Abstract
Rational CFT’s are classified by an integer ℓ, the number of zeroes of the Wronskian of their characters in moduli space. For ℓ = 0 they satisfy non-singular modularinvariant differential equations, while for ℓ > 0 the corresponding equations have singularities. We survey CFT’s with two characters and ℓ = 0, 2, 3, 4 and verify the consistency, at the level of characters, of some candidate theories with ℓ = 0. For ℓ = 2 there are seven consistents sets of characters. We identify specific combinations of level-1 current algebras that are potential symmetries of the corresponding CFT’s.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. Anderson and G.W. Moore, Rationality in Conformal Field Theory, Commun. Math. Phys. 117 (1988) 441 [INSPIRE].
S.D. Mathur, S. Mukhi and A. Sen, Differential equations for correlators and characters in arbitrary rational Conformal Field Theories, Nucl. Phys. B 312 (1989) 15 [INSPIRE].
S.D. Mathur, S. Mukhi and A. Sen, On the classification of rational Conformal Field Theories, Phys. Lett. B 213 (1988) 303 [INSPIRE].
S.D. Mathur, S. Mukhi and A. Sen, Reconstruction of Conformal Field Theories from modular geometry on the torus, Nucl. Phys. B 318 (1989) 483 [INSPIRE].
S.G. Naculich, Differential equations for rational conformal characters, Nucl. Phys. B 323 (1989) 423 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York U.S.A. (1997).
A.N. Schellekens, Meromorphic C = 24 conformal field theories, Commun. Math. Phys. 153 (1993) 159 [hep-th/9205072] [INSPIRE].
E. Witten, Three-dimensional gravity revisited, arXiv:0706.3359 [INSPIRE].
Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc 9 (1996) 237.
M.R. Gaberdiel, Constraints on extremal self-dual CFTs, JHEP 11 (2007) 087 [arXiv:0707.4073] [INSPIRE].
M.R. Gaberdiel and C.A. Keller, Modular differential equations and null vectors, JHEP 09 (2008) 079 [arXiv:0804.0489] [INSPIRE].
E.B. Kiritsis, Fuchsian differential equations for characters on the torus: a classification, Nucl. Phys. B 324 (1989) 475 [INSPIRE].
T. Eguchi and H. Ooguri, Differential equations for characters of Virasoro and affine Lie algebras, Nucl. Phys. B 313 (1989) 492 [INSPIRE].
A. Milas, On certain automorphic forms associated to rational vertex operator algebras, in Moonshine — the first quarter century and beyond, J. Lepowsky, J. McKay and M.P. Tuite eds., London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge U.K. (2010).
E.P. Verlinde, Fusion rules and modular transformations in 2D Conformal Field Theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].
G.W. Moore and N. Seiberg, Polynomial equations for rational conformal field theories, Phys. Lett. B 212 (1988) 451 [INSPIRE].
P. Bantay and T. Gannon, Vector-valued modular functions for the modular group and the hypergeometric equation, Commun. Num. Theor. Phys. 1 (2007) 651 [INSPIRE].
T. Gannon, The theory of vector-modular forms for the modular group, Contrib. Math. Comput. Sci. 8 (2014) 247 [arXiv:1310.4458] [INSPIRE].
S. Hellerman, A universal inequality for CFT and quantum gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE].
C.A. Keller and A. Maloney, Poincaré series, 3D gravity and CFT spectroscopy, JHEP 02 (2015) 080 [arXiv:1407.6008] [INSPIRE].
A. Castro, M.R. Gaberdiel, T. Hartman, A. Maloney and R. Volpato, The gravity dual of the Ising model, Phys. Rev. D 85 (2012) 024032 [arXiv:1111.1987] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1510.04478
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Hampapura, H.R., Mukhi, S. On 2d Conformal Field Theories with two characters. J. High Energ. Phys. 2016, 5 (2016). https://doi.org/10.1007/JHEP01(2016)005
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2016)005