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Extremal chiral \( \mathcal{N} \) = 4 SCFT with c = 24

  • Sarah M. Harrison
Open Access
Regular Article - Theoretical Physics

Abstract

We construct an extremal chiral \( \mathcal{N} \) = 4 superconformal field theory with central charge 24 from a \( {\mathbb{Z}}_2 \) orbifold of the chiral bosonic theory with target \( {\mathbb{R}}^{24}/\varLambda \), where Λ is the Niemeier lattice with root system A 2 12 . This construction is analogous to constructions of extremal chiral \( \mathcal{N} \) = 1 and \( \mathcal{N} \) = 2 CFTs with c = 24, where Λ = ΛLeech and the Niemeier lattice with root system A 1 24 , respectively. The theory has a discrete symmetry group related to the sporadic group M 11.

Keywords

AdS-CFT Correspondence Conformal Field Theory Models of Quantum Gravity 

Notes

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.

References

  1. [1]
    O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    E. Witten, Three-Dimensional Gravity Revisited, arXiv:0706.3359 [INSPIRE].
  3. [3]
    G. Hoehn, Conformal Designs based on Vertex Operator Algebras, math.QA/0701626 [INSPIRE].
  4. [4]
    I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras And The Monster, Pure and Applied Mathematics, volume 134, Academic Press, Boston Massachusetts U.S.A. (1988) [INSPIRE].
  5. [5]
    J. Duncan, Super-moonshine for Conway’s largest sporadic group, math.RT/0502267.
  6. [6]
    J.F.R. Duncan and S. Mack-Crane, The Moonshine Module for Conway’s Group, SIGMA 3 (2015) e10 [arXiv:1409.3829] [INSPIRE].
  7. [7]
    L.J. Dixon, P.H. Ginsparg and J.A. Harvey, Beauty and the Beast: Superconformal Symmetry in a Monster Module, Commun. Math. Phys. 119 (1988) 221 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    X. Yin, On Non-handlebody Instantons in 3D Gravity, JHEP 09 (2008) 120 [arXiv:0711.2803] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    X. Yin, Partition Functions of Three-Dimensional Pure Gravity, Commun. Num. Theor. Phys. 2 (2008) 285 [arXiv:0710.2129] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    D. Gaiotto and X. Yin, Genus two partition functions of extremal conformal field theories, JHEP 08 (2007) 029 [arXiv:0707.3437] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    M.R. Gaberdiel, Constraints on extremal self-dual CFTs, JHEP 11 (2007) 087 [arXiv:0707.4073] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  13. [13]
    M.R. Gaberdiel, C.A. Keller and R. Volpato, Genus Two Partition Functions of Chiral Conformal Field Theories, Commun. Num. Theor. Phys. 4 (2010) 295 [arXiv:1002.3371] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    M.R. Gaberdiel and C.A. Keller, Modular differential equations and null vectors, JHEP 09 (2008) 079 [arXiv:0804.0489] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    D. Gaiotto, Monster symmetry and Extremal CFTs, arXiv:0801.0988 [INSPIRE].
  16. [16]
    W. Li, W. Song and A. Strominger, Chiral Gravity in Three Dimensions, JHEP 04 (2008) 082 [arXiv:0801.4566] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    M.R. Gaberdiel, S. Gukov, C.A. Keller, G.W. Moore and H. Ooguri, Extremal N = (2, 2) 2D Conformal Field Theories and Constraints of Modularity, Commun. Num. Theor. Phys. 2 (2008) 743 [arXiv:0805.4216] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    T. Kawai, Y. Yamada and S.-K. Yang, Elliptic genera and N = 2 superconformal field theory, Nucl. Phys. B 414 (1994) 191 [hep-th/9306096] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  19. [19]
    R. Dijkgraaf, J.M. Maldacena, G.W. Moore and E.P. Verlinde, A Black hole Farey tail, hep-th/0005003 [INSPIRE].
  20. [20]
    M.C.N. Cheng, X. Dong, J.F.R. Duncan, S. Harrison, S. Kachru and T. Wrase, Mock Modular Mathieu Moonshine Modules, Res. Math. Sci. 2 (2015) 13 [arXiv:1406.5502] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    N. Benjamin, E. Dyer, A.L. Fitzpatrick and S. Kachru, An extremal \( \mathcal{N} \) = 2 superconformal field theory, J. Phys. A 48 (2015) 495401 [arXiv:1507.00004] [INSPIRE].MathSciNetMATHGoogle Scholar
