On the combinatorics of partition functions in AdS3/LCFT2

  • Yannick Mvondo-She
  • Konstantinos ZoubosEmail author
Open Access
Regular Article - Theoretical Physics


Three-dimensional Topologically Massive Gravity at its critical point has been conjectured to be holographically dual to a Logarithmic CFT. However, many details of this correspondence are still lacking. In this work, we study the 1-loop partition function of Critical Cosmological Topologically Massive Gravity, previously derived by Gaberdiel, Grumiller and Vassilevich, and show that it can be usefully rewritten as a Bell polynomial expansion. We also show that there is a relationship between this Bell polynomial expansion and the plethystic exponential. Our reformulation allows us to match the TMG partition function to states on the CFT side, including the multi-particle states of t (the logarithmic partner of the CFT stress tensor) which had previously been elusive. We also discuss the appearance of a ladder action between the different multi-particle sectors in the partition function, which induces an interesting sl(2) structure on the n-particle components of the partition function.


AdS-CFT Correspondence Classical Theories of Gravity Conformal and W Symmetry Conformal Field Models in String Theory 


Open Access

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  1. [1]
    M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    S. Deser, R. Jackiw and S. Templeton, Three-dimensional massive gauge theories, Phys. Rev. Lett. 48 (1982) 975 [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    S. Deser, R. Jackiw and S. Templeton, Topologically massive gauge theories, Annals Phys. 140 (1982) 372 [Annals Phys. 281 (2000) 409].Google Scholar
  6. [6]
    E. Witten, Three-dimensional gravity revisited, arXiv:0706.3359 [INSPIRE].
  7. [7]
    A. Maloney and E. Witten, Quantum gravity partition functions in three dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    W. Li, W. Song and A. Strominger, Chiral gravity in three dimensions, JHEP 04 (2008) 082 [arXiv:0801.4566] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    D. Grumiller and N. Johansson, Instability in cosmological topologically massive gravity at the chiral point, JHEP 07 (2008) 134 [arXiv:0805.2610] [arXiv:arXiv:0910.1706] [INSPIRE].
  10. [10]
    V. Gurarie, Logarithmic operators in conformal field theory, Nucl. Phys. B 410 (1993) 535 [hep-th/9303160] [INSPIRE].
  11. [11]
    M. Flohr, Bits and pieces in logarithmic conformal field theory, Int. J. Mod. Phys. A 18 (2003) 4497 [hep-th/0111228] [INSPIRE].
  12. [12]
    M.R. Gaberdiel, An algebraic approach to logarithmic conformal field theory, Int. J. Mod. Phys. A 18 (2003) 4593 [hep-th/0111260] [INSPIRE].
  13. [13]
    D. Grumiller and N. Johansson, Consistent boundary conditions for cosmological topologically massive gravity at the chiral point, Int. J. Mod. Phys. D 17 (2009) 2367 [arXiv:0808.2575] [INSPIRE].
  14. [14]
    M. Henneaux, C. Martinez and R. Troncoso, Asymptotically anti-de Sitter spacetimes in topologically massive gravity, Phys. Rev. D 79 (2009) 081502 [arXiv:0901.2874] [INSPIRE].
  15. [15]
    A. Maloney, W. Song and A. Strominger, Chiral gravity, log gravity and extremal CFT, Phys. Rev. D 81 (2010) 064007 [arXiv:0903.4573] [INSPIRE].
  16. [16]
    M. Henneaux, C. Martinez and R. Troncoso, More on asymptotically Anti-de Sitter spaces in topologically massive gravity, Phys. Rev. D 82 (2010) 064038 [arXiv:1006.0273] [INSPIRE].
  17. [17]
