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The Palatini- or first order formulation of the Einstein-Hilbert action for gravity is
where
$(R^{a b})$ is the curvature of an $\mathfrak{so}(n,1)$-connection
$(e^a)$ is the vielbein.
This is reminiscent of the form of the action functional in BF theory
Various proposals for extensions of this action functional have been made that feature $B$ as an independent field as indicated but then include some dynamical constraint which ensures that on-shell one has $B^{a b} = \epsilon_{a b c d} e^c \wedge e^d$.
This is also related to the Plebanski formulation of gravity.
The blog entry
recalls the construction of
and provides some noteworthy comments.
Approaches using the spin group instead of the rotation group include
and
For that spinorial approach see also
See also
Related is also the construction in
A blog discussion about this and possible interpretations in higher category theory is at