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 of a ZN -connection and to explore its structure. It is well known that one of the basic structures of the algebraic approach to the theory of connections is a graded differential algebra with differential d satisfying d2 = 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 Nth root of unity, with N-differential d satisfying dN = 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.
1. Mathai, V. and Quillen, D. Superconnections, Thom classes and equivariant differential forms. Topology, 1986, 25, 85–110.
https://doi.org/10.1016/0040-9383(86)90007-8
2. Berline, N., Getzler, E. and Vergne, M. Heat Kernels and Dirac Operators. Springer, 2004.
https://doi.org/10.1016/0393-0440(90)90023-V
3. Atiyah, M. F. and Jeffrey, L. Topological Lagrangians and cohomology. J. Geom. Phys., 1990, 7, 119–136.
https://doi.org/10.1007/BF01223371
4. Witten, E. Topological quantum field theory. Comm. Math. Phys, 1988, 117, 353–386.
5. Dubois-Violette, M. and Kerner, R. Universal q-differential calculus and q-analog of 14 homological algebra. Acta Math. Univ. Comenian., 1996, 65, 175–188.
6. Dubois-Violette, M. Lectures on differentials, generalized differentials and on some examples related to theoretical physics. Math. QA/0005256.
7. Dubois-Violette, M. Lectures on graded differential algebras and noncommutative geometry. In Noncommutative Differential Geometry and Its Applications to Physics: Proceedings of the Workshop (Maeda, Y. et al., eds). Math. Phys. Stud., 2001, 23, 245–306.
https://doi.org/10.1007/978-94-010-0704-7_15
8. Abramov, V. and Kerner, R. Exterior differentials of higher order and their covariant generalization. J. Math. Phys., 2000, 41, 5598–5614.
https://doi.org/10.1063/1.533428
9. Abramov, V. On agraded q-differential algebra. Math. QA/0509481.
10. Kapranov, M. M. On the q-analog of homological algebra. Math. QA/9611005.
11. Kerner, R. Gradedgaugetheory. Commun. Math. Phys., 1983, 91 ,213–234.
https://doi.org/10.1007/BF01211159
12. De Azcárraga, J. A. and Macfarlane, A. J. Group theoretical foundations of fractional supersymmetry. J. Math. Phys., 1996, 37, 1115–1127.
https://doi.org/10.1063/1.531451
