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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
Textbooks include
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
The result that gravity is not renormalizable is due to
Review includes