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International Conference dedicated to the 60-th birthday of Boris Feigin "Representation Theory and applications to Combinatorics, Geometry and Quantum Physics"
December 14, 2013 12:35–13:25, Moscow, Independent University of Moscow

Frobenious manifold structure for Douglas string equation and the correlation numbers for Minimal Liouville gravity

A. A. Belavin
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A. A. Belavin

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Abstract: I am going to present a reviw of some results of the joint works with my colleagues about so-called $(p,q)$ Minimal Liouville gravity.
I will argue that the generating function of the correlators in genus zero in Minimal Liouville gravity (MLG) is nothing but logarithm of the Sato tau-function for dispersionless Gefand–Dikii hierarchy with the special initial condition given by Douglas string equation.
The correlators of Minimal Liouville gravity are not equal to the expansion coefficients of log of the tau-function in respect to KdV times as in Matrix models. Instead the correlators of MLG are the expansion coefficients of Log of the tau-function in respect to the new variables connected with KdV variables by a special noliniear “resonance” transformation.
These correlators of MLG satisfy to the necessary conformal and fusion rules as it should be $M(p/q)$ conformal minimal models. I will use the connection between Minimal Liouville gravity and Frobenious manifolds to get an explicit and useful expression for log Sato tau-function corresponding to Douglas string equation in dispersionless limit.

Language: English

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