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W-algebras for Argyres–Douglas theories

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Abstract

The Schur index of the \((A_1, X_n)\)-Argyres–Douglas theory is conjecturally a character of a vertex operator algebra. Here such vertex algebras are found for the \(A_{\text {odd}}\) and \(D_{\text {even}}\)-type Argyres–Douglas theories. The vertex operator algebra corresponding to \(A_{2p-3}\)-Argyres–Douglas theory is the logarithmic -algebra of Creutzig et al. (Lett Math Phys 104(5):553–583, 2014), while the one corresponding to \(D_{2p}\), denoted by , is realized as a non-regular quantum Hamiltonian reduction of \(L_{k}(\mathfrak {sl}_{p+1})\) at level . For all n one observes that the quantum Hamiltonian reduction of the vertex operator algebra of \(D_n\)-Argyres–Douglas theory is the vertex operator algebra of \(A_{n-3}\)-Argyres–Douglas theory. As a corollary, one realizes the singlet and triplet algebras (the vertex algebras associated to the best understood logarithmic conformal field theories) as quantum Hamiltonian reductions as well. Finally, characters of certain modules of these vertex operator algebras and the modular properties of their meromorphic continuations are given.

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Notes

  1. We thank Tomoyuki Arakawa for pointing out that is similar to chiral Hecke algebras of \({\mathfrak {sl}}_2\).

References

  1. Adamović, D.: A construction of admissible \(A_1^{(1)}\)-modules of level \(-4/3\). J. Pure Appl. Algebra 196(2–3), 119–134 (2005)

    Article  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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Adamović, D.: The vertex algebras \(R(p)\) and their logarithmic representations (in preparation)

  4. Adamović, D., Kac, V.G., Möseneder Frajria, P., Papi, P., Perše, O.: Finite vs. infinite decompositions in conformal embeddings. Comm. Math. Phys. 348(2), 445–473 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Adamović, D., Kac, V.G., Möseneder Frajria, P., Papi, P., Perše, O.: Conformal embeddings of affine vertex algebras in minimal \(W\)-algebras I: structural results. J. Algebra (2017). doi:10.1016/j.jalgebra.2016.12.005

  6. Adamović, D., Kac, V.G., Möseneder Frajria, P., Papi, P., Perše, O.: Conformal embeddings of affine vertex algebras in minimal \(W\)-algebras II: decompositions (2016) (Japan J. Math. submitted). arXiv:1604.00893

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Adamović, D., Milas, A.: On \(W\)-algebras associated to \((2, p)\) minimal models and their representations. Int. Math. Res. Not. IMRN 2010(20), 3896–3934 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Alfes, C., Creutzig, T.: The mock modular data of a family of superalgebras. Proc. Amer. Math. Soc. 142(7), 2265–2280 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Arakawa, T.: Introduction to \(W\)-algebras and their representation theory (2016). arXiv:1605.00138

  14. Arakawa, T., Creutzig, T., Kawasetsu, K., Linshaw, A.R.: Orbifolds and cosets of minimal \({\cal{W}}\)-algebras. Comm. Math. Phys. (2017). doi:10.1007/s00220-017-2901-2

    MathSciNet  MATH  Google Scholar 

  15. Arakawa, T., Molev, A.: Explicit generators in rectangular affine \({\cal{W}}\)-algebras of type \(A\). Lett. Math. Phys. 107(1), 47–59 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Argyres, P.C., Douglas, M.R.: New phenomena in \({\rm SU}(3)\) supersymmetric gauge theory. Nuclear Phys. B 448(1–2), 93–126 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Auger, J., Creutzig, T., Kanade, S., Rupert, M.: Work in progress

  18. Beem, C., Lemos, M., Liendo, P., Peelaers, W., Rastelli, L., van Rees, B.C.: Infinite chiral symmetry in four dimensions. Comm. Math. Phys. 336(3), 1359–1433 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Beem, C., Peelaers, W., Rastelli, L., van Rees, B.S.: Chiral algebras of class \({\cal{S}}\). J. High Energy Phys. 2015(5), # 20 (2015)

    Article  MathSciNet  Google Scholar 

  20. Beem, C., Rastelli, L.: Higgs branches, vertex operator algebras and modular differential equations (in preparation)

  21. Bringmann, K., Creutzig, T., Rolen, L.: Negative index Jacobi forms and quantum modular forms. Res. Math. Sci. 1, # 11 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Córdova, C., Gaiotto, D., Shao, S.-H.: Infrared computations of defect Schur indices. J. High Energy Phys. 2016(11), # 106 (2016)

    Article  MathSciNet  Google Scholar 

  24. Córdova, C., Gaiotto, D., Shao, S.-H.: Surface defect indices and chiral algebras (2017). arXiv:1704.01955

  25. Córdova, C., Shao, S.-H.: Schur indices, BPS particles, and Argyres–Douglas theories. J. High Energy Phys. 2016(1), # 40 (2016)

    Article  MathSciNet  Google Scholar 

  26. Creutzig, T., Gaiotto, D.: Work in progress

  27. Creutzig, T., Gannon, T.: Logarithmic conformal field theory, log-modular tensor categories and modular forms (2016). arXiv:1605.04630

  28. Creutzig, T., Kanade, S., Linshaw, A.R.: Simple current extensions beyond semi-simplicity (2015). arXiv:1511.08754

  29. Creutzig, T., Kanade, S., Linshaw, A.R., Ridout, D.: Schur–Weyl duality for Heisenberg cosets (2016). arXiv:1611.00305

  30. Creutzig, T., Kanade, S., McRae, R.: Tensor categories for vertex operator superalgebra extensions (2017). arXiv:1705.05017

