A MATRIX INTEGRAL SOLUTION TO 2-DIMENSIONAL W(P)-GRAVITY

The p(th) Gel'fand-Dickey equation and the string equation [L, P] = 1 have a common solution-tau expressible in terms of an integral over n x n Hermitean matrices (for large n), the integrand being a perturbation of a Gaussian, generalizing Kontsevich's integral beyond the KdV-case; it is equivalent to showing that tau is a vacuum vector for a W(p)+-algebra, generated from the coefficients of the vertex operator. This connection is established via a quadratic identity involving the wave function and the vertex operator, which is a disguised differential version of the Fay identity. The latter is also the key to the spectral theory for the two compatible symplectic structures of KdV in terms of the stress-energy tensor associated with the Virasoro algebra. Given a differential operator L = D(P) + q2(t) D(p-2) + ... + q(p)(t), with D = partial derivative/dx, t = (t1, t2, t3,...), x = t1, consider the deformation equations 1 partial derivative L/partial derivative t(n) = [(L(n/p)+, L] n = 1, 2, ..., n not-equal 0(mod p) (0.1) (p-reduced KP-equation) of L, for which there exists a differential operator P (possibly of infinite order) such that [L, P] = 1 (string equation). (0.2) In this note, we give a complete solution to this problem. In section 1 we give a brief survey of useful facts about the I-function T (t), the wave function-PSI (t, z), solution of partial derivative-PSI/partial derivative t(n) = (L(n/p))x PSI and L1/p-PSI = z-PSI, and the corresponding infinite-dimensional plane V0 of formal power series in z (for large z) V0 = span {PSI(t, z) for all t is-an-element-of C(infinity)}(~)[GRAPHICS]