Abstract
Liouville field theory approach to 2-dimensional gravity possesses the duality (b ↔ b−1). The matrix counterpart of minimal gravity ℳ(q, p) (q < p co-prime) is effectively described on Aq−1 Frobenius manifold, which may exhibit a similar duality p ↔ q, and allow a description on Ap−1 Frobenius manifold. We have positive results from the bulk one-point and the bulk-boundary two-point correlations on disk that the dual description of the Frobenius manifold works for the unitary series ℳ(q, q + 1). However, for the Lee-Yang series ℳ(2, 2q + 1) on disk the duality is checked only partially. The main difficulty lies in the absence of a canonical description of trace in the continuum limit.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. Dorn and H.J. Otto, Two and three point functions in Liouville theory, Nucl. Phys. B 429 (1994) 375 [hep-th/9403141] [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].
J. Teschner, On the Liouville three point function, Phys. Lett. B 363 (1995) 65 [hep-th/9507109] [INSPIRE].
V. Fateev, A.B. Zamolodchikov and A.B. Zamolodchikov, Boundary Liouville field theory. 1. Boundary state and boundary two point function, hep-th/0001012 [INSPIRE].
J. Teschner, Remarks on Liouville theory with boundary, hep-th/0009138 [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].
P.H. Ginsparg, M. Goulian, M.R. Plesser and J. Zinn-Justin, (p, q) string actions, Nucl. Phys. B 342 (1990) 539 [INSPIRE].
K. Aleshkin and V. Belavin, On the construction of the correlation numbers in Minimal Liouville Gravity, JHEP 11 (2016) 142 [arXiv:1610.01558] [INSPIRE].
V. Belavin and Yu. Rud, Matrix model approach to minimal Liouville gravity revisited, J. Phys. A 48 (2015) 18FT01 [arXiv:1502.05575] [INSPIRE].
M.R. Douglas, Strings in Less Than One-dimension and the Generalized K − D − V Hierarchies, Phys. Lett. B 238 (1990) 176 [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.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Fractal Structure of 2D Quantum Gravity, Mod. Phys. Lett. A 3 (1988) 819 [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].
G. Ishiki and C. Rim, Boundary correlation numbers in one matrix model, Phys. Lett. B 694 (2011) 272 [arXiv:1006.3906] [INSPIRE].
N. Seiberg and D. Shih, Branes, rings and matrix models in minimal (super)string theory, JHEP 02 (2004) 021 [hep-th/0312170] [INSPIRE].
K. Hosomichi, Bulk boundary propagator in Liouville theory on a disc, JHEP 11 (2001) 044 [hep-th/0108093] [INSPIRE].
J.-E. Bourgine, G. Ishiki and C. Rim, Bulk-boundary correlators in the hermitian matrix model and minimal Liouville gravity, Nucl. Phys. B 854 (2012) 853 [arXiv:1107.4186] [INSPIRE].
K. Aleshkin, V. Belavin and C. Rim, Minimal gravity and Frobenius manifolds: bulk correlation on sphere and disk, JHEP 11 (2017) 169 [arXiv:1708.06380] [INSPIRE].
V. Belavin, Unitary Minimal Liouville Gravity and Frobenius Manifolds, JHEP 07 (2014) 129 [arXiv:1405.4468] [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: 1801.10328
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bawane, A., Muraki, H. & Rim, C. Dual Frobenius manifolds of minimal gravity on disk. J. High Energ. Phys. 2018, 134 (2018). https://doi.org/10.1007/JHEP03(2018)134
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2018)134