W-algebras as coset vertex algebras

  • Tomoyuki Arakawa
  • Thomas CreutzigEmail author
  • Andrew R. Linshaw


We prove the long-standing conjecture on the coset construction of the minimal series principal W-algebras of ADE types in full generality. We do this by first establishing Feigin’s conjecture on the coset realization of the universal principal W-algebras, which are not necessarily simple. As consequences, the unitarity of the “discrete series” of principal W-algebras is established, a second coset realization of rational and unitary W-algebras of type A and D are given and the rationality of Kazama–Suzuki coset vertex superalgebras is derived.



This work started when we visited Perimeter Institute for Theoretical Physics, Canada, for the conference “Exact operator algebras in superconformal field theories” in December 2016. We thank the organizers of the conference and the institute. The first named author would like to thank MIT for its hospitality during his visit from February 2016 to January 2018.


  1. 1.
    Adamovic, D., Kac, V.G., Frajria, P.M., Papi, P., Perse, O.: Finite vs infinite decompositions in conformal embeddings. Commun. Math. Phys. 348, 445–473 (2016)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aganagic, M., Frenkel, E., Okounkov, A.: Quantum q-Langlands correspondence. Trans. Mosc. Math. Soc. 79, 1–83 (2018)MathSciNetGoogle Scholar
  3. 3.
    Alday, L.F., Gaiotto, D., Tachikawa, Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91(2), 167–197 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Altschuler, D., Bauer, M., Itzykson, C.: The branching rules of conformal embeddings. Commun. Math. Phys. 132(2), 349–364 (1990)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Aomoto, K., Kita, M.: Theory of Hypergeometric Functions. Springer Monographs in Mathematics. Springer, Tokyo (2011). With an appendix by Toshitake Kohno, Translated from the Japanese by Kenji IoharaGoogle Scholar
  6. 6.
    Arakawa, T.: Vanishing of cohomology associated to quantized Drinfeld–Sokolov reduction. Int. Math. Res. Not. 15, 730–767 (2004)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Arakawa, T.: Representation theory of $W$-algebras. Invent. Math. 169(2), 219–320 (2007)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Arakawa, T.: A remark on the $C_2$ cofiniteness condition on vertex algebras. Math. Z. 270(1–2), 559–575 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Arakawa, T.: Two-sided BGG resolution of admissible representations. Represent. Theory 18(3), 183–222 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Arakawa, T.: Associated varieties of modules over Kac–Moody algebras and $C_2$-cofiniteness of W-algebras. Int. Math. Res. Not. 11605–11666, 2015 (2015)zbMATHGoogle Scholar
  11. 11.
    Arakawa, T.: Rationality of W-algebras: principal nilpotent cases. Ann. Math. 182(2), 565–694 (2015)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Arakawa, T.: Rationality of admissible affine vertex algebras in the category $\cal{O}$. Duke Math. J. 165(1), 67–93 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Arakawa, T.: Introduction to W-algebras and their representation theory. In: Callegaro, F., Carnovale, G., Caselli, F., De Concini, C., De Sole, A. (eds.) Perspectives in Lie Theory. Springer INdAM Series, vol. 19. Springer, Cham (2017)Google Scholar
  14. 14.
    Arakawa, T., Creutzig, T., Kawasetsu, K., Linshaw, A.R.: Orbifolds and cosets of minimal $W$-algebras. Commun. Math. Phys. 355(1), 339–372 (2017)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Arakawa, T., Jiang, C.: Coset vertex operator algebras and W-algebras. Sci. China Math. 61(2), 191–206 (2017)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Arakawa, T., Lam, C.H., Yamada, H.: Parafermion vertex operator algebras and W-algebras. Trans. Am. Math. Soc. 371(6), 4277–4301 (2019)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Bais, F.A., Bouwknegt, P., Surridge, M., Schoutens, K.: Coset construction for extended Virasoro algebras. Nucl. Phys. B 304(2), 371–391 (1988)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Bakalov, B., Milanov, T.: $\cal{W}$-constraints for the total descendant potential of a simple singularity. Compos. Math. 149(5), 840–888 (2013)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Beilinson, A., Feigin, B., Mazur, B.: Introduction to algebraic field theory on curves. PreprintGoogle Scholar
  20. 20.
