nLab gravity as a BF theory

Idea

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

$$\epsilon_{a b c d} \int_X R^{a b} \wedge e^c \wedge e^d$$

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

$$\int_X F^{a b} \wedge B_{a b} \,.$$

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 $B^{a b} = \epsilon_{a b c d} e^c \wedge e^d$.

This is also related to the Plebanski formulation of gravity.

References

The blog entry

• Jacques Distler, BF (blog)

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)

For that spinorial approach see also

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

See also

• R. Capovilla, M. Montesinos, V. A. Prieto, E. Rojas, BF gravity and the Immirzi parameter (arXiv)

Related is also the construction in

• Michael P. Reisenberger, Classical Euclidean general relativity from “left-handed area = right-handed area” (arXiv)

A blog discussion about this and possible interpretations in higher category theory is at

• Urs Schreiber, 2-Palatini (blog)