A Lorentzian SU(3)-covariant noncommutative KP hierarchy and hypercomplex gauge fields

TL;DR

This paper develops a Lorentzian, SU(3)-covariant noncommutative KP hierarchy using quaternionic and Clifford algebra structures, linking integrable systems with nonabelian gauge theories in four dimensions.

Contribution

It introduces a novel noncommutative KP hierarchy framework that is Lorentz invariant and SU(3)-covariant, incorporating hypercomplex gauge fields and extending integrable systems.

Findings

Hierarchy is Lorentz invariant and SU(3) covariant.

Framework connects integrable systems with nonabelian gauge theories.

Uses quaternionic and Clifford algebra structures for coefficients.

Abstract

We propose a formal framework for a noncommutative Kadomtsev--Petviashvili (KP) hierarchy which is covariant under the action of $SU(3)$ and compatible with a Lorentzian structure encoded in a twisted quaternionic (or Clifford) algebra. The starting point is a formal pseudodifferential operator $L$ built from an abstract derivation $D$ of Dirac type and coefficients in an associative algebra $\A$ that combines spin degrees of freedom (twisted quaternions, Clifford algebras) and color degrees of freedom (an internal $SU(3)$ factor, possibly realized via the octonions). In this way we obtain a hierarchy of formal partial differential equations which are Lorentz invariant and $SU(3)$ covariant and can be interpreted as integrable sectors of nonabelian gauge theories in $(3+1)$ dimensions and of their dimensional reductions.