    Belavin, A.A.: KdV-type equations and $W$-algebras. Integrable Systems in Quantum Field Theory. Advanced Studies in Pure Mathematics, vol. 19, pp. 117–125. Academic Press, San Diego (1989)Google Scholar
  21. 21.
    Braverman, A., Finkelberg, M., Nakajima, H.: Instanton moduli spaces and $\cal{W}$-algebras. Astérisque 385, 128 (2016)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241(2), 333–380 (1984)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Carnahan, S., Miyamoto, M.: Regularity of fixed-point vertex operator subalgebras. arXiv:1603.05645 Google Scholar
  24. 24.
    Creutzig, T., Kanade, S., McRae, R.: Tensor categories for vertex operator superalgebra extensions. arXiv:1705.05017 Google Scholar
  25. 25.
    Creutzig, T., Linshaw, A.R.: Cosets of affine vertex algebras inside larger structures. J. Algebra 517, 396–438 (2019)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Creutzig, T., Gaiotto, D.: Vertex Algebras for S-duality. arXiv:1708.00875 Google Scholar
  27. 27.
    Creutzig, T., Gaiotto, D., Linshaw, A.R.: S-duality for the large $N=4$ superconformal algebra. arXiv:1804.09821 Google Scholar
  28. 28.
    Creutzig, T., Hikida, Y., Ronne, P.B.: Higher spin $\text{ AdS }_3$ supergravity and its dual CFT. JHEP 1202, 109 (2012)zbMATHGoogle Scholar
  29. 29.
    Creutzig, T., Hikida, Y., Ronne, P.B.: $N=1$ supersymmetric higher spin holography on AdS$_3$. JHEP 1302, 019 (2013)zbMATHGoogle Scholar
  30. 30.
    Dickey, L.A.: Soliton Equations and Hamiltonian Systems. Advanced Series in Mathematical Physics, vol. 26, 2nd edn. World Scientific Publishing, Singapore (2003)zbMATHGoogle Scholar
  31. 31.
    De Sole, A., Kac, V.G.: Finite vs affine $W$-algebras. Jpn. J. Math. 1(1), 137–261 (2006)MathSciNetzbMATHGoogle Scholar
  32. 32.
    De Sole, A., Kac, V.G., Valeri, D.: Classical $\mathscr {W}$-algebras and generalized Drinfeld–Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras. Commun. Math. Phys. 323(2), 663–711 (2013)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Dong, C., Lin, X.: Unitary vertex operator algebras. J. Algebra 397, 252–277 (2014)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Dong, C., Mason, G.: Rational vertex operator algebras and the effective central charge. Int. Math. Res. Not. 56, 2989–3008 (2004)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Feigin, B.: Semi-infinite homology of Lie, Kac–Moody and Virasoro algebras. Uspekhi Mat. Nauk. 39(2(236)), 195–196 (1984)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Feigin, B., Frenkel, E.: Quantization of the Drinfeld–Sokolov reduction. Phys. Lett. B 246(1–2), 75–81 (1990)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Feigin, B., Frenkel, E.: Affine Kac–Moody algebras and semi-infinite flag manifolds. Commun. Math. Phys. 128(1), 161–189 (1990)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Feigin, B., Frenkel, E.: Duality in $W$-algebras. Int. Math. Res. Not. 6, 75–82 (1991)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Feigin, B., Frenkel, E.: Affine Kac–Moody algebras at the critical level and Gelfand–Dikii algebras. In: Infinite Analysis, Part A, B (Kyoto, 1991). Advances Series in Mathematical Physics, vol. 16, pp. 197–215. World Scientific Publishing, River Edge, NJ (1992)Google Scholar
  40. 40.
    Feigin, B., Frenkel, E.: Integrals of motion and quantum groups. In: Integrable Systems and Quantum Groups (Montecatini Terme, 1993). Lecture Notes in Mathematics, vol. 1620, pp. 349–418. Springer, Berlin (1996)Google Scholar
  41. 41.
