Nonabelian KP hierarchy with Moyal algebraic coefficients

TL;DR

This paper introduces a higher-dimensional nonabelian KP hierarchy using Moyal algebraic coefficients, connecting it to large-N limits of multi-component KP hierarchies and Moyal deformations of selfdual gravity, with classical limits described by Poisson structures.

Contribution

It constructs two new hierarchies: one with commuting flows extending KP hierarchies, and another with noncommuting flows related to Moyal deformations of gravity.

Findings

Two hierarchies with Moyal algebraic coefficients are constructed.

Both hierarchies have quasi-classical limits with Poisson structures.

W-infinity algebra unifies the theoretical framework.

Abstract

A higher dimensional analogue of the KP hierarchy is presented. Fundamental constituents of the theory are pseudo-differential operators with Moyal algebraic coefficients. The new hierarchy can be interpreted as large-$N$ limit of multi-component ($\gl(N)$ symmetric) KP hierarchies. Actually, two different hierarchies are constructed. The first hierarchy consists of commuting flows and may be thought of as a straightforward extension of the ordinary and multi-component KP hierarchies. The second one is a hierarchy of noncommuting flows, and related to Moyal algebraic deformations of selfdual gravity. Both hierarchies turn out to possess quasi-classical limit, replacing Moyal algebraic structures by Poisson algebraic structures. The language of W-infinity algebras provides a unified point of view to these results.