Z 3-Graded exterior differential calculus and gauge theories of higher order

We present a possible generalization of the exterior differential calculus, based on the operator d such that d³=0, but d²0. The entities dxⁱ and d²x^k generate an associative algebra; we shall suppose that the products dxⁱ dx^k are independent of dx^k dxⁱ, while theternary products will satisfy the relation: dxⁱ dx^k dx^m=jdx^k dx^m dxⁱ=j²dx^m dx^m dxⁱ dx^k, complemented by the relation dxⁱ d²x^k=jd²x^k dxⁱ, withj:=e^2i/3.We shall attribute grade 1 to the differentials dxⁱ and grade 2 to the second differentials d²x^k; under the associative multiplication law the grades add up modulo 3.We show how the notion ofcovariant derivation can be generalized with a 1-formA so thatD:=d+A, and we give the expression in local coordinates of thecurvature 3-form defined as :=d²A+d(A²)+AdA+A³.Finally, the introduction of notions of a scalar product and integration of theZ₃-graded exterior forms enables us to define the variational principle and to derive the differential equations satisfied by the 3-form . The Lagrangian obtained in this way contains the invariants of the ordinary gauge field tensorF_ik and its covariant derivativesD_iF_km.