Abstract
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.
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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)
Aganagic, M., Frenkel, E., Okounkov, A.: Quantum q-Langlands correspondence. Trans. Mosc. Math. Soc. 79, 1–83 (2018)
Alday, L.F., Gaiotto, D., Tachikawa, Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91(2), 167–197 (2010)
Altschuler, D., Bauer, M., Itzykson, C.: The branching rules of conformal embeddings. Commun. Math. Phys. 132(2), 349–364 (1990)
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 Iohara
Arakawa, T.: Vanishing of cohomology associated to quantized Drinfeld–Sokolov reduction. Int. Math. Res. Not. 15, 730–767 (2004)
Arakawa, T.: Representation theory of $W$-algebras. Invent. Math. 169(2), 219–320 (2007)
Arakawa, T.: A remark on the $C_2$ cofiniteness condition on vertex algebras. Math. Z. 270(1–2), 559–575 (2012)
Arakawa, T.: Two-sided BGG resolution of admissible representations. Represent. Theory 18(3), 183–222 (2014)
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)
Arakawa, T.: Rationality of W-algebras: principal nilpotent cases. Ann. Math. 182(2), 565–694 (2015)
Arakawa, T.: Rationality of admissible affine vertex algebras in the category $\cal{O}$. Duke Math. J. 165(1), 67–93 (2016)
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)
Arakawa, T., Creutzig, T., Kawasetsu, K., Linshaw, A.R.: Orbifolds and cosets of minimal $W$-algebras. Commun. Math. Phys. 355(1), 339–372 (2017)
Arakawa, T., Jiang, C.: Coset vertex operator algebras and W-algebras. Sci. China Math. 61(2), 191–206 (2017)
Arakawa, T., Lam, C.H., Yamada, H.: Parafermion vertex operator algebras and W-algebras. Trans. Am. Math. Soc. 371(6), 4277–4301 (2019)
Bais, F.A., Bouwknegt, P., Surridge, M., Schoutens, K.: Coset construction for extended Virasoro algebras. Nucl. Phys. B 304(2), 371–391 (1988)
Bakalov, B., Milanov, T.: $\cal{W}$-constraints for the total descendant potential of a simple singularity. Compos. Math. 149(5), 840–888 (2013)
Beilinson, A., Feigin, B., Mazur, B.: Introduction to algebraic field theory on curves. Preprint
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)
Braverman, A., Finkelberg, M., Nakajima, H.: Instanton moduli spaces and $\cal{W}$-algebras. Astérisque 385, 128 (2016)
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)
Carnahan, S., Miyamoto, M.: Regularity of fixed-point vertex operator subalgebras. arXiv:1603.05645
Creutzig, T., Kanade, S., McRae, R.: Tensor categories for vertex operator superalgebra extensions. arXiv:1705.05017
Creutzig, T., Linshaw, A.R.: Cosets of affine vertex algebras inside larger structures. J. Algebra 517, 396–438 (2019)
Creutzig, T., Gaiotto, D.: Vertex Algebras for S-duality. arXiv:1708.00875
Creutzig, T., Gaiotto, D., Linshaw, A.R.: S-duality for the large $N=4$ superconformal algebra. arXiv:1804.09821
Creutzig, T., Hikida, Y., Ronne, P.B.: Higher spin $\text{ AdS }_3$ supergravity and its dual CFT. JHEP 1202, 109 (2012)
Creutzig, T., Hikida, Y., Ronne, P.B.: $N=1$ supersymmetric higher spin holography on AdS$_3$. JHEP 1302, 019 (2013)
Dickey, L.A.: Soliton Equations and Hamiltonian Systems. Advanced Series in Mathematical Physics, vol. 26, 2nd edn. World Scientific Publishing, Singapore (2003)
De Sole, A., Kac, V.G.: Finite vs affine $W$-algebras. Jpn. J. Math. 1(1), 137–261 (2006)
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)
Dong, C., Lin, X.: Unitary vertex operator algebras. J. Algebra 397, 252–277 (2014)
Dong, C., Mason, G.: Rational vertex operator algebras and the effective central charge. Int. Math. Res. Not. 56, 2989–3008 (2004)
Feigin, B.: Semi-infinite homology of Lie, Kac–Moody and Virasoro algebras. Uspekhi Mat. Nauk. 39(2(236)), 195–196 (1984)
Feigin, B., Frenkel, E.: Quantization of the Drinfeld–Sokolov reduction. Phys. Lett. B 246(1–2), 75–81 (1990)
Feigin, B., Frenkel, E.: Affine Kac–Moody algebras and semi-infinite flag manifolds. Commun. Math. Phys. 128(1), 161–189 (1990)
Feigin, B., Frenkel, E.: Duality in $W$-algebras. Int. Math. Res. Not. 6, 75–82 (1991)
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)
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)
Frenkel, E., Gaiotto, D.: Quantum Langlands dualities of boundary conditions, D-modules, and conformal blocks. arXiv:1805.00203 [hep-th]
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)
Fiebig, P.: The combinatorics of category $\cal{O}$ over symmetrizable Kac–Moody algebras. Transform. Groups 11(1), 29–49 (2006)
