Sato, R.: Duality via convolution of W-algebras, arXiv:2203.01843 [math.QA]
  • Creutzig, T., Liu, T., Ridout, D., Wood, S.: Unitary and non-unitary N = 2 minimal models. J. High Energ. Phys. 24, 2019 (2019). https://doi.org/10.1007/JHEP06(2019)024
  • Creutzig, T., Ridout, D.: Modular data and Verlinde formulae for fractional level WZW models I. Nucl. Phys. B 865(1), 83–114 (2012). https://doi.org/10.1016/j.nuclphysb.2012.07.018
  • Creutzig, T., Ridout, D.: Modular data and Verlinde formulae for fractional level WZW models II. Nucl. Phys. B 875(2), 423–458 (2013). https://doi.org/10.1016/j.nuclphysb.2013.07.008
  • De Sole, A., Kac, V.: Finite vs affine W-algebras. Jpn. J. Math. 1, 137–261 (2006). https://doi.org/ 10.1007/s11537-006-0505-2
  • Dhillon, G.: Semi-infinite cohomology and the linkage principle for W -algebras. Adv. Math. 381, 107625 (2021). https://doi.org/10.1016/j.aim.2021.107625
  • Dhillon, G., Raskin, S.: Localization for affine W -algebras. Adv. Math. 413, 108837 (2023). https:// doi.org/10.1016/j.aim.2022.108837
  • di Francesco, P., Mathieu, P., Sénéchal, D.: Conformal Field Theory, Graduate Texts in Contemp. Phys., (1997)
  • Di Vecchia, P., Petersen, J.L., Yu, M., Zheng, H.B.: Explicit Construction of Unitary Representations of the N = 2 Superconformal Algebra....
  • Dong, C., Lin, X.: Unitary vertex operator algebras. J. Algebra 397, 252–277 (2014). https://doi.org/ 10.1016/j.jalgebra.2013.09.007
  • 47. charges c < 1. Comm. Math. Phys. 340, 613–637 (2015). https://doi.org/10.1007/s00220-015-2468- 8
  • Dong, C., Li, H., Mason, G.: Compact automorphism groups of vertex operator algebras. Int. Math. Res. Not. 18, 913–921 (1996). https://doi.org/10.1155/S1073792896000566
  • Dong, C., Li, H., Mason, G.: Twisted representations of vertex operator algebras. Math. Ann. 310(3), 571–600 (1998). https://doi.org/10.1007/s002080050161
  • Dong, C., Mason, G.: Rational vertex operator algebras and the effective central charge. Int. Math. Res. Not. 56, 2989–3008 (2004). https://doi.org/10.1155/S1073792804140968
  • Dong, C., Nagatomo, K.: Representations of vertex operator algebra V + L for rank one lattice L. Comm. Math. Phys. 202, 169–195 (1999)....
  • Elashvili, A., Kac, V.: Classification of good gradings of simple Lie algebras. Amer. Math. Soc. Transl. Ser. 2(213), 85–104 (2005). https://doi.org/10.1090/trans2/213/05
  • Fasquel, J.: Rationality of the exceptional W-algebras Wk (sp4, fsubreg) associated with subregular nilpotent elements of sp4. Commun. Math....
  • Fasquel, J.: OPEs of rank two W-algebras, Contemp. Math., 813(2), (2025), https://doi.org/10.1090/ conm/813
  • Fasquel, J., Linshaw, A., Nakatsuka, S.: work in progress
  • Feigin, B.: Semi-infinite homology of Lie, Kac-Moody and Virasoro algebras. Uspekhi Mat. Nauk 39(2), 195–196 (1984). https://doi.org/10.1070/RM1984v039n02ABEH003112
  • Fehily, Z.: Subregular W-algebras of type A, Commun. Contemp. Math., 2250049, (2022), https:// doi.org/10.1142/S0219199722500493
  • Fehily, Z.: Inverse reduction for hook-type W-algebras. Commun. Math. Phys. 405, 214 (2024). https://doi.org/10.1007/s00220-024-05082-8
  • Fehily, Z., Ridout, D.: Modularity of Bershadsky-Polyakov minimal models. Lett. Math. Phys. 112, 46 (2022). https://doi.org/10.1007/s11005-022-01536-z
  • Feigin, B., Frenkel, E.: A family of representations of affine Lie algebras. Uspekhi Mat. Nauk 43(5), 227–228 (1988). https://doi.org/10.1070/RM1988v043n05ABEH001935
  • Feigin, B., Frenkel, E.: Affine Kac-Moody algebras and semi-infinite flag manifolds. Comm. Math. Phys. 128, 161–189 (1990). https://doi.org/10.1007/BF02097051
  • Feigin, B., Frenkel, E.: Quantization of the Drinfeld-Sokolov reduction. Phys. Lett. B246(1–2), 75–81 (1990). https://doi.org/10.1016/0370-2693(90)91310-8
  • Feigin, B., Frenkel, E.: Duality in W-algebras. Int. Math. Res. Not. 6, 75–82 (1991). https://doi.org/ 10.1155/S1073792891000119
  • Feigin, B., Frenkel, E.: Affine Kac–Moody algebras at the critical level and Gelfand–Dikii algebras, Infinite Analysis, Part A, B (Kyoto,...
