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

Jean-Pierre Magnot

TL;DR

This work constructs a formal noncommutative Kadomtsev–Petviashvili hierarchy built from a Dirac type derivation acting on a hypercomplex coefficient algebra that blends spin( Lorentz) and SU(3) color degrees of freedom. The hierarchy uses a Lax operator $L = D + \sum_k U_k D^{-k}$ with quaternionic time variables that generate a rich web of covariances and reductions to complex and real time directions, recovering SU(3) covariant KdV, KP II and Boussinesq type equations as well as their embeddings into larger quaternionic time and gauge structures. The framework supports BKP and CKP like reductions and offers a gauge theoretic perspective where constrained $SU(3)$ gauge–Dirac systems yield integrable evolutions, with potential links to self dual Yang–Mills type configurations and flux tube solitons. Altogether, the paper provides a formal, algebraic platform for exploring integrable sectors of nonabelian gauge theories in four dimensions through hypercomplex, quaternionic time, and SU(3) covariant structures.

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.

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

TL;DR

This work constructs a formal noncommutative Kadomtsev–Petviashvili hierarchy built from a Dirac type derivation acting on a hypercomplex coefficient algebra that blends spin( Lorentz) and SU(3) color degrees of freedom. The hierarchy uses a Lax operator with quaternionic time variables that generate a rich web of covariances and reductions to complex and real time directions, recovering SU(3) covariant KdV, KP II and Boussinesq type equations as well as their embeddings into larger quaternionic time and gauge structures. The framework supports BKP and CKP like reductions and offers a gauge theoretic perspective where constrained gauge–Dirac systems yield integrable evolutions, with potential links to self dual Yang–Mills type configurations and flux tube solitons. Altogether, the paper provides a formal, algebraic platform for exploring integrable sectors of nonabelian gauge theories in four dimensions through hypercomplex, quaternionic time, and SU(3) covariant structures.

Abstract

We propose a formal framework for a noncommutative Kadomtsev--Petviashvili (KP) hierarchy which is covariant under the action of and compatible with a Lorentzian structure encoded in a twisted quaternionic (or Clifford) algebra. The starting point is a formal pseudodifferential operator built from an abstract derivation of Dirac type and coefficients in an associative algebra that combines spin degrees of freedom (twisted quaternions, Clifford algebras) and color degrees of freedom (an internal factor, possibly realized via the octonions). In this way we obtain a hierarchy of formal partial differential equations which are Lorentz invariant and covariant and can be interpreted as integrable sectors of nonabelian gauge theories in dimensions and of their dimensional reductions.
Paper Structure (68 sections, 2 theorems, 127 equations)

This paper contains 68 sections, 2 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: hypercomplex structures, $SU(3)$ and Dirac-type derivations
  3. Quaternions and Lorentzian twists
  4. The quaternion algebra
  5. A Lorentzianly twisted quaternionic structure
  6. Complexified and Clifford-realized quaternions
  7. Octonions and the internal $SU(3)$ symmetry
  8. The octonion algebra and its automorphisms
  9. Decomposition O = C ⊕ C³ and $SU(3)$ as a stabilizer
  10. Tensor product algebras and internal symmetry
  11. Spinorial and color factors
  12. Automorphisms and inner actions
  13. Dirac-type operators and (twisted) derivations
  14. Derivations on associative algebras
  15. Dirac-type operators in quaternionic and Clifford guise
  16. ...and 53 more sections

Key Result

Proposition 6.1

Assume that a Lax operator $L\in\Psi(\mathcal{A},D)$ satisfies at some initial time (for all time variables). Then, for each $n\ge 1$, the generalized KP flows preserve this constraint: In particular, the KP flows restrict to a B-type invariant submanifold of Lax operators, and the restricted flows define a generalized BKP-like hierarchy in the hypercomplex, $SU(3)$-covariant setting.

Theorems & Definitions (6)

  • Definition 2.1
  • Definition 2.2
  • Proposition 6.1: B-type constraint and induced hierarchy
  • Remark 6.2
  • Proposition 6.3: C-type constraint and induced hierarchy
  • Remark 6.4