Abstract
We continue to study minimal Liouville gravity (MLG) using a dual approach based on the idea that the MLG partition function is related to the tau function of the A q integrable hierarchy via the resonance transformations, which are in turn fixed by conformal selection rules. One of the main problems in this approach is to choose the solution of the Douglas string equation that is relevant for MLG. The appropriate solution was recently found using connection with the Frobenius manifolds. We use this solution to investigate three- and four-point correlators in the unitary MLG models. We find an agreement with the results of the original approach in the region of the parameters where both methods are applicable. In addition, we find that only part of the selection rules can be satisfied using the resonance transformations. The physical meaning of the nonzero correlators, which before coupling to Liouville gravity are forbidden by the selection rules, and also the modification of the dual formulation that takes this effect into account remains to be found.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.M. Polyakov, Quantum Geometry of Bosonic Strings, Phys. Lett. B 103 (1981) 207 [INSPIRE].
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
A.B. Zamolodchikov, Three-point function in minimal Liouville gravity, Theor. Math. Phys. 142 (2005) 183 [hep-th/0505063] [INSPIRE].
A. Belavin and A. Zamolodchikov, Integrals over moduli spaces, ground ring, and four-point function in minimal Liouville gravity, Theor. Math. Phys 147 (2006) 729 [hep-th/0510214] [INSPIRE].
A. Zamolodchikov, Gravitational Yang-Lee Model: Four Point Function, Theor. Math. Phys. 151 (2007) 439 [hep-th/0604158] [INSPIRE].
V. Belavin, Torus Amplitudes in Minimal Liouville Gravity and Matrix Models, Phys. Lett. B 698 (2011) 86 [arXiv:1010.5508] [INSPIRE].
V.A. Kazakov, A.A. Migdal and I.K. Kostov, Critical Properties of Randomly Triangulated Planar Random Surfaces, Phys. Lett. B 157 (1985) 295 [INSPIRE].
V.A. Kazakov, Ising model on a dynamical planar random lattice: Exact solution, Phys. Lett. A 119 (1986) 140 [INSPIRE].
V.A. Kazakov, The Appearance of Matter Fields from Quantum Fluctuations of 2D Gravity, Mod. Phys. Lett. A 4 (1989) 2125 [INSPIRE].
M. Staudacher, The Yang-lee Edge Singularity on a Dynamical Planar Random Surface, Nucl. Phys. B 336 (1990) 349 [INSPIRE].
E. Brézin and V.A. Kazakov, Exactly Solvable Field Theories of Closed Strings, Phys. Lett. B 236 (1990) 144 [INSPIRE].
M.R. Douglas and S.H. Shenker, Strings in Less Than One-Dimension, Nucl. Phys. B 335 (1990) 635 [INSPIRE].
D.J. Gross and A.A. Migdal, Nonperturbative Two-Dimensional Quantum Gravity, Phys. Rev. Lett. 64 (1990) 127 [INSPIRE].
M.R. Douglas, Strings in Less Than One-dimension and the Generalized K − D − V Hierarchies, Phys. Lett. B 238 (1990) 176 [INSPIRE].
I.K. Kostov, Strings with discrete target space, Nucl. Phys. B 376 (1992) 539 [hep-th/9112059] [INSPIRE].
I.K. Kostov, Gauge invariant matrix model for the A-D-E closed strings, Phys. Lett. B 297 (1992) 74 [hep-th/9208053] [INSPIRE].
V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Fractal Structure of 2D Quantum Gravity, Mod. Phys. Lett. A 3 (1988) 819 [INSPIRE].
G.W. Moore, N. Seiberg and M. Staudacher, From loops to states in 2 − D quantum gravity, Nucl. Phys. B 362 (1991) 665 [INSPIRE].
I.K. Kostov and V.B. Petkova, Non-rational 2 − D quantum gravity. I. World sheet CFT, Nucl. Phys. B 770 (2007) 273 [hep-th/0512346] [INSPIRE].
I.K. Kostov and V.B. Petkova, Non-Rational 2D Quantum Gravity II. Target Space CFT, Nucl. Phys. B 769 (2007) 175 [hep-th/0609020] [INSPIRE].
I.K. Kostov and V.B. Petkova, Bulk correlation functions in 2 − D quantum gravity, Theor. Math. Phys. 146 (2006) 108 [hep-th/0505078] [INSPIRE].
A.A. Belavin and A.B. Zamolodchikov, On Correlation Numbers in 2D Minimal Gravity and Matrix Models, J. Phys. A 42 (2009) 304004 [arXiv:0811.0450] [INSPIRE].
A. Belavin, B. Dubrovin and B. Mukhametzhanov, Minimal Liouville Gravity correlation numbers from Douglas string equation, JHEP 01 (2014) 156 [arXiv:1310.5659] [INSPIRE].
A.A. Belavin and V.A. Belavin, Frobenius manifolds, Integrable Hierarchies and Minimal Liouville Gravity, JHEP 09 (2014) 151 [arXiv:1406.6661] [INSPIRE].
V. Belavin, Unitary Minimal Liouville Gravity and Frobenius Manifolds, JHEP 07 (2014) 129 [arXiv:1405.4468] [INSPIRE].
L. Spodyneiko, Minimal Liouville Gravity on the Torus via Matrix Models, arXiv:1407.3546 [INSPIRE].
I. Krichever, The Dispersionless Lax equations and topological minimal models, Commun. Math. Phys. 143 (1992) 415.
B. Dubrovin, Integrable systems in topological field theory, Nucl. Phys. B 379 (1992) 627 [INSPIRE].
R. Dijkgraaf, H.L. Verlinde and E.P. Verlinde, Topological strings in d < 1, Nucl. Phys. B 352 (1991) 59 [INSPIRE].
P.H. Ginsparg, M. Goulian, M.R. Plesser and J. Zinn-Justin, (p, q) string actions, Nucl. Phys. B 342 (1990) 539 [INSPIRE].
P. Di Francesco and D. Kutasov, Unitary minimal models coupled to 2 − D quantum gravity, Nucl. Phys. B 342 (1990) 589 [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: 1412.4245
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
Belavin, V. Correlation functions in unitary minimal Liouville gravity and Frobenius manifolds. J. High Energ. Phys. 2015, 52 (2015). https://doi.org/10.1007/JHEP02(2015)052
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2015)052