Geometric approach to BRST-symmetry and ZN-generalization of superconnection

We propose a geometric approach to the BRST-symmetries of the Lagrangian of a topological quantum field theory on a four dimensional manifold based on the formalism of superconnections. Making use of a graded q-differential algebra, where q is a primitive N -th root of unity, we also propose a notion of ZN -connection which is a generalization of a superconnection. In our approach the Lagrangian of a topological field theory is presented as the value of the curvature of a superconnection evaluated at an appropriate section of a vector bundle. Since this value of the curvature satisfies the Bianchi identity and representing the Bianchi identity in this case in the form of an operator applied to the mentioned above value of the curvature we obtain an operator which gives zero when applied to the Lagrangian. We show that this operator generates the BRST-transformations of the fields of a topological field theory on a four dimensional manifold.