First-order representation of the Faddeev formulation of gravity

We study the Faddeev formulation of gravity in which the metric is composed of a ten-dimensional tetrad fAλ. Here, A = 1, …, 10 refers to an Euclidean (or Minkowsky) ten-dimensional spacetime and λ = 1, 2, 3, 4 is the usual world index. A remarkable unique feature of the Faddeev formulation is that the action remains finite for the discontinuous tetrad. If the spacetime is composed of flat microblocks (in the discrete version of gravity), this means that these microblocks in quantum theory do not coincide on their common faces, that is, they are independent. We propose a representation of the Faddeev formulation analogous to the Cartan–Weyl formalism. It is based on extending the set of variables by introducing an infinitesimal SO(10) connection. Excluding this connection via equations of motion we reproduce the original Faddeev action. A peculiar feature of this representation is the occurrence of a local SO(10) symmetry violating condition so that the SO(10) symmetry is only global one in full correspondence with the fact that the original Faddeev formulation possesses just the global SO(10) symmetry. We also consider an analogue of the Barbero–Immirzi parameter which can be naturally introduced in the considered representation. (paper)