    K. Skenderis, M. Taylor and B.C. van Rees, Topologically massive gravity and the AdS/CFT correspondence, JHEP 09 (2009) 045 [arXiv:0906.4926] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    D. Grumiller and I. Sachs, AdS 3 /LCFT 2correlators in cosmological topologically massive gravity, JHEP 03 (2010) 012 [arXiv:0910.5241] [INSPIRE].
  19. [19]
    M.R. Gaberdiel, D. Grumiller and D. Vassilevich, Graviton 1-loop partition function for 3-dimensional massive gravity, JHEP 11 (2010) 094 [arXiv:1007.5189] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    S. Giombi, A. Maloney and X. Yin, One-loop partition functions of 3D gravity, JHEP 08 (2008) 007 [arXiv:0804.1773] [INSPIRE].
  21. [21]
    R. Gopakumar, R.K. Gupta and S. Lal, The heat kernel on AdS, JHEP 11 (2011) 010 [arXiv:1103.3627] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    D. Grumiller, W. Riedler, J. Rosseel and T. Zojer, Holographic applications of logarithmic conformal field theories, J. Phys. A 46 (2013) 494002 [arXiv:1302.0280] [INSPIRE].
  23. [23]
    S. Benvenuti, B. Feng, A. Hanany and Y.-H. He, Counting BPS operators in gauge theories: quivers, syzygies and plethystics, JHEP 11 (2007) 050 [hep-th/0608050] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    B. Feng, A. Hanany and Y.-H. He, Counting gauge invariants: the plethystic program, JHEP 03 (2007) 090 [hep-th/0701063] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    E.A. Bergshoeff, O. Hohm and P.K. Townsend, Massive gravity in three dimensions, Phys. Rev. Lett. 102 (2009) 201301 [arXiv:0901.1766] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    D. Grumiller and O. Hohm, AdS 3 /LCFT 2 : correlators in new massive gravity, Phys. Lett. B 686 (2010) 264 [arXiv:0911.4274] [INSPIRE].
  27. [27]
    M. Alishahiha and A. Naseh, Holographic renormalization of new massive gravity, Phys. Rev. D 82 (2010) 104043 [arXiv:1005.1544] [INSPIRE].
  28. [28]
    A. Bagchi, S. Lal, A. Saha and B. Sahoo, Topologically massive higher spin gravity, JHEP 10 (2011) 150 [arXiv:1107.0915] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    A. Bagchi, S. Lal, A. Saha and B. Sahoo, One loop partition function for Topologically Massive Higher Spin Gravity, JHEP 12 (2011) 068 [arXiv:1107.2063] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  30. [30]
    G.E. Andrews, The theory of partitions, Encyclopedia of Mathematics Volume 2, Addison-Wesley Publishing Company, U.S.A. (1976).Google Scholar
  31. [31]
    A.A. Bytsenko and M. Chaichian, Multipartite generating functions and infinite products for quantum invariants, arXiv:1702.02208 [INSPIRE].
  32. [32]
    E.T. Bell, Exponential polynomials, Ann. Math. (1934) 258.Google Scholar
  33. [33]
    J. Riordan, An introduction to combinatorial analysis, John Wiley & Sons, Reading U.S.A. (1958).Google Scholar
  34. [34]
    F. Faà di Bruno, Sullo sviluppo delle funzioni, Ann. Sci. Mat. Fis. 6 (1855) 479.Google Scholar
  35. [35]
    F. Faà di Bruno, Note sur une nouvelle formule de calcul differentiel, Quart. J. Pure App. Math 1 (1857) 359.Google Scholar
  36. [36]
    J. Riordan, Derivatives of composite functions, Bull. Amer. Math. Soc. 52 (1946) 664.Google Scholar
  37. [37]
    M.R. Gaberdiel, R. Gopakumar, T. Hartman and S. Raju, Partition Functions of Holographic Minimal Models, JHEP 08 (2011) 077 [arXiv:1106.1897] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    P. Pouliot, Molien function for duality, JHEP 01 (1999) 021 [hep-th/9812015] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    D. Forcella, BPS partition functions for quiver gauge theories: Counting fermionic operators, arXiv:0705.2989 [INSPIRE].