  22. [22]
    N. Benjamin, S.M. Harrison, S. Kachru, N.M. Paquette and D. Whalen, On the elliptic genera of manifolds of Spin(7) holonomy, Ann. Henri Poincaré 17 (2016) 2663 [arXiv:1412.2804] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    M.C.N. Cheng, S.M. Harrison, S. Kachru and D. Whalen, Exceptional Algebra and Sporadic Groups at c = 12, arXiv:1503.07219 [INSPIRE].
  24. [24]
    M. Eichler and D. Zagier, The theory of Jacobi forms, Birkhäuser Boston (1985).Google Scholar
  25. [25]
    A.N. Schellekens, Meromorphic c = 24 conformal field theories, Commun. Math. Phys. 153 (1993) 159 [hep-th/9205072] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    P.S. Montague, Orbifold constructions and the classification of selfdual c = 24 conformal field theories, Nucl. Phys. B 428 (1994) 233 [hep-th/9403088] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    J.H. Conway and N.J.A. Sloane, Sphere packings, lattices and groups, Grundlehren der mathematischen Wissenschaften, volume 290, Springer-Verlag, New York U.S.A. (2013).Google Scholar
  28. [28]
    L. Dolan, P. Goddard and P. Montague, Conformal Field Theory of Twisted Vertex Operators, Nucl. Phys. B 338 (1990) 529 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    J. Distler, Notes on N = 2 σ-models, in proceedings of the Spring School on String Theory and Quantum Gravity (to be followed by Workshop), ICTP, Trieste, Italy, 30 March-10 April 1992, J. Harvey, R. Iengo, K.S. Narain, S. Randjbar-Daemi and H. Verlinde eds., World Scientific (1993), pp. 234-256 [PUPT-1365] [hep-th/9212062] [INSPIRE].
  30. [30]
    J.M. Maldacena, G.W. Moore and A. Strominger, Counting BPS black holes in toroidal Type II string theory, hep-th/9903163 [INSPIRE].
  31. [31]
    A. Sevrin, W. Troost and A. Van Proeyen, Superconformal Algebras in Two-Dimensions with N = 4, Phys. Lett. B 208 (1988) 447 [INSPIRE].
  32. [32]
    S. Gukov, E. Martinec, G.W. Moore and A. Strominger, An Index for 2-D field theories with large N = 4 superconformal symmetry, hep-th/0404023 [INSPIRE].
  33. [33]
    S. Zwegers, Mock Theta Functions, arXiv:0807.4834 [INSPIRE].
  34. [34]
    A. Dabholkar, S. Murthy and D. Zagier, Quantum Black Holes, Wall Crossing and Mock Modular Forms, arXiv:1208.4074 [INSPIRE].
  35. [35]
    M.C.N. Cheng, J.F.R. Duncan and J.A. Harvey, Umbral Moonshine and the Niemeier Lattices, arXiv:1307.5793 [INSPIRE].
  36. [36]
    M.C.N. Cheng and J.F.R. Duncan, On Rademacher Sums, the Largest Mathieu Group and the Holographic Modularity of Moonshine, Commun. Num. Theor. Phys. 6 (2012) 697 [arXiv:1110.3859] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    N.M. Paquette, D. Persson and R. Volpato, Monstrous BPS-Algebras and the Superstring Origin of Moonshine, arXiv:1601.05412 [INSPIRE].
  38. [38]
    T. Eguchi and A. Taormina, Unitary representations of the N = 4 superconformal algebra, Phys. Lett. B 196 (1987) 75 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    T. Eguchi and A. Taormina, Character Formulas for the N = 4 Superconformal Algebra, Phys. Lett. B 200 (1988) 315 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    T. Eguchi and A. Taormina, On the Unitary Representations of N = 2 and N = 4 Superconformal Algebras, Phys. Lett. B 210 (1988) 125 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    V.K. Dobrev, Characters of the Unitarizable Highest Weight Modules Over the N = 2 Superconformal Algebras, Phys. Lett. B 186 (1987) 43 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    E. Kiritsis, Character Formulae and the Structure of the Representations of the N = 1, N =2 Superconformal Algebras,Int. J. Mod. Phys. A 3(1988) 1871 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    T. Eguchi, H. Ooguri and Y. Tachikawa, Notes on the K3 Surface and the Mathieu group M 24, Exper. Math. 20 (2011) 91 [arXiv:1004.0956] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Center for the Fundamental Laws of Nature, Department of PhysicsHarvard UniversityCambridgeU.S.A.

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