  • Feigin, B., Semikhatov, A.: W(2) n -algebras. Nucl. Phys. B 698(3), 409–449 (2004). https://doi.org/ 10.1016/j.nuclphysb.2004.06.056
  • Feigin, B., Semikhatov, A., Sirota, V., Tipunin, Y.: Resolutions and characters of irreducible representations of the N = 2 superconformal...
  • Feigin, B., Semikhatov, A., Tipunin, Y.: Equivalence between chain categories of representations of affine sl(2) and N = 2 superconformal...
  • Fiebig, P.: The combinatorics of category O over symmetrizable Kac-Moody algebras. Transform. Groups 11, 29–49 (2006). https://doi.org/10.1007/s00031-005-1103-8
  • Frenkel, E.: Langlands correspondence for loop groups. Cambridge Studies in Advanced Mathematics, vol. 103. Cambridge University Press, Cambridge...
  • Frenkel, E., Ben-Zvi, D.: Vertex algebras and algebraic curves, Mathematical Surveys and Monographs, 88. American Mathematical Society, Providence,...
  • Frenkel, I., Garland, H., Zuckerman, G.: Semi-infinite cohomology and string theory. Proc. Nat. Acad. Sci. U.S.A. 83(22), 8442–8446 (1986)....
  • Frenkel, E., Kac, V., Wakimoto, M.: Characters and fusion rules for W-algebras via quantized Drinfeld-Sokolov reduction. Comm. Math. Phys....
  • Frenkel, I., Zhu, Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66(1), 123–168...
  • Friedan, D., Martinec, E., Shenker, S.: Conformal invariance, supersymmetry and string theory. Nuclear Phys. B 271(1), 93–165 (1986). https://doi.org/10.1016/S0550-3213(86)80006-2
  • Fuchs, J.: Simple WZW currents. Commun. Math. Phys. 136(2), 345–356 (1991). https://doi.org/10. 1007/BF02100029
  • Fulton, W., Harris, J.: Representation theory, Graduate Texts in Mathematics. A first course, Readings in Mathematics, vol. 129. Springer-Verlag,...
  • Gaiotto, D., Rapˇcák, M.: Vertex Algebras at the Corner, J. High Energ. Phys., 01(160), (2019), https:// doi.org/10.1007/JHEP01(2019)160
  • Gaitsgory, D.: Quantum Langlands correspondence, arXiv:1601.05279 [math.AG]
  • Genra, N.: Screening operators for W-algebras. Sel. Math. New Ser. 23, 2157–2202 (2017). https:// doi.org/10.1007/s00029-017-0315-9
  • Genra, N.: Screening operators and parabolic inductions for affine W-algebras. Adv. Math. 369, 107179 (2020). https://doi.org/10.1016/j.aim.2020.107179
  • Goddard, P., Kent, A., Olive, D.I.: Unitary Representations of the Virasoro and Supervirasoro Algebras. Commun. Math. Phys. 103, 105–119 (1986)....
  • Huang, Y.Z.: Rigidity and modularity of vertex tensor categories. Comm. Contemp. Math. 10, 871– 911 (2008). https://doi.org/10.1142/S0219199708003083
  • Huang, Y.Z., Lepowsky, J.: A theory of tensor products for module categories for a vertex operator algebra I, II, III. Sel. Math. New Ser....
  • Huang, Y.Z., Lepowsky, J.: A theory of tensor products for module categories for a vertex operator algebra I, II, III. Sel. Math. New Ser....
  • Huang, Y.Z., Lepowsky, J.: A theory of tensor products for module categories for a vertex operator algebra I, II, III. J. Pure Appl. Algebra...
  • Huang, Y.Z., Lepowsky, J., Zhang, L.: A logarithmic generalization of tensor product theory for modules for a vertex operator algebra. Internat....
  • Kac, V., Roan, S., Wakimoto, M.: Quantum reduction for affine superalgebras. Comm. Math. Phys. 241, 307–342 (2003). https://doi.org/10.1007/s00220-003-0926-1
  • Kac, V., Wakimoto, M.: Classification of modular invariant representations of affine algebras, Infinitedimensional Lie algebras and groups,...
