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

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.