Abstract
We determine the spectrum and correlation functions of Liouville theory with a central charge less than (or equal) one. This completes the definition of Liouville theory for all complex values of the central charge. The spectrum is always spacelike, and there is no consistent timelike Liouville theory. We also study the non-analytic conformal field theories that exist at rational values of the central charge. Our claims are supported by numerical checks of crossing symmetry. We provide Python code for computing Virasoro conformal blocks, and correlation functions in Liouville theory and (generalized) minimal models.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Ribault, Conformal field theory on the plane, arXiv:1406.4290 [INSPIRE].
J. Teschner, On the Liouville three point function, Phys. Lett. B 363 (1995) 65 [hep-th/9507109] [INSPIRE].
V. Schomerus, Rolling tachyons from Liouville theory, JHEP 11 (2003) 043 [hep-th/0306026] [INSPIRE].
A.B. Zamolodchikov, Three-point function in the minimal Liouville gravity, hep-th/0505063 [INSPIRE].
I.K. Kostov and V.B. Petkova, Bulk correlation functions in 2D quantum gravity, Theor. Math. Phys. 146 (2006) 108 [hep-th/0505078] [INSPIRE].
B. Duplantier, Conformal random geometry, math-ph/0608053.
G. Delfino and J. Viti, On three-point connectivity in two-dimensional percolation, J. Phys. A 44 (2011) 032001 [arXiv:1009.1314] [INSPIRE].
R.M. Ziff, J.J.H. Simmons and P. Kleban, Factorization of correlations in two-dimensional percolation on the plane and torus, J. Phys. A 44 (2011) 065002 [arXiv:1011.1101] [INSPIRE].
M. Picco, R. Santachiara, J. Viti and G. Delfino, Connectivities of Potts Fortuin-Kasteleyn clusters and time-like Liouville correlator, Nucl. Phys. B 875 (2013) 719 [arXiv:1304.6511] [INSPIRE].
A. Strominger and T. Takayanagi, Correlators in time-like bulk Liouville theory, Adv. Theor. Math. Phys. 7 (2003) 369 [hep-th/0303221] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE].
D. Harlow, J. Maltz and E. Witten, Analytic continuation of Liouville theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].
L. Hadasz, Z. Jaskolski and P. Suchanek, Modular bootstrap in Liouville field theory, Phys. Lett. B 685 (2010) 79 [arXiv:0911.4296] [INSPIRE].
R. Santachiara and J. Viti, Local logarithmic correlators as limits of Coulomb gas integrals, Nucl. Phys. B 882 (2014) 229 [arXiv:1311.2055] [INSPIRE].
I. Runkel and G.M.T. Watts, A nonrational CFT with c = 1 as a limit of minimal models, JHEP 09 (2001) 006 [hep-th/0107118] [INSPIRE].
W. McElgin, Notes on Liouville theory at c ≤ 1, Phys. Rev. D 77 (2008) 066009 [arXiv:0706.0365] [INSPIRE].
B. Eynard and S. Ribault, Lax matrix solution of c = 1 conformal field theory, JHEP 02 (2014) 059 [arXiv:1307.4865] [INSPIRE].
V. Schomerus and P. Suchanek, Liouville’s imaginary shadow, arXiv:1210.1856 [INSPIRE].
A. Bawane, G. Bonelli, M. Ronzani and A. Tanzini, \( \mathcal{N}=2 \) supersymmetric gauge theories on S 2 × S 2 and Liouville Gravity, JHEP 07 (2015) 054 [arXiv:1411.2762] [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: 1503.02067
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
Ribault, S., Santachiara, R. Liouville theory with a central charge less than one. J. High Energ. Phys. 2015, 109 (2015). https://doi.org/10.1007/JHEP08(2015)109
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2015)109