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The Vertex Algebras \(\mathcal {R}^{(p)}\) and \(\mathcal {V}^{({p})}\)

Abstract

The vertex algebras \(\mathcal {V}^{(p)}\) and \(\mathcal R^{(p)}\) introduced in Adamović (Transform Groups 21(2):299–327, 2016) are very interesting relatives of the well-known triplet algebras of logarithmic CFT. The algebra \(\mathcal {V}^{(p)}\) (respectively, \(\mathcal {R}^{(p)}\)) is a large extension of the simple affine vertex algebra \(L_{k}(\mathfrak {sl}_{2})\) (respectively, \(L_{k}(\mathfrak {sl}_{2})\) times a Heisenberg algebra), at level \(k=-2+1/p\) for positive integer p. Particularly, the algebra \(\mathcal {V}^{(2)}\) is the simple small \(N=4\) superconformal vertex algebra with \(c=-9\), and \(\mathcal {R}^{(2)}\) is \(L_{-3/2}(\mathfrak {sl}_3)\). In this paper, we derive structural results of these algebras and prove various conjectures coming from representation theory and physics. We show that SU(2) acts as automorphisms on \(\mathcal {V}^{({p})}\) and we decompose \(\mathcal {V}^{({p})}\) as an \(L_{k}(\mathfrak {sl}_{2})\)-module and \(\mathcal {R}^{({p})}\) as an \(L_k(\mathfrak {gl}_2)\)-module. The decomposition of \(\mathcal {V}^{({p})}\) shows that \(\mathcal {V}^{({p})}\) is the large level limit of a corner vertex algebra appearing in the context of S-duality. We also show that the quantum Hamiltonian reduction of \(\mathcal {V}^{({p})}\) is the logarithmic doublet algebra \(\mathcal {A}^{({p})}\) introduced in Adamović and Milas (Contemp Math 602:23–38, 2013), while the reduction of \(\mathcal {R}^{({p})}\) yields the \(\mathcal {B}^{({p})}\)-algebra of Creutzig et al. (Lett Math Phys 104(5):553–583, 2014). Conversely, we realize \(\mathcal {V}^{({p})}\) and \(\mathcal {R}^{({p})}\) from \(\mathcal {A}^{({p})}\) and \(\mathcal {B}^{({p})}\) via a procedure that deserves to be called inverse quantum Hamiltonian reduction. As a corollary, we obtain that the category \(KL_{k}\) of ordinary \(L_{k}(\mathfrak {sl}_{2})\)-modules at level \(k=-2+1/p\) is a rigid vertex tensor category equivalent to a twist of the category \(\text {Rep}(SU(2))\). This finally completes rigid braided tensor category structures for \(L_{k}(\mathfrak {sl}_{2})\) at all complex levels k. We also establish a uniqueness result of certain vertex operator algebra extensions and use this result to prove that both \(\mathcal {R}^{({p})}\) and \(\mathcal {B}^{({p})}\) are certain non-principal \(\mathcal {W}\)-algebras of type A at boundary admissible levels. The same uniqueness result also shows that \(\mathcal {R}^{({p})}\) and \(\mathcal {B}^{({p})}\) are the chiral algebras of Argyres-Douglas theories of type \((A_1, D_{2p})\) and \((A_1, A_{2p-3})\).

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References

  1. Adamović, D.: Classification of irreducible modules of certain subalgebras of free boson vertex algebra. J. Algorithm 270, 115–132 (2003)

    MathSciNet  MATH  Google Scholar 

  2. Adamović, D.: A realization of certain modules for the \(N=4\) superconformal algebra and the affine Lie algebra \(A_{2}^{(1)}\). Transform. Groups 21(2), 299–327 (2016)

    MathSciNet  MATH  Google Scholar 

  3. Adamović, D., Creutzig, T., Genra, N., Yang, J.: Inverse reduction, in preparation

  4. Adamović, D.: Realizations of simple affine vertex algebras and their modules: the cases \(\widehat{sl(2)}\) and \(\widehat{osp(1,2)}\). Commun. Math. Phys. 366(3), 1025–1067 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  5. Adamović, D., Kac, V.G., Frajria, P.M., Papi, P., Per\(\breve{\text{s}}\)e, O.: Finite vs infinite decompositions in conformal embeddings. Commun. Math. Phys. 348(2), 445–473 (2016)

  6. Adamović, D., Kac, V.G., Frajria, P.M., Papi, P., Per\(\breve{\text{ s }}\)e, O.: Conformal embeddings of affine vertex algebras in minimal W-algebras I: structural results. J. Alg. 500, 117–152 (2018)

