A matrix integral solution to two-dimensionalWp-gravity
A matrix integral solution to two-dimensionalWp-gravity
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DOI：
10.1007/bf02099527
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发表时间：
1992-06
影响因子：
2.4
通讯作者：
M. Adler;P. Moerbeke;P. Moerbeke
中科院分区：
文献类型：
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作者：
M. Adler;P. Moerbeke;P. Moerbeke
pthGel'fand-Dickey 方程和弦方程 [L, P]=1 有一个公共解 τ，可以用 n×nHermitean 矩阵（对于largen）的积分来表示，被积函数是高斯的扰动，将 Kontsevich 积分推广到 KdV 情况之外；它相当于表明 τ 是 aW∿p+ 的真空向量，由顶点算子的系数生成。这种联系是通过涉及波函数和顶点算子的二次恒等式建立的，这是 Fay 恒等式的变相微分版本。后者也是 KdV 的两个兼容辛结构在与 Virasoro 代数相关的应力-能量张量方面的谱理论的关键。给定一个微分算子 $$L = D^p + q_2 (t) D^{p - 2} + \cdots + q_p (t)，其中 D = \frac{\partial }{{dx}},t = (t_1 ,t_2 ,t_3 ,...),x \equiv t_1 ,$$考虑变形方程1$$\begin{gathered} \frac{{\partial L}}{{\partial t_n }} = [(L^{n/p} )_ + ,L] n = 1,2,...,n + - 0(mod p) \hfill \\ (p - 简化的 KP - 方程) \hfill \\ \end{gathered} $$ ofL，其中存在一个微分算子P（可能是无限阶）使得$$[L,P] = 1（弦方程）。$$在这篇笔记中，我们给出了这个问题的完整解决方案。在第 1 节中，我们简要概述了有关 I 函数 τ(t)、波函数 Ψ(t,z)、∂Ψ/∂tn=(Ln/p)xΨ 和 L1/pΨ=zΨ 的解，以及形式幂级数 inz 的相应无限维平面 V0（对于大 z）$$V^0 = span \{ \Psi (t,z) 对于所有 t \in 的有用事实佐藤格拉斯曼式的 \mathbb{C}^\infty \} $$。以下三个定理构成了本文的核心；他们的证明将在随后的部分中给出，每个部分都有自己的权利。
ThepthGel'fand-Dickey equation and the string equation [L, P]=1 have a common solution τ expressible in terms of an integral overn×nHermitean 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$$\begin{gathered} \frac{{\partial L}}{{\partial t_n }} = [(L^{n/p} )_ + ,L] n = 1,2,...,n + - 0(mod p) \hfill \\ (p - reduced KP - equation) \hfill \\ \end{gathered} $$ ofL, for which there exists a differential operatorP(possibly of infinite order) such that $$[L,P] = 1 (string equation).$$ 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/∂tn=(Ln/p)xΨ andL1/pΨ=zΨ, and the corresponding infinitedimensional planeV0of 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.