    Frenkel, E., Gaiotto, D.: Quantum Langlands dualities of boundary conditions, D-modules, and conformal blocks. arXiv:1805.00203 [hep-th]Google Scholar
  42. 42.
    Frenkel, I.B., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Am. Math. Soc. 104(494), viii+64 (1993)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Fiebig, P.: The combinatorics of category $\cal{O}$ over symmetrizable Kac–Moody algebras. Transform. Groups 11(1), 29–49 (2006)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Feigin, B., Jimbo, M., Miwa, T., Mukhin, E.: Branching rules for quantum toroidal $\mathfrak{gl}_n$. Adv. Math. 300, 229–274 (2016)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Frenkel, E., Ben-Zvi, D.: Vertex Algebras and Algebraic Curves. Mathematical Surveys and Monographs, vol. 88, 2nd edn. American Mathematical Society, Providence, RI (2004)zbMATHGoogle Scholar
  46. 46.
    Frenkel, I.B., Kac, V.G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math. 62(1), 23–66 (1980/81)Google Scholar
  47. 47.
    Frenkel, E., Kac, V., Wakimoto, M.: Characters and fusion rules for $W$-algebras via quantized Drinfeld–Sokolov reduction. Commun. Math. Phys. 147(2), 295–328 (1992)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Fateev, V.A., Lykyanov, S.L.: The models of two-dimensional conformal quantum field theory with $Z_n$ symmetry. Int. J. Modern Phys. A 3(2), 507–520 (1988)MathSciNetGoogle Scholar
  49. 49.
    Frenkel, I.B.: Representations of affine Lie algebras, Hecke modular forms and Korteweg–de Vries type equations. In: Lie Algebras and Related Topics (New Brunswick, NJ, 1981). Lecture Notes in Mathematics, vol. 933, pp. 71–110. Springer, Berlin (1982)Google Scholar
  50. 50.
    Frenkel, E.: $\mathscr {W}$-algebras and Langlands–Drinfeld correspondence. In: New Symmetry Principles in Quantum Field Theory (Cargèse, 1991). NATO Advanced Study Institute, Series B: Physics, vol. 295, pp. 433–447. Plenum, New York (1992)Google Scholar
  51. 51.
    Frenkel, E.: Wakimoto modules, opers and the center at the critical level. Adv. Math. 195(2), 297–404 (2005)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Frenkel, E.: Langlands Correspondence for Loop Groups. Cambridge Studies in Advanced Mathematics, vol. 103. Cambridge University Press, Cambridge (2007)zbMATHGoogle Scholar
  53. 53.
    Frenkel, I.B., Zhu, Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66(1), 123–168 (1992)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Gaitsgory, D.: Quantum Langlands correspondence. arXiv:1601.05279 [math.AG]Google Scholar
  55. 55.
    Gaitsgory, D.: The master chiral algebras. Talk at Perimeter Institute.
  56. 56.
    Gepner, D.: Space-time supersymmetry in compactified string theory and superconformal models. Nucl. Phys. B 296, 757 (1988)MathSciNetGoogle Scholar
  57. 57.
    Genra, N.: Screening operators for W-algebras. Sel. Math. New Ser. 23(3), 2157–2202 (2017)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Gaberdiel, M.R., Gopakumar, R.: An ${A}d{S}_3$ dual for minimal model CFTs. Phys. Rev. D 83, 066007 (2011)Google Scholar
  59. 59.
    Goddard, P., Kent, A., Olive, D.: Unitary representations of the Virasoro and super-Virasoro algebras. Commun. Math. Phys. 103(1), 105–119 (1986)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Huang, Y.-Z., Kirillov Jr., A., Lepowsky, J.: Braided tensor categories and extensions of vertex operator algebras. Commun. Math. Phys. 337, 1143–1159 (2015)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Jiang, C., Lam, C.H.: Level-rank duality for vertex operator algebras of types B and D. arXiv:1703.04889 Google Scholar
  62. 62.
    Kac, V.G.: Infinite-Dimensional Lie Algebras, third edn. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  63. 63.
