On the Faddeev gravity on the piecewise flat manifold

Abstract

We study the Faddeev formulation of gravity in which the metric is composed of vector fields. This system is reducible with the help of the equations of motion to the general relativity. The Faddeev action is evaluated for the piecewise flat ansatz for these fields when the metric corresponds to the flat interior of the 4-simplices of the general simplicial complex. Thereby an analogue of the Regge action in the usual general relativity is obtained. A peculiar feature of the Faddeev gravity is finiteness of the action on the discontinuous fields, and this means possibility of the complete independence of the fields in the different 4-simplices or incoincidence of the 4-simplices on their common faces. The earlier introduced analogue of the Barbero-Immirzi parameter for the Faddeev gravity is taken into account. There is some freedom in defining the Faddeev action on the piecewise flat manifold, and the task is set to make use of this freedom to ensure that this discrete system be reducible with the help of the {\it discrete} equations of motion to the analogous discrete general relativity (Regge calculus).