The conformal characters

  • Antoine Bourget
  • Jan Troost
Open Access
Regular Article - Theoretical Physics


We revisit the study of the multiplets of the conformal algebra in any dimension. The theory of highest weight representations is reviewed in the context of the Bernstein-Gelfand-Gelfand category of modules. The Kazhdan-Lusztig polynomials code the relation between the Verma modules and the irreducible modules in the category and are the key to the characters of the conformal multiplets (whether finite dimensional, infinite dimensional, unitary or non-unitary). We discuss the representation theory and review in full generality which representations are unitarizable. The mathematical theory that allows for both the general treatment of characters and the full analysis of unitarity is made accessible. A good understanding of the mathematics of conformal multiplets renders the treatment of all highest weight representations in any dimension uniform, and provides an overarching comprehension of case-by-case results. Unitary highest weight representations and their characters are classified and computed in terms of data associated to cosets of the Weyl group of the conformal algebra. An executive summary is provided, as well as look-up tables up to and including rank four.


Conformal and W Symmetry Conformal Field Theory Supersymmetry and Duality 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    S. Ferrara and C. Fronsdal, Conformal fields in higher dimensions, in Recent developments in theoretical and experimental general relativity, gravitation and relativistic field theories. Proceedings, 9th Marcel Grossmann Meeting, MG’9, Rome, Italy, July 2-8, 2000. Pts. A-C, pp. 508-527 (2000) [hep-th/0006009] [INSPIRE].
  2. [2]
    J. Penedones, E. Trevisani and M. Yamazaki, Recursion Relations for Conformal Blocks, JHEP 09 (2016) 070 [arXiv:1509.00428] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    J. Troost, Models for modules: The story of O, J. Phys. A 45 (2012) 415202 [arXiv:1202.1935] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  4. [4]
    J.E. Humphreys, Reflection groups and Coxeter groups, vol. 29, Cambridge University Press (1992).Google Scholar
  5. [5]
    J.E. Humphreys, Representations of semisimple Lie algebras in the BGG category O, vol. 94, American Mathematical Soc. (2008).Google Scholar
  6. [6]
    P.A.M. Dirac, A Remarkable representation of the 3 + 2 de Sitter group, J. Math. Phys. 4 (1963) 901 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    C. Fronsdal, Elementary Particles in a Curved Space, Rev. Mod. Phys. 37 (1965) 221 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    N. Evans, Discrete series for the universal covering group of the 3 + 2 de sitter group, J. Math. Phys. 8 (1967) 170.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    V. Dobrev and E. Sezgin, Spectrum and character formulae of so (3, 2) unitary representations, in Differential Geometry, Group Representations, and Quantization, Springer (1991), pp. 227-238.Google Scholar
  10. [10]
    I. Bernstein, I. Gelfand and S. Gelfand, Structure of representations generated by highest weight vectors, Funct. Anal. Appl 5 (1971) 1 [Funktsional. Anal. i Prilozhen. 5 (1971) 1].Google Scholar
  11. [11]
    J.C. Jantzen, Moduln mit einem höchsten gewicht, in Moduln mit einem höchsten Gewicht, Springer (1979), pp. 11-41.Google Scholar
  12. [12]
    D. Kazhdan and G. Lusztig, Representations of coxeter groups and hecke algebras, Invent. Math. 53 (1979) 165.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    A. Beilinson and J. Bernstein, Localisation de g-modules, CR Acad. Sci. Paris 292 (1981) 15.MathSciNetzbMATHGoogle Scholar
  14. [14]
    J.-L. Brylinski and M. Kashiwara, Kazhdan-lusztig conjecture and holonomic systems, Invent. Math. 64 (1981) 387.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    V.V. Deodhar, On some geometric aspects of bruhat orderings ii. the parabolic analogue of kazhdan-lusztig polynomials, J. Algebra 111 (1987) 483.Google Scholar
  16. [16]
    E. Cartan, Sur les domaines bornés homogènes de l’espace den variables complexes, in Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, vol. 11, Springer (1935), pp. 116-162.Google Scholar
