Z(3)-GRADED DIFFERENTIAL-CALCULUS AND GAU GE-THEORIES OF HIGHER-ORDER

We present a generalization of the exterior differential calculus, bas ed on the operator d such that a(3) = 0, but d(2) not equal 0. The ent ities dx(i) and d(2) x(k) generate an associative algebra; we shall as sume that the products dx(i) dx(k) are independent of dx(k) dx(i), whi le the ternary products will satisfy the relation: dx(i) dx(k) dx(m) = jdx(k) dx(m) dx(i) = j(2) dx(m) dx(i) dx(k), complemented by the rela tion dx(i) d(2) x(k) = jd(2) x(k) dx(i), with j := e(2 pi i/3). We sho w how the covariant derivation can be generalized with a 1-form A so t hat D Phi := d Phi + A Phi, and we give the expression in local coordi nates of the curvature 3-form Omega := d(2) A+d(A(2))+AdA+A(3).