Encoding simplicial quantum geometry in group field theories

An extended group field theory formalism for quantum gravity, based on a field that is a function of both group variables, interpreted as discretized connection, and Lie algebra variables, interpreted as discretized triads, has been proposed recently as an attempt to define models with a clearer link with simplicial geometry. In the context of such a formalism, we introduce a new symmetry requirement on the field. This leads, in 3D, to Feynman amplitudes interpreted as simplicial path integrals based on the Regge action, to a proper relation between the discrete connection and the triad vectors appearing in the Regge action, and to a much more satisfactory and transparent encoding of simplicial geometry already at the level of the group field theory action.