  31. Creutzig, T., Linshaw, A.R.: Cosets of affine vertex algebras inside larger structures (2014). arXiv:1407.8512v3

  32. Creutzig, T., Milas, A.: False theta functions and the Verlinde formula. Adv. Math. 262, 520–545 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Creutzig, T., Milas, A.: Higher rank partial and false theta functions and representation theory (2016) (Adv. Math. submitted). arXiv:1607.08563

  34. Creutzig, T., Milas, A., Rupert, M.: Logarithmic link invariants of \(\overline{U}^H_{q}({\mathfrak{sl}}_2)\) and asymptotic dimensions of singlet vertex algebras (2016). arXiv:1605.05634

  35. Creutzig, T., Ridout, D.: Modular data and Verlinde formulae for fractional level WZW models I. Nuclear Phys. B 865(1), 83–114 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Creutzig, T., Ridout, D.: Relating the archetypes of logarithmic conformal field theory. Nuclear Phys. B 872(3), 348–391 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Creutzig, T., Ridout, D.: Modular data and Verlinde formulae for fractional level WZW models II. Nuclear Phys. B 875(2), 423–458 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Feigin, B.L., Semikhatov, A.M.: \({\cal{W}}^{(2)}_n\) algebras. Nuclear Phys. B 698(3), 409–449 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  40. Fredrickson, L., Pei, D., Yan, W., Ye, K.: Argyres–Douglas theories, chiral algebras and wild Hitchin characters (2017). arXiv:1701.08782

  41. Frenkel, E., Ben-Zvi, D.: Vertex Algebras and Algebraic Curves. Mathematical Surveys and Monographs, vol. 88, 2nd edn. American Mathematical Society, Providence (2004)

    Book  MATH  Google Scholar 

  42. Feigin, B.L., Tipunin, I.Y.: Logarithmic CFTs connected with simple Lie algebras (2010). arXiv:1002.5047

  43. Gadde, A., Rastelli, L., Razamat, S.S., Yan, W.: Four dimensional superconformal index from \(q\)-deformed two dimensional Yang–Mills theory. Phys. Rev. Lett. 106, # 241602 (2011)

    Article  Google Scholar 

  44. Gaiotto, D.: Twisted compactifications of \(3d\) \(N = 4\) theories and conformal blocks (2016). arXiv:1611.01528

  45. Genra, N.: Screening operators for \({\cal{W}}\)-algebras. Selecta Math. (N.S.) (2017). doi:10.1007/s00029-017-0315-9

  46. Gorelik, M., Kac, V.: On simplicity of vacuum modules. Adv. Math. 211(2), 621–677 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  47. Kac, V.G., Kazhdan, D.A.: Structure of representations with highest weight of infinite-dimensional Lie algebras. Adv. Math. 34(1), 97–108 (1979)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  49. Kac, V.G., Wakimoto, M.: Modular invariant representations of infinite-dimensional Lie algebras and superalgebras. Proc. Nat. Acad. Sci. USA 85(14), 4956–4960 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  50. Kac, V.G., Wakimoto, M.: Integrable highest weight modules over affine superalgebras and number theory. In: Brylinski, J.-L., Brylinski, R., Guillemin, V., Kac, V. (eds.) Lie Theory and Geometry. Progress in Mathematics, vol. 123, pp. 415–456. Birkhäuser, Boston (1994)

    Chapter  Google Scholar 

  51. Kac, V.G., Wakimoto, M.: Integrable highest weight modules over affine superalgebras and Appell’s function. Comm. Math. Phys. 215(3), 631–682 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  52. Kac, V.G., Wakimoto, M.: Quantum reduction and representation theory of superconformal algebras. Adv. Math. 185(2), 400–458 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  53. Kac, V.G., Wakimoto, M.: A remark on boundary level admissible representations (2016). arXiv:1612.07423

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

    Article  MathSciNet  MATH  Google Scholar 

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

  56. Ridout, D.: \(\widehat{\mathfrak{sl}}(2)_{-1/2}\): a case study. Nuclear Phys. B 814(3), 485–521 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  57. Ridout, D.: \(\widehat{\mathfrak{sl}}(2)_{-1/2}\) and the triplet model. Nuclear Phys. B 835(3), 314–342 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  58. Tachikawa, Y.: On some conjectures on VOAs (2013). http://member.ipmu.jp/yuji.tachikawa

  59. Tsuchiya, A., Wood, S.: The tensor structure on the representation category of the \({\cal{W}}_{p}\) triplet algebra. J. Phys. A 46(44), # 445203 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  60. Xie, D., Yan, W., Yau, S.-T.: Chiral algebra of Argyres–Douglas theory from M5 brane (2016). arXiv:1604.02155

  61. Zwegers, S.P.: Mock Theta Functions. Ph.D. thesis, Utrecht University (2002)

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Acknowledgements

Most of all, I am very grateful to Shu-Heng Shao for many discussions on this topic. I also would like to thank Tomoyuki Arakawa, Chris Beem, Leonardo Rastelli and Davide Gaiotto for explanations on the relation of vertex operator algebras and four-dimensional quantum field theory. I thank Dražen Adamović for his useful comments on the draft and discussion. Finally I very much appreciate discussions with Wenbin Yan and Ke Ye and sharing [40] with me.

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  1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Thomas Creutzig

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  1. Thomas Creutzig
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Correspondence to Thomas Creutzig.

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The author is supported by NSERC RES0020460 and has benefitted from the workshop Exact Operator Algebras in Superconformal Field Theories on the topic at the Perimeter Institute.

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Creutzig, T. W-algebras for Argyres–Douglas theories. European Journal of Mathematics 3, 659–690 (2017). https://doi.org/10.1007/s40879-017-0156-2

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