On Torsion and Nieh–Yan Form

Abstract

Using the well-known Chern-Weil formula and its generalization, we systematically construct the Chern-Simons forms and their generalization induced by torsion as well as the Nieh-Yan (N-Y) forms. We also give an argument on the vanishing of integration of N-Y form on any compact manifold without boundary. A systematic construction of N-Y forms in D=4n dimension is also given.

Generalizations of the Nieh-Yan topological invariant

“…Bearing in mind that ξ is an arbitrary parameter, we can equate the terms with the same powers of ξ on both sides of the identity (13). By doing this, from the zero, first, second, and third powers of ξ we obtain, respectively, the following identities:…”

“…Subsequently, in Refs. [12,13] it was shown how to systematically construct Nieh-Yan-like invariants in 4k (k ∈ Z) dimensions. Another natural, and perhaps more interesting, direction to generalize the Nieh-Yan topological invariant is to modify its structure but remaining in four-dimensional manifolds.…”