  17. [17]
    B.D. Boe, Kazhdan-lusztig polynomials for hermitian symmetric spaces, Trans. Am. Math. Soc. 309 (1988) 279.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    F. Brenti, Parabolic Kazhdan-Lusztig polynomials for Hermitian symmetric pairs, Trans. Am. Math. Soc. 361 (2009) 1703.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    M. Kashiwara and T. Tanisaki, Characters of irreducible modules with non-critical highest weights over affine lie algebras, math/9903123.
  20. [20]
    J.C. Jantzen, Character formulae from Hermann Weyl to the present, in Groups and analysis, Lond. Math. Soc. Lect. Notes Ser. 354 (2008) 232.Google Scholar
  21. [21]
    N. Bourbaki, Groupes et algebres de Lie. Chapitre iv-vi, Hermann, Paris (1968), Actualités Scientifiques et Industrielles (1972).Google Scholar
  22. [22]
    J.E. Humphreys, Introduction to Lie algebras and representation theory, vol. 9, Springer Science & Business Media (2012).Google Scholar
  23. [23]
    A.L. Onishchik and E.B. Vinberg, Lie groups and algebraic groups, Springer (1990).Google Scholar
  24. [24]
    B. Harish-Chandra, Representations of semisimple Lie groups: IV, Proc. Nat. Acad. Sci. 37 (1951) 691.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    B. Harish-Chandra, Representations of semisimple Lie groups. V, Am. J. Math 78 (1956) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    A.W. Knapp, Lie groups beyond an introduction, vol. 140, Springer Science and Business Media (2013).Google Scholar
  27. [27]
    T. Enright, R. Howe and N. Wallach, A classification of unitary highest weight modules, in Representation theory of reductive groups, Springer (1983), pp. 97-143.Google Scholar
  28. [28]
    H.P. Jakobsen, Hermitian symmetric spaces and their unitary highest weight modules, J. Funct. Anal. 52 (1983) 385.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    R. Parthasarathy, Criteria for the unitarizability of some highest weight modules, Proc. Indian Acad. Sci 89 (1980) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    M. Beccaria, X. Bekaert and A.A. Tseytlin, Partition function of free conformal higher spin theory, JHEP 08 (2014) 113 [arXiv:1406.3542] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    T. Basile, X. Bekaert and N. Boulanger, Mixed-symmetry fields in de Sitter space: a group theoretical glance, JHEP 05 (2017) 081 [arXiv:1612.08166] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    O.V. Shaynkman, I.Yu. Tipunin and M.A. Vasiliev, Unfolded form of conformal equations in M dimensions and o(M + 2) modules, Rev. Math. Phys. 18 (2006) 823 [hep-th/0401086] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    A. Geraschenko, Properties of the longest element in a Weyl group, Mathematics Stack Exchange, version: 2011-08-26, (2011).
  34. [34]
    G. Mack, All unitary ray representations of the conformal group SU(2, 2) with positive energy, Commun. Math. Phys. 55 (1977) 1 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 783 [hep-th/9712074] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    F.A. Dolan, Character formulae and partition functions in higher dimensional conformal field theory, J. Math. Phys. 47 (2006) 062303 [hep-th/0508031] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    A. Barabanschikov, L. Grant, L.L. Huang and S. Raju, The Spectrum of Yang-Mills on a sphere, JHEP 01 (2006) 160 [hep-th/0501063] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    M. Beccaria, G. Macorini and A.A. Tseytlin, Supergravity one-loop corrections on AdS 7 and AdS 3 , higher spins and AdS/CFT, Nucl. Phys. B 892 (2015) 211 [arXiv:1412.0489] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  39. [39]
    D.-N. Verma, Structure of certain induced representations of complex semisimple lie algebras, Bull. Am. Math. Soc. 74 (1968) 160.MathSciNetCrossRefzbMATHGoogle Scholar

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© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of PhysicsUniversidad de OviedoOviedoSpain
  2. 2.Département de Physique, École Normale SupérieureCNRS, PSL Research University, Sorbonne UniversitésParisFrance

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