Z(3)-GRADED EXTERIOR DIFFERENTIAL-CALCULUS AND GAUGE-THEORIES OF HIGHER-ORDER

We present a possible generalization of the exterior differential calc ulus, based on the operator d such that d(3) = 0, but d(2) not equal 0 . The entities dx(i) and d(2)x(k) generate an associative algebra; we shall suppose that the products dx(i) dx(k) are independent of dx(k) d x(i), while the ternary products will satisfy the relation: dx(i) dx(k ) dx(m) = j dx(k) dx(m) dx(i) = j(2)dx(m) dx(i) dx(k), complemented by the relation dx(i) d(2)x(k) = jd(2)x(k)dx(i), with j := e(2 pi i/3). We shall attribute grade 1 to the differentials dx(i) and grade 2 to t he 'second differentials' d(2)x(k); under the associative multiplicati on law the grades add up module 3. We show how the notion of covariant derivation can be generalized with a 1-form A so that D Phi := d Phi + A Phi, and we give the expression in local coordinates of the curvat ure 3-form defined as Omega := d(2)A + d(A(2)) + A dA + A(3). Finally, the introduction of notions of a scalar product and integration of th e Z(3)-graded exterior forms enables us to define the variational prin ciple and to derive the differential equations satisfied by the 3-form Omega. The Lagrangian obtained in this way contains the invariants of the ordinary gauge field tensor F-ik and its covariant derivatives Di Fkm.