Lovelock-Cartan theory of gravity

The most general theory of gravity with torsion in D dimensions is discussed. The Lagrangian is required to be: (i) a D-form scalar under local Lorentz transformations; (ii) a local polynomial of the vielbein, the spin connection, and their exterior derivatives; (iii) constructed without the Hodge dual (*-operation). Besides the purely metric Lovelock action, there is a series of torsion terms related to the Pontryagin classes in a way analogous to the relation between the Lovelock action and the Euler classes. Also, a family of global invariants of the differentiable structure of the manifold, constructed with the torsion, is identified. Relaxing the condition of invariance of the Lagrangian under local Lorentz rotations, but still requiring the action to be invariant, another family of torsional actions that generalizes the Chern-Simons theories is found. Systematic algorithms for the explicit construction of these actions are provided.