Non-Local Matrix Generalizations of W-Algebras

Abstract

There is a standard way to define two symplectic (hamiltonian) structures, the first and second Gelfand-Dikii brackets, on the space of ordinary linear differential operators of order $m$, $L=-d^m+U_1{d}^{m-1}+U_2{d}^{m-2}+\dots +U_m$. In this paper, I consider in detail the case where the $U_k$ are $n\times n$-matrix-valued functions, with particular emphasis on the (more interesting) second Gelfand-Dikii bracket. Of particular interest is the reduction to the symplectic submanifold $U_1=0$. This reduction gives rise to matrix generalizations of (the classical version of) the {\it non-linear} $W_m$-algebras, called $V_{m,n}$-algebras. The non-commutativity of the matrices leads to {\it non-local} terms in these $V_{m,n}$-algebras. I show that these algebras contain a conformal Virasoro subalgebra and that combinations $W_k$ of the $U_k$ can be formed that are $n\times n$-matrices of conformally primary fields of spin $k$, in analogy with the scalar case $n=1$. In general however, the $V_{m,n}$-algebras have a much richer structure than the $W_m$-algebras as can be seen on the examples of the {\it non-linear} and {\it non-local} Poisson brackets of any two matrix elements of $U_2$ or $W_3$ which I work out explicitly for all $m$ and $n$. A matrix Miura transformation is derived, mapping these complicated second Gelfand-Dikii brackets of the $U_k$ to a set of much simpler Poisson brackets, providing the analogue of the free-field realization of the $W_m$-algebras.

43 pages, a reference and a remark on the conformal properties for $U_1\ne 0$ added

References in corpus (3)

• W-symmetry in Conformal Field Theory
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• Multi-Component KdV Hierarchy, V-Algebra and Non-Abelian Toda Theory

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• W-algebras for non-abelian Toda systems
• On the Kadomtsev-Petviashvili hierarchy in an extended class of formal pseudo-differential operators
• On the Classical $W_4^{(2)}$ Algebra
• W$(2,4)$, Linear and Non-local \W-Algebras in Sp(4) Particle Model
• Nonlocal Matrix Generalizations of N=2 Super Virasoro Algebra