Palatini-Lovelock-Cartan gravity-Bianchi identities for stringy fluxes

A Palatini-type action for Einstein and Gauss-Bonnet gravity with non-trivial torsion is proposed. A three-form flux is incorporated via a deformation of the Riemann tensor, and consistency of the Palatini variational principle requires the flux to be covariantly constant and to satisfy a Jacobi identity. Studying gravity actions of third order in the curvature leads to a conjecture about general Palatini-Lovelock-Cartan gravity. We point out potential relations to string-theoretic Bianchi identities and, using the Schouten-Nijenhuis bracket, derive a set of Bianchi identities for the non-geometric Q- and R-fluxes which include derivative and curvature terms. Finally, the problem of relating torsional gravity to higher order corrections of the bosonic string-effective action is revisited.