# Idea

The Palatini- or first order formulation of the Einstein-Hilbert action for gravity is

${ϵ}_{abcd}{\int }_{X}{R}^{ab}\wedge {e}^{c}\wedge {e}^{d}$\epsilon_{a b c d} \int_X R^{a b} \wedge e^c \wedge e^d

where

• $\left({R}^{ab}\right)$ is the curvature of an $\mathrm{𝔰𝔬}\left(n,1\right)$-connection

• $\left({e}^{a}\right)$ is the vielbein.

This is reminiscent of the form of the action functional in BF theory

${\int }_{X}{F}^{ab}\wedge {B}_{ab}\phantom{\rule{thinmathspace}{0ex}}.$\int_X F^{a b} \wedge B_{a b} \,.

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 ${B}^{ab}={ϵ}_{abcd}{e}^{c}\wedge {e}^{d}$.

This is also related to the Plebanski formulation of gravity.

# References

The blog entry

recalls the construction of

• Laurent Freidel, Artem Starodubtsev, Quantum gravity in terms of topological observables (arXiv)

and provides some noteworthy comments.

Approaches using the spin group instead of the rotation group include

• Jerzy Lewandowski, Andrzej Okolow, 2-Form Gravity of the Lorentzian signature (arXiv)

and

• Han-Ying Guo, Yi Ling, Roh-Suan Tung, Yuan-Zhong Zhang, Chern-Simons Term for BF Theory and Gravity as a Generalized Topological Field Theory in Four Dimensions (arXiv)

• Roh Suan Tung, Ted Jacobson, Spinor One-forms as Gravitational Potentials (arXiv)