The Lanczos potential and Chern-Simons theory

In electromagnetism, the Faraday tensor F-ab can be constructed from the vector potential A(a), it is possible to add term to the Lagrangian depending on A(a) but not its derivatives called Chern-Simons terms. In gravitation, the Weyl tensor C-abcd can be constructed from the Lanczos potential H-abc, I pursue the analogy to see if terms of Chern-Simons form can be added to the Lagrangian. A new tensor D is introduced which is constructed from the Lanczos potential and is of the same form as that of the Weyl tensor C expressed in terms of the Lanczos potential except that covariant differentiation is replaced by transvection with a vector v. The new tensor has associated invariants C . D and D-2, the first of these can be interpreted as a Chern-Simons term for Weyl C-2 gravity. Both invariants allow various tensors to be constructed and some of their properties are investigated by using exact examples.