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Group Field Theories, a generalization of matrix models for 2d gravity,represent a 2nd quantization of both loop quantum gravity and simplicialquantum gravity. In this paper, we construct a new class of Group Field Theorymodels, for any choice of spacetime dimension and signature, whose Feynmanamplitudes are given by path integrals for clearly identified discrete gravityactions, in 1st order variables. In the 3-dimensional case, the correspondingdiscrete action is that of 1st order Regge calculus for gravity (generalized toinclude higher order corrections), while in higher dimensions, they correspondto a discrete BF theory (again, generalized to higher order) with an imposedorientation restriction on hinge volumes, similar to that characterizingdiscrete gravity. The new models shed also light on the large distance orsemi-classical approximation of spin foam models. This new class of group fieldtheories may represent a concrete unifying framework for loop quantum gravityand simplicial quantum gravity approaches.

A New Class of Group Field Theories for 1st Order Discrete Quantum Gravity