  40. [40]
    A. Hanany and R. Kalveks, Quiver theories for moduli spaces of classical group nilpotent orbits, JHEP 06 (2016) 130 [arXiv:1601.04020] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    Y. Mvondo-She, to appear.Google Scholar
  42. [42]
    J. Steffensen, The poweroid, an extension of the mathematical notion of power, Acta Mathematica 73 (1941) 333.MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    G. Dattoli, P. Ottaviani, A. Torre and L. Vázquez, Evolution operator equations: Integration with algebraic and finite difference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory, Riv. Nuovo Cim. 20 (1997) 3.Google Scholar
  44. [44]
    G. Dattoli, Hermite-Bessel and Laguerre-Bessel functions: a by-product of the monomiality principle, Proc. Melfi Sch. Adv. Top. Math. Phys 1 (1999) 147.MathSciNetzbMATHGoogle Scholar
  45. [45]
    Y.B. Cheikh, Some results on quasi-monomiality, Appl. Math. Comput. 141 (2003) 63.MathSciNetzbMATHGoogle Scholar
  46. [46]
    Y.B. Cheikh, On obtaining dual sequences via quasi-monomiality, Georgian Math. J. 9 (2002) 413.MathSciNetzbMATHGoogle Scholar
  47. [47]
    S. Roman, The umbral calculus, Springer, Germany (2005).Google Scholar
  48. [48]
    P. Blasiak, G. Dattoli, A. Horzela and K. Penson, Representations of monomiality principle with Sheffer-type polynomials and boson normal ordering, Phys. Lett. A 352 (2006) 7 [quant-ph/0504009].
  49. [49]
    J. Riordan, Combinatorial identities, Wiley, U.S.A. (1968).Google Scholar
  50. [50]
    P. Feinsilver, J. Kocik and R. Schott, Representations of the Schrödinger algebra and Appell systems, Fort. Physik 52 (2004) 343.CrossRefzbMATHGoogle Scholar
  51. [51]
    B. Chen, J. Long and J.-B. Wu, Spin-3 topologically massive gravity, Phys. Lett. B 705 (2011) 513 [arXiv:1106.5141] [INSPIRE].
  52. [52]
    B. Chen and J. Long, High spin topologically massive gravity, JHEP 12 (2011) 114 [arXiv:1110.5113] [INSPIRE].ADSzbMATHGoogle Scholar
  53. [53]
    P. Bouwknegt and K. Schoutens, W symmetry in conformal field theory, Phys. Rept. 223 (1993) 183 [hep-th/9210010] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  54. [54]
    M.R. Gaberdiel, R. Gopakumar and A. Saha, Quantum W -symmetry in AdS 3, JHEP 02 (2011) 004 [arXiv:1009.6087] [INSPIRE].
  55. [55]
    M. Bertin, D. Grumiller, D. Vassilevich and T. Zojer, Generalised massive gravity one-loop partition function and AdS/(L)CFT, JHEP 06 (2011) 111 [arXiv:1103.5468] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    T. Zojer, On gravity one-loop partition functions of three-dimensional critical gravities, Class. Quant. Grav. 30 (2013) 075005 [arXiv:1210.6887] [INSPIRE].
  57. [57]
    J. Lucietti and M. Rangamani, Asymptotic counting of BPS operators in superconformal field theories, J. Math. Phys. 49 (2008) 082301 [arXiv:0802.3015] [INSPIRE].
  58. [58]
    P.A. Pearce, J. Rasmussen and J.-B. Zuber, Logarithmic minimal models, J. Stat. Mech. 0611 (2006) P11017 [hep-th/0607232] [INSPIRE].
  59. [59]
    T. Creutzig and D. Ridout, Logarithmic conformal field theory: beyond an introduction, J. Phys. A 46 (2013) 4006 [arXiv:1303.0847] [INSPIRE].
  60. [60]
    A. Castro et al., The gravity dual of the Ising model, Phys. Rev. D 85 (2012) 024032 [arXiv:1111.1987] [INSPIRE].

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© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of PretoriaPretoriaSouth Africa
  2. 2.National Institute for Theoretical Physics (NITheP)GautengSouth Africa

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