The Lanczos potential and Chern–Simons theory

Abstract

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