  7. Adamović, D., Kac, V.G., Frajria, P.M., Papi, P., Per\(\breve{\text{ s }}\)e, O.: Conformal embeddings of affine vertex algebras in minimal W-algebras II: decompositions. Jpn. J. Math. 12(2), 261–315 (2017)

  8. Adamović, D., Lin, X., Milas, A.: ADE subalgebras of the triplet vertex algebra \(W(p)\): \(A\)-series. Commun. Contemp. Math. 15(6), 1350028 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Adamović, D., Milas, A.: Logarithmic intertwining operators and \(W(2,2p-1)\)-algebras. J. Math. Phys. 48(7), 073503 (2007)

    ADS  MathSciNet  MATH  Google Scholar 

  10. Adamović, D., Milas, A.: On the triplet vertex algebra \(W(p)\). Adv. Math. 217(6), 2664–2699 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Adamović, D., Milas, A.: The structure of Zhu’s algebras for certain \(\cal{W}\)-algebras. Adv. Math. 227(6), 2425–2456 (2011)

    MathSciNet  MATH  Google Scholar 

  12. Adamović, D., Milas, A.: The doublet vertex operator superalgebras \({\cal{A}}(p)\) and \({\cal{A}}_{2, p}\). Contemp. Math. 602, 23–38 (2013)

    MATH  Google Scholar 

  13. Adamović, D., Milas, A.: Vertex operator (super)algebras and LCFT. J. Phys. A 46(49), 494005 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Adamović, D., Milas, A.: \(C_2\)-cofinite vertex algebras and their logarithmic modules, in: Conformal Field Theories and Tensor Categories, Proceedings of a Workshop Held at Beijing International Center for Mathematics Research, ed. C. Bai, J. Fuchs, Y.-Z. Huang, L. Kong, I. Runkel and C. Schweigert, Mathematical Lectures from Beijing University, Vol. 2, Springer, New York, 249–270 (2014)

  15. Auger, J., Creutzig, T., Kanade, S., Rupert, M.: Braided Tensor Categories related to \({\cal{B}}_{p}\) Vertex Algebras, Commun. Math. Phys. 378 (2020) no. 1, 219–260

  16. Arakawa, T., Creutzig, T., Linshaw, A.R.: W-algebras as coset vertex algebras. Invent. Math. 218(1), 145–195 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  17. Argyres, P.C., Douglas, M.R.: New phenomena in \(SU(3)\) supersymmetric gauge theory. Nucl. Phys. B 448, 93–126 (1995)

    ADS  MathSciNet  MATH  Google Scholar 

  18. Aganagic, M., Frenkel, E., Okounkov, A.: Quantum \(q\)-Langlands Correspondence. Trans. Moscow Math. Soc. 79, 1–83 (2018)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  20. Arakawa, T.: Representation theory of \(W\)-algebras. Invent. Math. 169(2), 219–320 (2007)

    ADS  MathSciNet  MATH  Google Scholar 

  21. Arakawa, T., Creutzig, T., Kawasetsu, K., Linshaw, A.R.: Orbifolds and cosets of minimal \(W\)-algebras. Commun. Math. Phys. 355(1), 339–372 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  22. Frenkel, E., Ben-Zvi, D.: Vertex algebras and algebraic curves. Second edition. Mathematical Surveys and Monographs, 88. American Mathematical Society, Providence, RI, xiv+400 pp (2004)

  23. Buican, M., Nishinaka, T.: On the superconformal index of Argyres-Douglas theories. J. Phys. A 49(1), 015401 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  24. Buican, M., Nishinaka, T.: On irregular singularity wave functions and superconformal indices. JHEP 1709, 066 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  25. Beem, C., Lemos, M., Liendo, P., Peelaers, W., Rastelli, L., van Rees, B.C.: Infinite Chiral Symmetry in Four Dimensions. Commun. Math. Phys. 336(3), 1359–1433 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  26. Creutzig, T.: \(W\)-algebras for Argyres-Douglas theories. Euro. J. Math. 3(3), 659–690 (2017)

    MathSciNet  MATH  Google Scholar 

  27. Creutzig, T.: Fusion categories for affine vertex algebras at admissible levels. Selecta Math 25(2), 21 (2019)

    MathSciNet  MATH  Google Scholar 

  28. Creutzig, T.: Logarithmic W-algebras and Argyres-Douglas theories at higher rank. JHEP 1811, 188 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  29. Creutzig, T., Gainutdinov, A. M., Runkel, I.: A quasi-Hopf algebra for the triplet vertex operator algebra, Comm. Contemp. Math. 22, 1950024 (2019), arXiv:1712.07260

