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JHEP, 2014, Volume 18, Issue 1, 156, 50 pp. (Mi jhep12)  

Minimal Liouville gravity correlation numbers from Douglas string equation

A. Belavinabc, B. Dubrovindef, B. Mukhametzhanovbg

a Institute for Information Transmission Problems, Bol’shoy karetni pereulok 19, 127994, Moscow, Russia
b L.D. Landau Institute for Theoretical Physics, prospect academica Semenova 1a, 142432 Chernogolovka, Russia
c Moscow Institute of Physics and Technology, Insitutsky pereulok 9, 141700 Dolgoprudny, Russia
d International School of Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy
e N.N. Bogolyubov Laboratory for Geometrical Methods in Mathematical Physics, Moscow State University “M.V.Lomonosov”, Leninskie Gory 1, 119899 Moscow, Russia
f V.A. Steklov Mathematical Institute, Gubkina street 8, 119991 Moscow, Russia
g Department of Physics, Harvard University, 17 Oxford street, 02138 Cambridge, U.S.A.

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 $\tau_{m,n}$ 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 $\tau_{m,n}$ are related in a non-linear fashion to the natural coupling constants $\lambda_{m,n}$ 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].

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-90614
12-01-00836
Ministry of Education and Science of the Russian Federation 8528
8410
2010-220-01-077
European Research Council Advanced Grant FroM-PDE
PRIN 2010-11
The work of A.B. and B.M. was supported by RFBR grants no. 13-01-90614, 12-01-00836-a and by the Russian Ministry of Education and Science under the grants no. 8528 and no. 8410. The work of B.D. was partially supported by the European Research Council Advanced Grant FroM-PDE, by the Russian Federation Government Grant No. 2010-220-01-077 and by PRIN 2010-11 Grant “Geometric and analytic theory of Hamiltonian systems in finite and infinite dimensions” of Italian Ministry of Universities and Researches.


DOI: https://doi.org/10.1007/JHEP01(2014)156


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Received: 05.11.2013
Revised: 16.12.2013
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