T. Shiota

A Lax Representation for the Vertex Operator and the Central Extension

Integrable hierarchies, viewed as isospectral deformations of an operatorL may admit symmetries; they are time-dependent vector fields, transversal to and commuting with the hierarchy and forming an algebra. In this work, the commutation relations for the symmetries are shown to be based on a non-commutative Lie algebra splitting theorem. The symmetries, viewed as vector fields onL, are expressed in terms of a Lax pair.This study introduces a “generating symmetry”, a generating function for symmetries, both of the KP equation (continuous), and the two-dimensional Toda lattice (discrete), in terms ofL and an operatorM, introduced by Orlov and Schulman, such that [L, M] = 1. This “generating symmetry”, acting on the wave function (or wave vector) lifts to a vertex operatorà la Date-Jimbo-Kashiwara-Miwa, acting on the τ-function (or τ-vector). Lifting the algebra of symmetries, acting on wave functions, to an algebra of symmetries, acting on τ-functions, amounts to passing from an algebra to its central extension.This provides a handy technology to find the constraints satisfied by various matrix integrals, arising in the context of 2d-quantum gravity and moduli space topology. The point is to first prove the vanishing of symmetries at the Lax pair level, which usually turns out to be elementary and conceptual, and then use the lifting above to get the subalgebra of vanishing symmetries for the τ-function (or τ-vectors).

Random Matrices, Vertex Operators and the Virasoro-algebra

This work aims at a new approach to the theory of random matrices, inspired by recent work on matrix models. Introducing an appropriate time t-dependence in the probability distribution of the matrix ensemble leads to vertex operator expressions for the n-point correlation functions (probabilities of an eigenvalue in infinitesimal intervals) and the corresponding Fredholm determinants (probabilities of no eigenvalue in an interval), the latter satisfy Virasoro-like constraints, which, upon setting t = 0, lead to a new hierarchy of PDEs for the P (no eigenvalue is an element of J), where J = boolean OR(i=1)(r) [A(2i-1), A(2i)], in terms of the endpoints A(i). In the single interval case, the first equation in the hierarchy recovers the Painleve distributions for the classical ensembles. This is done in Section 1 for polynomial ensembles, i.e., the probabilities are given by explicit matrix integrals, and in Section 3 for ensembles, defined by more general kernels. Examples are given in Section 4. From the point of view of the KP and Toda symmetries and their Virasoro (or W)-counterparts on tau, as studied by us previously, the probabilities above are expressed in terms of a tau-function tau(t, A), depending on the integrable directions t(j) and the endpoints A(i) of the intervals J. The Virasoro vector fields on tau move the endpoints (motion in moduli space) according to the simple (decoupled) differential equations A(i) = A(i)(k+1).

Schur function expansion for normal matrix model and associated discrete matrix models

We consider Schur function expansion for the partition function of the model of normal matrices. This expansion coincides with Takasakis expansion for tau functions of Toda lattice hierarchy. We show that the partition function of the model of normal matrices is, at the same time, a partition function of certain discrete models, which can be solved by the method of orthogonal polynomials. We obtain discrete versions of various known matrix models: models of non-negative matrices, unitary matrices, normal matrices. We also introduce Hermitian and unitary two-matrix models with generalized interaction terms in continuous and discrete versions.

From the ω∞-algebra to its central extension: a τ-function approach

Abstract The KP hierarchy, deformations of pseudo-differential operators L of order one, admits a ω∞-algebra of symmetries Y z α ( ∂ ∂z ) β , which are vector fields transversal to and commuting with the KP hierarchy. Expressed in terms of L and another pseudo-differential operator M (introduced by Orlov and coworkers) satisfying [L,M] = 1, these vector fields act on the wave function Ψ (a properly normalized eigenfunction of L) as Y z α ( ∂ ∂z β Ψ ≡ −(MβLα) _ Ψ. Introducing a generating function YN Ψ = N_ Ψ, with N ≡ (gm − λ) exp [(μ − λ) M] δ (λ, L), for the algebra of symmetries ω∞ on Ψ and taking into account the well-known representation of Ψ (t, z) = [e −η τ( t τ( t)] exp (Σ 1 ∞ t i z i ) , in terms of the τ-function, where η = Σi=1∞(z−i/i)(∂/∂ti. We show a precise relationship between YN and the Date-Jimbo-Kashiwara-Miwa vertex operator X(t,λ,μ) ≡ exp [Σ 1 ∞ (μ i − λ i )t i ] exp [Σ 1 ∞ (λ −i − μ −i )( 1 i )(∂/∂t i )] , a generating function of the W∞-algebra of symmetries (with central extension) on τ, to wit YN log Ψ = (e−η − 1) X log τ, where YN log and X log act on Ψ and τ as logarithmic derivatives, with respect to the vector fields YN and X.

A matrix integral solution to [P,Q]=P and matrix Laplace transforms

In this paper we solve the following problems: (i) find two differential operatorsP andQ satisfying [P, Q]=P, whereP flows according to the KP hierarchy ϖP/ϖtn=[(Pn/p)+,P], withp:=ordP≥2; (ii) find a matrix a integral representation for the associated τ-function. First we construct an infinite dimensional spaceW= spanℂ{ψ0(z,ψ1(z,...)} of functions ofzεℂ invariant under the action of two operators, multiplication byzp andAc:=zϖ/ϖz−z+c. This requirement is satisfied, for arbitraryp, ifψ0 is a certain function generalizing the classical Hänkel function (forp=2); our representation of the generalized Hänkel function as adouble Laplace transform of a simple function, which was unknown even for thep=2 case, enables us to represent the τ-function associated with the KP time evolution of the spaceW as a “double matrix Laplace transform” in two different ways. One representation involves an integration over the space of matrices whose spectrum belongs to a wedge-shaped contourγ≔γ-+γ- ⊂ℂ defined byγ± = ℝ+e±πi/p. The new integrals above relate to matrix Laplace transforms, in contrast with matrix Fourier transforms, which generalize the Kontsevich integrals and solve the operator equation [P, Q]=1.

New matrix model solutions to the Kac-Schwarz problem

We examine the Kac-Schwarz problem of specification of point in Grassmannian in the restricted case of gap-one first-order differential Kac-Schwarz operators. While the pair of constraints satisfying

CKP Hierarchy, Bosonic Tau Function and Bosonization Formulae

[{\cal K}_1,W] = 1

Pfaff

always leads to Kontsevich type models, in the case of

\tau

[{\cal K}_1,W] = W

-functions

the corresponding KP