    Kac, V.G., Frajria, P.M., Papi, P., Xu, F.: Conformal embeddings and simple current extensions. IMRN 14, 5229–5288 (2015)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Kac, V.G., Kazhdan, D.A.: Structure of representations with highest weight of infinite-dimensional Lie algebras. Adv. Math. 34(1), 97–108 (1979)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Kac, V., Raina, A.: Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras. World Scientific, Singapore (1987)zbMATHGoogle Scholar
  66. 66.
    Kac, V., Roan, S.-S., Wakimoto, M.: Quantum reduction for affine superalgebras. Commun. Math. Phys. 241(2–3), 307–342 (2003)MathSciNetzbMATHGoogle Scholar
  67. 67.
    Kac, V., Wakimoto, M.: Classification of modular invariant representations of affine algebras. In: Infinite-Dimensional Lie Algebras and Groups (Luminy-Marseille, 1988). Advances Series in Mathematical Physics, vol. 7, pp. 138–177. World Scientific Publishing, Teaneck, NJ (1989)Google Scholar
  68. 68.
    Kac, V.G., Wakimoto, M.: Branching functions for winding subalgebras and tensor products. Acta Appl. Math. 21(1–2), 3–39 (1990)MathSciNetzbMATHGoogle Scholar
  69. 69.
    Kac, V.G., Wakimoto, M.: On rationality of $W$-algebras transform. Groups 13(3–4), 671–713 (2008)MathSciNetzbMATHGoogle Scholar
  70. 70.
    Kazama, Y., Suzuki, H.: New $\text{ N }=2$ superconformal field theories and superstring compactification. Nucl. Phys. B 321, 232 (1989)MathSciNetGoogle Scholar
  71. 71.
    Li, H.: Abelianizing vertex algebras. Commun. Math. Phys. 259(2), 391–411 (2005)MathSciNetzbMATHGoogle Scholar
  72. 72.
    Matsuo, A., Nagatomo, K., Tsuchiya, A.: Quasi-finite algebras graded by Hamiltonian and vertex operator algebras. In: Moonshine: The First Quarter Century and Beyond. London Mathematical Society Lecture Note Series, vol. 372, pp. 282–329. Cambridge University Press, Cambridge (2010)Google Scholar
  73. 73.
    Nakanishi, T., Tsuchiya, A.: Level-rank duality of WZW models in conformal field theory. Commun. Math. Phys. 144(2), 351–372 (1992)MathSciNetzbMATHGoogle Scholar
  74. 74.
    Ostrik, V., Sun, M.: Level-rank duality via tensor categories. Commun. Math. Phys. 326, 49–61 (2014)MathSciNetzbMATHGoogle Scholar
  75. 75.
    Schiffmann, O., Vasserot, E.: Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on $\mathbf{A}^2$. Publ. Math. Inst. Hautes Études Sci. 118, 213–342 (2013)MathSciNetzbMATHGoogle Scholar
  76. 76.
    Tsuchiya, A., Kanie, Y.: Fock space representations of the Virasoro algebra. Intertwining operators. Publ. Res. Inst. Math. Sci. 22(2), 259–327 (1986)MathSciNetzbMATHGoogle Scholar
  77. 77.
    Voronov, A.A.: Semi-infinite homological algebra. Invent. Math. 113(1), 103–146 (1993)MathSciNetzbMATHGoogle Scholar
  78. 78.
    Voronov, A.A.: Semi-infinite induction and Wakimoto modules. Am. J. Math. 121(5), 1079–1094 (1999)MathSciNetzbMATHGoogle Scholar
  79. 79.
    Walton, M.A.: Conformal branching rules and modular invariants. Nucl. Phys. B 322, 775–790 (1989)MathSciNetGoogle Scholar
  80. 80.
    Wang, W.: Rationality of Virasoro vertex operator algebras. Int. Math. Res. Not. 7, 197–211 (1993)MathSciNetzbMATHGoogle Scholar
  81. 81.
    Zhu, Y.: Modular invariance of characters of vertex operator algebras. J. Am. Math. Soc. 9(1), 237–302 (1996)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Tomoyuki Arakawa
    • 1
  • Thomas Creutzig
    • 2
    Email author
  • Andrew R. Linshaw
    • 3
  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan
  2. 2.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  3. 3.Department of MathematicsUniversity of DenverDenverUSA

Personalised recommendations