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On the structure of Normal Matrix Model
We study the structure of the normal matrix model (NMM). We show that all
correlation functions of the model with axially symmetric potentials can be
expressed in terms of holomorphic functions of one variable. This observation
is used to demonstrate the exact solvability of the model. The two-point
correlation function is calculated in the scaling limit by solving the BBGKY
chain of equations. The answer is shown to be universal (i.e. potential
independent up to a change of the scale). We then develop a two-dimensional
free fermion formalism and construct a family of completely integrable
hierarchies (which we call the extended-KP(N) hierarchies) of non-linear
differential equations. The well-known KP hierarchy is a lower-dimensional
reduction of this family. The extended-KP(1) hierarchy contains the
(2+1)-dimensional Burgers equations. The partition function of the N*N NMM is
the tau function of the extended-KP(N) hierarchy invariant with respect to a
subalgebra of an algebra of all infinitesimal diffeomorphisms of the plane.Comment: 43 pages, no figure
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