Types of quantum field thories
The action functional of gravity was originally conceived as a functional on the space of pseudo-Riemannian metrics of a manifold . Later on it was realized that it may alternatively naturally be thought of as a functional on the space of connections with values in the Poincare Lie algebra – essentially the Levi-Civita connection. This is known as the first order formalism or the Palatini formalism. The field strength of gravity – the Riemann tensor – is the curvature of Levi-Civita connection. Typically this is referred to a spin connection in this context.
The first order formalism is also the formalism in which supergravity has to be formulated.
A decent introduction is in sections 4 and 5 of
A detailed account is in section I.4.1 of