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In a recent work, a dual formulation of group field theories asnon-commutative quantum field theories has been proposed, providing an exactduality between spin foam models and non-commutative simplicial path integralsfor constrained BF theories. In light of this new framework, we define a modelfor 4d gravity which includes the Immirzi parameter gamma. It reproduces theBarrett-Crane amplitudes when gamma goes to infinity, but differs from existingmodels otherwise; in particular it does not require any rationality conditionfor gamma. We formulate the amplitudes both as BF simplicial path integralswith explicit non-commutative B variables, and in spin foam form in terms ofWigner 15j-symbols. Finally, we briefly discuss the correlation betweenneighboring simplices, often argued to be a problematic feature, for example,in the Barrett-Crane model.

Group field theory and simplicial gravity path integrals: A model for Holst-Plebanski gravity