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

In a recent work, a dual formulation of group field theories as noncommutative quantum field theories has been proposed, providing an exact duality between spin foam models and noncommutative simplicial path integrals for constrained BF theories. In light of this new framework, we define a model for four-dimensional gravity which includes the Immirzi parameter γ. It reproduces the Barrett-Crane amplitudes when γ=∞, but differs from existing models otherwise; in particular, it does not require any rationality condition for γ. We formulate the amplitudes both as BF simplicial path integrals with explicit noncommutative B variables, and in spin foam form in terms of Wigner $15j$ symbols. Finally, we briefly discuss the correlation between neighboring simplices, often argued to be a problematic feature, for example, in the Barrett-Crane model.