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Types of quantum field thories
A field configuration of the physical theory of gravity on a spacetime $X$ is equivalently
a vielbein field, hence a reduction of the structure group of the tangent bundle along $\mathbf{B} O(n-1,1) \to \mathbf{B}GL(n)$, defining a pseudo-Riemannian metric;
a connection that is locally a Lie algebra-valued 1-form with values in the Poincare Lie algebra.
such that this is a Cartan connection.
(This parameterization of the gravitational field is called the first-order formulation of gravity.) The component $E$ of the connection is the vielbein that encodes a pseudo-Riemannian metric $g = E \cdot E$ on $X$ and makes $X$ a pseudo-Riemannian manifold. Its quanta are the gravitons.
The non-propagating field? $\Omega$ is the spin connection.
The action functional on the space of such connection which defines the classical field theory of gravity is the Einstein-Hilbert action.
More generally, supergravity is a gauge theory over a supermanifold $X$ for the super Poincare group. The field of supergravity is a Lie-algebra valued form with values in the super Poincare Lie algebra.
The additional fermionic field $\Psi$ is the gravitino field.
So the configuration space of gravity on some $X$ is essentially the moduli space of Riemannian metrics on $X$.
for the moment see D'Auria-Fre formulation of supergravity for further details
gravitational entropy
The theory of gravity based on the standard Einstein-Hilbert action may be regarded as just an effective quantum field theory, which makes some of its notorious problems be non-problems:
The (reduced) covariant phase space of gravity (presented for instance by its BV-BRST complex, see there fore more details) is discussed for instance in
which is surveyed in
Careful discussion of observables in gravity is in