  • Kac, V., Wakimoto, M.: Branching functions for winding subalgebras and tensor products. Acta Appl. Math. 21(1–2), 3–39 (1990). https://doi.org/10.1007/BF00053290
  • Kac, V., Wakimoto, M.: Quantum reduction and representation theory of superconformal algebras. Adv. Math. 185(2), 400–458 (2004). https://doi.org/10.1016/j.aim.2003.12.005
  • Kac, V., Wakimoto, M.: On rationality of W-algebras. Transform. Groups 13, 671–713 (2008). https:// doi.org/10.1007/s00031-008-9028-7
  • Kac, V., Wakimoto, M.: On Free Field Realization of Quantum Affine W-Algebras. Commun. Math. Phys. 395, 571–600 (2022). https://doi.org/10.1007/s00220-022-04436-4
  • Kanade, S., Linshaw, A.R.: Universal two-parameter even spin W∞-algebra. Adv. Math. 315, 106774 (2019). https://doi.org/10.1016/j.aim.2019.106774
  • Kazama, Y., Suzuki, H.: New N = 2 superconformal field theories and superstring compactification. Nucl. Phys. B 321(1), 232–268 (1989)....
  • Lam, C., Lam, N., Yamauchi, H.: Extension of unitary Virasoro vertex operator algebra by a simple module. Int. Math. Res. Not. 11, 577–611...
  • Lam, C., Yamada, H.: Tricritical 3-state Potts model and vertex operator algebras constructed from ternary codes. Comm. Algebra 32(11), 4197–4219...
  • Li, H.: Symmetric invariant bilinear forms on vertex operator algebras. J. Pure Appl. Algebra 96(3), 279–297 (1994). https://doi.org/10.1016/0022-4049(94)90104-X
  • Li, H.: The physics superselection principle in vertex operator algebra theory. J. Algebra 196(2), 436–457 (1997). https://doi.org/10.1006/jabr.1997.7126
  • Losev, I.: Finite-dimensional representations of W-algebras. Duke Math. J. 159(1), 99–143 (2011). https://doi.org/10.1215/00127094-1384800
  • Losev, I., Tsymbaliuk, A.: Infinitesimal Cherednik algebras as W-algebras. Transform. Groups 19(1), 495–526 (2014). https://doi.org/10.1007/s00031-014-9261-1
  • Mason, G.: Lattice subalgebras of strongly regular vertex operator algebras, Conformal field theory, automorphic forms and related topics,...
  • McRae, R.: On rationality for C2-cofinite vertex operator algebras, arXiv:2108.01898 [math. QA]
  • Moriwaki, Y.: Quantum coordinate ring in WZW model and affine vertex algebra extensions. Sel. Math. New Ser. 28, 68 (2022). https://doi.org/10.1007/s00029-022-00782-2
  • Nakatsuka, S.: On Miura maps for W-superalgebras. Transform. Groups 28, 375–399 (2023). https:// doi.org/10.1007/s00031-021-09679-4
  • Premet, A.: Special transverse slices and their enveloping algebras. Adv. Math. 170(1), 1–55 (2002). https://doi.org/10.1006/aima.2001.2063
  • Raskin, S.: W-algebras and Whittaker categories. Sel. Math. New Ser. 27, 46 (2021). https://doi.org/ 10.1007/s00029-021-00641-6
  • Sato, R.: Modular invariant representations over the N = 2 superconformal algebra. Int. Math. Res. Not. 24, 7659–7690 (2019). https://doi.org/10.1093/imrn/rny007
  • Semikhatov, A.: Inverting the Hamiltonian reduction in string theory, In 28th International Symposium on Particle Theory, Wendisch-Rietz,...
  • Stokman, J.: Hyperbolic beta integrals. Adv. Math. 190(1), 119–160 (2005). https://doi.org/10.1016/ j.aim.2003.12.003
  • Taubes, C.: Differential geometry, Oxford Graduate Texts in Mathematics, 23. Oxford University Press, Oxford (2011)
  • Tsymbaliuk, A.: Infinitesimal Hecke algebras of soN . J. Pure Appl. Algebra 219(6), 2046–2061 (2015). https://doi.org/10.1016/j.jpaa.2014.07.022
  • Wakimoto, M.: Fock representations of the affine Lie algebra A(1) 1 . Comm. Math. Phys. 104, 605–609 (1986). https://doi.org/10.1007/BF01211068
  • Yamada, H., Yamauchi, H.: Simple current extensions of tensor products of vertex operator algebras. Int. Math. Res. Not. 16, 12778–12807 (2021)....
  • Zhu, Y.: Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9(1), 237–302 (1996). https://doi.org/10.1090/S0894-0347-96-00182-8