Z_3-graded exterior differential calculus and gauge theories of higher order

TL;DR

This paper generalizes exterior calculus with a differential operator satisfying d^3=0, introduces Z_3 grading, and applies it to develop higher-order gauge theories with a new curvature 3-form and variational principles.

Contribution

It introduces a novel Z_3-graded exterior calculus with d^3=0, extending gauge theories to higher order forms and deriving associated curvature and field equations.

Findings

Defined a Z_3-graded differential calculus with d^3=0

Constructed a gauge theory with a curvature 3-form

Derived field equations from a variational principle

Abstract

We present a possible generalization of the exterior differential calculus, based on the operator d such that d^3=0, but d^2\not=0. The first and second order differentials generate an associative algebra; we shall suppose that there are no binary relations between first order differentials, while the ternary products will satisfy the cyclic relations based on the representation of cyclic group Z_3 by cubic roots of unity. We shall attribute grade 1 to the first order differentials and grade 2 to the second order differentials; under the associative multiplication law the grades add up modulo 3. We show how the notion of covariant derivation can be generalized with a 1-form A, and we give the expression in local coordinates of the curvature 3-form. Finally, the introduction of notions of a scalar product and integration of the Z_3-graded exterior forms enables us to define variational principle and to derive the differential equations satisfied by the curvature 3-form. The Lagrangian obtained in this way contains the invariants of the ordinary gauge field tensor F_{ik} and its covariant derivatives D_i F_{km}.