A matrix integral solution to two-dimensionalW _p-gravity

Abstract

Thep ^th Gel'fand-Dickey equation and the string equation [L, P]=1 have a common solution τ expressible in terms of an integral overn×n Hermitean matrices (for largen), the integrand being a perturbation of a Gaussian, generalizing Kontsevich's integral beyond the KdV-case; it is equivalent to showing that τ is a vacuum vector for aW ⁺_∿p , 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

$$L = D^p + q_2 (t) D^{p - 2} + \cdots + q_p (t), with D = \frac{\partial }{{dx}},t = (t_1 ,t_2 ,t_3 ,...),x \equiv t_1 ,$$

consider the deformation equations¹

$$\begin{gathered} \frac{{\partial L}}{{\partial t_n }} = [(L^{n/p} )_ + ,L] n = 1,2,...,n + - 0(mod p) \hfill \\ (p - reduced KP - equation) \hfill \\ \end{gathered} $$

((0.1))

ofL, for which there exists a differential operatorP (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 theI-function τ(t), the wave function Ψ(t,z), solution of ∂Ψ/∂t _n=(L ^n/p) _x Ψ andL ^1/pΨ=zΨ, and the corresponding infinitedimensional planeV ⁰ of formal power series inz (for largez)

$$V^0 = span \{ \Psi (t,z) for all t \in \mathbb{C}^\infty \} $$

in Sato's Grassmannian. The three theorems below form the core of the paper; their proof will be given in subseuqent sections, each of which lives on its own right.