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.
Highlights
The idea of fluctuating geometries first led to the development of the matrix-model (MM) approach to 2D gravity [7,8,9,10,11,12,13,14,15,16]
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 Aq 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
Summary
We restrict our attention to the series of unitary models Mq,q+1 coupled to Liouville gravity in the spherical topology. U∗q−1) that can be used to construct the generating function of the correlators in MLG It was shown in [23] that the parameters uα can be interpreted as the coordinates on the (q − 1)-dimensional Frobenius manifold such that the metric in these coordinates is given by. Flatness of the metric, existence of the Frobenius potential, etc.) can be checked in the initial coordinates uα [28] This interpretation of the parameters uα turns out to be very efficient. ) in the limit where the Liouville coupling constants are equal to zero This solution is unique, i.e., only this solution gives the generating function (defined below) for which the necessary selection rules are satisfied on the level of one- and two-point correlators: it gives zero vacuum expectation values of the physical fields (except unity), and the two-point correlators are diagonal
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