Gravitational duality, Palatini variation and boundary terms: a synopsis

We consider $f(R)$ gravity and Born-Infeld-Einstein (BIE) in formulations where the metric connection are treated independently integrate out to find corresponding models solely terms of connection, archetypical treatment being that Eddington-Schr\"odinger (ES) duality between cosmological Einstein Eddington theories. For dimensions $D\ne2$, we this requires have a specific form which makes model Weyl invariant, its reduction is then equivalent BIE with certain parameters. $D=2$ dimensions, ES not applicable, both invariant first order formulation bosonic string. also discuss boundary needed for variational principle be well defined on manifolds non-null boundaries. This modification Gibbons-Hawking-York (GHY) term gravity. means dualities consistent include terms.