  30. Creutzig, T., Huang, Y.-Z., Yang, J.: Braided tensor categories of admissible modules for affine Lie algebras. Commun. Math. Phys. 362(3), 827–854 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  31. Creutzig, T., Kanade, S., Linshaw, A.R.: Simple current extensions beyond semi-simplicity. Commun. Contemp. Math. 22, 1950001 (2019)

    MathSciNet  MATH  Google Scholar 

  32. Creutzig, T., Kanade, S., Linshaw, A.R., Ridout, D.: Schur-Weyl Duality for Heisenberg Cosets. Transform. Groups 24(2), 301–354 (2019)

    MathSciNet  MATH  Google Scholar 

  33. Creutzig, T., Kanade, S., McRae, R.: Tensor categories for vertex operator superalgebra extensions, arXiv:1705.05017

  34. Creutzig, T., Kanade, S., McRae, R.: Glueing vertex algebras, arXiv:1906.00119

  35. Creutzig, T., Gaiotto, D.: Vertex Algebras for S-duality, Commun. Math. Phys. 379 (2020) no. 3, 785–845.

  36. Creutzig, T., Gaiotto, D., Linshaw, A. R.: S-duality for the large \(N=4\) superconformal algebra, Comm. Math. Phys. 374 (2020) no. 3, 1787–1808.

  37. Creutzig, T., Gannon, T.: Logarithmic conformal field theory, log-modular tensor categories and modular forms. J. Phys. A 50(40), 404004 (2017)

    MathSciNet  MATH  Google Scholar 

  38. Creutzig, T., Linshaw, A.R.: Cosets of affine vertex algebras inside larger structures. J. Alg. 517, 396–438 (2019)

    MathSciNet  MATH  Google Scholar 

  39. Creutzig, T., Ridout, D.: Logarithmic conformal field theory: beyond an introduction. J. Phys. A 46, 494006 (2013)

    MathSciNet  MATH  Google Scholar 

  40. Creutzig, T., Ridout, D., Wood, S.: Coset constructions of logarithmic \((1, p)\)-models. Lett. Math. Phys. 104(5), 553–583 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  41. Creutzig, T., Milas, A.: False Theta Functions and the Verlinde formula. Adv. Math. 262, 520–545 (2014)

    MathSciNet  MATH  Google Scholar 

  42. Creutzig, T., Milas, A.: Higher rank partial and false theta functions and representation theory. Adv. Math. 314, 203–227 (2017)

    MathSciNet  MATH  Google Scholar 

  43. Creutzig, T., Milas, A., Wood, S.: On regularised quantum dimensions of the singlet vertex operator algebra and false theta functions. Int. Math. Res. Not. 5, 1390–1432 (2017)

    MathSciNet  MATH  Google Scholar 

  44. Cordova, C., Shao, S.H.: Schur Indices, BPS Particles, and Argyres-Douglas Theories. JHEP 1601, 040 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  45. Creutzig, T., Yang, J.: Tensor category of affine Lie algebras beyond admissble levels, arXiv:2002.05686 [math.RT]

  46. Creutzig, T., Jiang, C., Orosz Hunziker, F., Ridout, D., Yang, J.: Tensor categories arising from the Virasoro algebra, Advances in Mathematics, 380 (2021), 107601.

  47. Dong, C., Li, H., Mason, G.: Compact automorphism groups of vertex operator algebras. Int. Math. Res. Not. 913–921, (1996)

  48. Dong, C., Lepowsky, J.: Abelian intertwining algebras–A generalization of vertex operator algebras, in “Algebraic Groups and Generalizations, Proc. 1991 Amer. Math. Soc. Summer Research Institute (W. Haboush and B. Parshall, Eds.), Proceedings of Symposia in Pure Mathematics., American. Mathematical Society, Providence, (1993)

  49. Dong, C., Lepowsky, J.: Generalized Vertex Algebras and Relative Vertex Operators, Progress in Math, vol. 112. Birkhäuser, Boston (1993)

    MATH  Google Scholar 

  50. van Ekeren, J., Möller, S., Scheithauer, N.: Construction and classification of holomorphic vertex operator algebras. J. Reine Angew. Math. 759, 61–99 (2020)

    MathSciNet  MATH  Google Scholar 

  51. Feigin, B.L., Frenkel, E.: Quantization of Drinfel’d-Sokolov reduction. Phys. Lett. B 246(1–2), 75–81 (1990)

    ADS  MathSciNet  MATH  Google Scholar 

  52. Frenkel, E.: Lectures on Wakimoto modules, opers and the center at the critical level. Adv. Math 195, 297–404 (2005)

    MathSciNet  MATH  Google Scholar 

  53. Frenkel, E., Gaiotto, D.: Quantum Langlands dualities of boundary conditions, D-modules, and conformal blocks, arXiv:1805.00203

