INTEGRABLE STRUCTURES IN MATRIX MODELS AND PHYSICS OF 2D GRAVITY

Abstract

A review of the appearance of integrable structures in the matrix model description of 2D gravity is presented. Most of the ideas are demonstrated with technically simple but ideologically important examples. Matrix models are considered as a sort of “effective” description of continuum 2D field theory formulation. The main physical role in such a description is played by the Virasoro-W conditions, which can be interpreted as certain unitarity or factorization constraints. Both discrete and continuum (generalized Kontsevich) models are formulated as the solutions to those discrete (continuous) Virasoro-W constraints. Their integrability properties are proved, using mostly the determinant technique highly related to the representation in terms of free fields. The paper also contains some new observations connected with formulation of more-general-than-GKM solutions and deeper understanding of their relation to 2D gravity.