Feigin, B., Jimbo, M., Miwa, T., Mukhin, E.: Branching rules for quantum toroidal $\mathfrak{gl}_n$. Adv. Math. 300, 229–274 (2016)
Frenkel, E., Ben-Zvi, D.: Vertex Algebras and Algebraic Curves. Mathematical Surveys and Monographs, vol. 88, 2nd edn. American Mathematical Society, Providence, RI (2004)
Frenkel, I.B., Kac, V.G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math. 62(1), 23–66 (1980/81)
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)
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)
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)
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)
Frenkel, E.: Wakimoto modules, opers and the center at the critical level. Adv. Math. 195(2), 297–404 (2005)
Frenkel, E.: Langlands Correspondence for Loop Groups. Cambridge Studies in Advanced Mathematics, vol. 103. Cambridge University Press, Cambridge (2007)
Frenkel, I.B., Zhu, Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66(1), 123–168 (1992)
Gaitsgory, D.: Quantum Langlands correspondence. arXiv:1601.05279 [math.AG]
Gaitsgory, D.: The master chiral algebras. Talk at Perimeter Institute. https://www.perimeterinstitute.ca/videos/master-chiral-algebra
Gepner, D.: Space-time supersymmetry in compactified string theory and superconformal models. Nucl. Phys. B 296, 757 (1988)
Genra, N.: Screening operators for W-algebras. Sel. Math. New Ser. 23(3), 2157–2202 (2017)
Gaberdiel, M.R., Gopakumar, R.: An ${A}d{S}_3$ dual for minimal model CFTs. Phys. Rev. D 83, 066007 (2011)
Goddard, P., Kent, A., Olive, D.: Unitary representations of the Virasoro and super-Virasoro algebras. Commun. Math. Phys. 103(1), 105–119 (1986)
Huang, Y.-Z., Kirillov Jr., A., Lepowsky, J.: Braided tensor categories and extensions of vertex operator algebras. Commun. Math. Phys. 337, 1143–1159 (2015)
Jiang, C., Lam, C.H.: Level-rank duality for vertex operator algebras of types B and D. arXiv:1703.04889
Kac, V.G.: Infinite-Dimensional Lie Algebras, third edn. Cambridge University Press, Cambridge (1990)
Kac, V.G., Frajria, P.M., Papi, P., Xu, F.: Conformal embeddings and simple current extensions. IMRN 14, 5229–5288 (2015)
Kac, V.G., Kazhdan, D.A.: Structure of representations with highest weight of infinite-dimensional Lie algebras. Adv. Math. 34(1), 97–108 (1979)
Kac, V., Raina, A.: Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras. World Scientific, Singapore (1987)
Kac, V., Roan, S.-S., Wakimoto, M.: Quantum reduction for affine superalgebras. Commun. Math. Phys. 241(2–3), 307–342 (2003)
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)
Kac, V.G., Wakimoto, M.: Branching functions for winding subalgebras and tensor products. Acta Appl. Math. 21(1–2), 3–39 (1990)
Kac, V.G., Wakimoto, M.: On rationality of $W$-algebras transform. Groups 13(3–4), 671–713 (2008)
Kazama, Y., Suzuki, H.: New $\text{ N }=2$ superconformal field theories and superstring compactification. Nucl. Phys. B 321, 232 (1989)
Li, H.: Abelianizing vertex algebras. Commun. Math. Phys. 259(2), 391–411 (2005)
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)
Nakanishi, T., Tsuchiya, A.: Level-rank duality of WZW models in conformal field theory. Commun. Math. Phys. 144(2), 351–372 (1992)
Ostrik, V., Sun, M.: Level-rank duality via tensor categories. Commun. Math. Phys. 326, 49–61 (2014)
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)
Tsuchiya, A., Kanie, Y.: Fock space representations of the Virasoro algebra. Intertwining operators. Publ. Res. Inst. Math. Sci. 22(2), 259–327 (1986)
Voronov, A.A.: Semi-infinite homological algebra. Invent. Math. 113(1), 103–146 (1993)
Voronov, A.A.: Semi-infinite induction and Wakimoto modules. Am. J. Math. 121(5), 1079–1094 (1999)
Walton, M.A.: Conformal branching rules and modular invariants. Nucl. Phys. B 322, 775–790 (1989)
Wang, W.: Rationality of Virasoro vertex operator algebras. Int. Math. Res. Not. 7, 197–211 (1993)
Zhu, Y.: Modular invariance of characters of vertex operator algebras. J. Am. Math. Soc. 9(1), 237–302 (1996)
Acknowledgements
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.
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This work was partially supported by JSPS KAKENHI Grants (#17H01086 and #17K18724 to T. Arakawa), an NSERC Discovery Grant (#RES0019997 to T. Creutzig), and a grant from the Simons Foundation (#318755 to A. Linshaw).
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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
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DOI: https://doi.org/10.1007/s00222-019-00884-3