gravity as a BF theory



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

ϵ abcd XR abe ce d \epsilon_{a b c d} \int_X R^{a b} \wedge e^c \wedge e^d


  • (R ab)(R^{a b}) is the curvature of an 𝔰𝔬(n,1)\mathfrak{so}(n,1)-connection

  • (e a)(e^a) is the vielbein.

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

XF abB ab. \int_X F^{a b} \wedge B_{a b} \,.

Various proposals for extensions of this action functional have been made that feature BB as an independent field as indicated but then include some dynamical constraint which ensures that on-shell one has B ab=ϵ abcde ce dB^{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

  • 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)


  • 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

Revised on September 22, 2010 16:29:00 by Urs Schreiber (