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

Abstract

We present a possible generalization of the exterior differential calculus, based on the operator d such that d³=0, but d²≠0. The entities dx ⁱ and d² x ^k generate an associative algebra; we shall suppose that the products dx ⁱ dx ^k are independent of dx ^k dx ⁱ, while theternary products will satisfy the relation: dx ⁱ dx ^k dx ^m=jdx ^k dx ^m dx ⁱ=j ²dx ^m dx ^m dx ⁱ dx ^k, complemented by the relation dx ⁱ d² x ^k=jd² x ^k dx ⁱ, withj:=e^2πi/3.

We shall attribute grade 1 to the differentials dx ⁱ and grade 2 to the ‘second differentials’ d² x ^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 thatDΦ:=dΦ+AΦ, and we give the expression in local coordinates of thecurvature 3-form defined as Ω:=d² A+d(A ²)+AdA+A ³.

Finally, the introduction of notions of a scalar product and integration of theZ ₃-graded exterior forms enables us to define the variational principle and to derive the differential equations satisfied by the 3-form Ω. The Lagrangian obtained in this way contains the invariants of the ordinary gauge field tensorF _ik and its covariant derivativesD _i F _km .