Abstract
We continue the study of (q, p) Minimal Liouville Gravity with the help of Douglas string equation. We generalize the results of [1,2], where Lee-Yang series (2, 2s + 1) was studied, to (3, 3s + p 0) Minimal Liouville Gravity, where p 0 = 1, 2. We demonstrate that there exist such coordinates τ m,n on the space of the perturbed Minimal Liouville Gravity theories, in which the partition function of the theory is determined by the Douglas string equation. The coordinates τ m,n are related in a non-linear fashion to the natural coupling constants λ m,n of the perturbations of Minimal Lioville Gravity by the physical operators O m,n . We find this relation from the requirement that the correlation numbers in Minimal Liouville Gravity must satisfy the conformal and fusion selection rules. After fixing this relation we compute three- and four-point correlation numbers when they are not zero. The results are in agreement with the direct calculations in Minimal Liouville Gravity available in the literature [3–5].
References
G.W. Moore, N. Seiberg and M. Staudacher, From loops to states in 2D quantum gravity, Nucl. Phys. B 362 (1991) 665 [INSPIRE].
A. Belavin and A. Zamolodchikov, On correlation numbers in 2D minimal gravity and matrix models, J. Phys. A 42 (2009) 304004 [arXiv:0811.0450] [INSPIRE].
M. Goulian and M. Li, Correlation functions in Liouville theory, Phys. Rev. Lett. 66 (1991) 2051 [INSPIRE].
A.B. Zamolodchikov, Three-point function in the minimal Liouville gravity, Theor. Math. Phys. 142 (2005) 183 [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 [Teor. Mat. Fiz. 147 (2006) 339] [INSPIRE].
A.M. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981) 207 [INSPIRE].
V. Knizhnik, A.M. Polyakov and A. Zamolodchikov, Fractal structure of 2D quantum gravity, Mod. Phys. Lett. A 3 (1988) 819 [INSPIRE].
A. Belavin, A.M. Polyakov and A. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
A. Zamolodchikov, Higher equations of motion in Liouville field theory, Int. J. Mod. Phys. A 19S2 (2004) 510 [hep-th/0312279] [INSPIRE].
V. Kazakov, A.A. Migdal and I. Kostov, Critical properties of randomly triangulated planar random surfaces, Phys. Lett. B 157 (1985) 295 [INSPIRE].
V. Kazakov, Ising model on a dynamical planar random lattice: exact solution, Phys. Lett. A 119 (1986) 140 [INSPIRE].
V. 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. 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].
P.H. Ginsparg and G.W. Moore, Lectures on 2D gravity and 2D string theory, hep-th/9304011 [INSPIRE].
P. Di Francesco, P.H. Ginsparg and J. Zinn-Justin, 2D gravity and random matrices, Phys. Rept. 254 (1995) 1 [hep-th/9306153] [INSPIRE].
P. Di Francesco and D. Kutasov, World sheet and space-time physics in two-dimensional (super)string theory, Nucl. Phys. B 375 (1992) 119 [hep-th/9109005] [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. Krichever, The dispersionless Lax equations and topological minimal models, Commun. Math. Phys. 143 (1992) 415 [INSPIRE].
B. Dubrovin, Integrable systems in topological field theory, Nucl. Phys. B 379 (1992) 627 [INSPIRE].
B. Dubrovin, Geometry of 2D topological field theories, in Integrable Systems and Quantum Groups, Montecatini Terme Italy 1993, M. Francaviglia and S. Greco eds., Springer Lect. Notes Math. 1620 (1996) 120 [hep-th/9407018] [INSPIRE].
G. Segal and G. Wilson, Loop groups and equations of KdV type, Publ. Math. IHES 61 (1985) 5.
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. Plesser and J. Zinn-Justin, (p, q) string actions, Nucl. Phys. B 342 (1990) 539 [INSPIRE].
I.M. Gelfand and L.A. Dickey, Fractional powers of operators and Hamiltonian systems, Funct. Anal. Appl. 10 (1976) 259.
M. Adler, On a trace functional for formal pseudodifferential operators and the symplectic structure of Korteweg-de Vries equations, Invent. Math. 50 (1979) 219 [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: 1310.5659
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Belavin, A., Dubrovin, B. & Mukhametzhanov, B. Minimal Liouville gravity correlation numbers from Douglas string equation. J. High Energ. Phys. 2014, 156 (2014). https://doi.org/10.1007/JHEP01(2014)156
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2014)156
Keywords
- 2D Gravity
- Conformal and W Symmetry
- Matrix Models