  54. Feigin, B.L., Gaĭnutdinov, A., Semikhatov, A., Yu Tipunin, I.: Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center. Commun. Math. Phys 265, 47–93 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

  55. Feigin, B.L., Gaĭnutdinov, A., Semikhatov, A., Yu Tipunin, I.: Kazhdan-Lusztig correspondence for the representation category of the triplet W-algebra in logarithmic CFT. Theor. Math. Phys. 148(3), 1210–1235 (2006)

    MATH  Google Scholar 

  56. Feigin, B. L., Yu Tipunin, I.: Logarithmic CFTs connected with simple Lie algebras, arXiv:1002.5047

  57. Friedan, D., Martinec, E., Shenker, S.: Conformal invariance, supersymmetry and string theory. Nucl. Phys. B 271, 93–165 (1986)

    ADS  MathSciNet  Google Scholar 

  58. Gaiotto, D., Rapcak, M.: Vertex Algebras at the Corner. JHEP 1901, 160 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  59. Huang, Y.-Z.: Vertex operator algebras and the Verlinde conjecture. Commun. Contemp. Math. 10, 103–154 (2008)

    MathSciNet  MATH  Google Scholar 

  60. Huang, Y.-Z.: Rigidity and modularity of vertex tensor categories. Commun. Contemp. Math. 10, 871–911 (2008)

    MathSciNet  MATH  Google Scholar 

  61. Huang, Y.Z., Kirillov, A., Lepowsky, J.: Braided tensor categories and extensions of vertex operator algebras. Commun. Math. Phys. 337(3), 1143–1159 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  62. Kumar, S.: Extension of the category \({\cal{O}}^{g}\) and a vanishing theorem for the Ext functor for Kac-Moody algebras. J. Alg. 108(2), 472–491 (1987)

    MATH  Google Scholar 

  63. Kac, V.G., Frajria, P.M., Papi, P., Xu, F.: Conformal embeddings and simple current extensions. Int. Math. Res. Not. 14, 5229–5288 (2015)

    MathSciNet  MATH  Google Scholar 

  64. Kazhdan, D., Lusztig, G.: Affine Lie algebras and quatum groups. Int. Math. Res. Not. (in Duke Math. J.) 2, 21–29 (1991)

    MATH  Google Scholar 

  65. Kazhdan, D., Lusztig, G.: Tensor structure arising from affine Lie algebras, I. J. Am. Math. Soc. 6, 905–947 (1993)

    MathSciNet  MATH  Google Scholar 

  66. Kazhdan, D., Lusztig, G.: Tensor structure arising from affine Lie algebras, II. J. Am. Math. Soc. 6, 949–1011 (1993)

    MathSciNet  MATH  Google Scholar 

  67. Kazhdan, D., Lusztig, G.: Tensor structure arising from affine Lie algebras, III. J. Am. Math. Soc. 7, 335–381 (1994)

    MathSciNet  MATH  Google Scholar 

  68. Kazhdan, D., Lusztig, G.: Tensor structure arising from affine Lie algebras, IV. J. Am. Math. Soc. 7, 383–453 (1994)

    MathSciNet  MATH  Google Scholar 

  69. Kapustin, A., Witten, E.: Electric-Magnetic Duality And The Geometric Langlands Program. Commun. Num. Theor. Phys. 1(1), 1–236 (2007)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  71. McRae, R.: On the tensor structure of modules for compact orbifold vertex operator algebras, arXiv: 1810.00747

  72. McRae, R.: Twisted modules and \(G\)-equivariantization in logarithmic conformal field theory, Commun. Math. Phys. (2020). https://doi.org/10.1007/s00220-020-03882-2, arXiv:1910.13226

  73. Rastelli, L.: Infinite Chiral Symmetry in Four and Six Dimensions, Seminar at Harvard University, November (2014)

  74. Tsuchiya, A., Wood, S.: The tensor structure on the representation category of the \(W_{p}\) triplet algebra. J. Phys. A 46, 445203 (2013)

    ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was done in part during the visit of D.A. to the University of Alberta. T.C. appreciates the many discussions with Boris Feigin and we thank the referee for his useful comments. D.A. is partially supported by the QuantiXLie Centre of Excellence, a project cofinanced by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme (KK.01.1.1.01.0004). T. C is supported by NSERC \(\#\)RES0020460. N. G is supported by JSPS Overseas Research Fellowships.

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Adamović, D., Creutzig, T., Genra, N. et al. The Vertex Algebras \(\mathcal {R}^{(p)}\) and \(\mathcal {V}^{({p})}\). Commun. Math. Phys. 383, 1207–1241 (2021). https://doi.org/10.1007/s00220-021-03950-1

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