Generalization of superconnection in noncommutative geometry

We propose the notion of a ZN -connection, where N � 2, which can be viewed as a generalization of the notion of a Z2-connection or superconnection. We use the algebraic approach to the theory of connections to give the definition o f a ZN -connection and to explore its structure. It is well known that one of the basic structur es of the algebraic approach to the theory of connections is a graded differential algebra with differential d satisfying d 2 = 0. In order to construct a ZN -generalization of a superconnection for any N > 2, we make use of a ZN -graded q-differential algebra, where q is a primitive N th root of unity, with N -differential d satisfying d N = 0. The concept of a graded q-differential algebra arises naturally within the framework of noncommutative geometry and the use of this algebra in our construction involves the appearance of q-deformed structures such as graded q-commutator, graded q-Leibniz rule, and q-binomial coefficients. Particularly, if N = 2,q = 1, then the notion of a ZN -connection coincides with the notion of a superconnection. We define the curvature of a ZN -connection and prove that it satisfies the Bianchi identity .