Within the framework of spinfoam models, we revisit the simplicity constraints reducing topological BF theory to 4d Riemannian gravity. We use the reformulation of SU(2) intertwiners and spin networks in terms of spinors, which has come out from both the recently developed U(N ) framework for SU(2) intertwiners and the twisted geometry approach to spin networks and spinfoam boundary states. Using these tools, we are able to perform a holomorphic/anti-holomorphic splitting of the simplicity constraints and define a new set of holomorphic simplicity constraints, which are equivalent to the standard ones at the classical level and which can be imposed strongly on intertwiners at the quantum level. We then show how to solve these new holomorphic simplicity constraints using coherent intertwiner states. We further define the corresponding coherent spin network functionals and introduce a new spinfoam model for 4d Riemannian gravity based on these holomorphic simplicity constraints and whose amplitudes are defined from the evaluation of the new coherent spin networks. * Electronic address: maite.dupuis@ens-lyon.fr † Electronic address: etera.livine@ens-lyon.fr 1 They also appear as the reality conditions in the self-dual Ashtekar formulation of general relativity as an SU(2) gauge theory. A. The Classical Spinor Framework for SU(2)Following the previous ideas on the U(N ) formalism for intertwiners [12,14,15] and on twisted geometries for loop gravity and spin foams [18,19], it was realized that loop quantum gravity's spin network states are the quantization of some classical spinor networks [17]. We review this formalism below.Spin networks, and thus spinor networks, are constructed on a given graph. Let us thus choose a closed oriented graph Γ with E edges and V vertices. We will label its vertices as v and its edges as e, calling s(e) and t(e) respectively 2 Actually, the requirement is slightly stronger and we require the vanishing of the matrix elements of the constraints between solution states. Calling Hs the Hilbert space of solution states and C the simplicity constraints, we ask [7,9]:∀ψ, φ ∈ Hs, ψ|C|φ = 0. This is stronger than simply requiring the vanishing of the expectation values ψ|C|ψ = 0, but in practice it amounts simply to vanishing of the expectation values with small (almost-minimal) uncertainty.Calling S[J] ≡ 2J(ln A(z) + 1) − (2J + 2) ln J the exponent, we can investigate its behavior and check whether it has any extremum:Thus S has a unique extremum, which is a maximum, for the value J 0 , which is approximately J 0 ∼ A(z) when A(z) ≫ 1. Thus for a large classical area A(z), we do recover that the probability distribution for J is peaked on this classical value J 0 ∼ |λ| 2 . Moreover, we have approximatively a Gaussian around this value:

In the context of loop quantum gravity and spinfoam models, the simplicity constraints are essential in that they allow to write general relativity as a constrained topological BF theory. In this work, we apply the recently developed U(N ) framework for SU(2) intertwiners to the issue of imposing the simplicity constraints to spin network states. More particularly, we focus on solving them on individual intertwiners in the 4d Euclidean theory. We review the standard way of solving the simplicity constraints using coherent intertwiners and we explain how these fit within the U(N ) framework. Then we show how these constraints can be written as a closed u(N ) algebra and we propose a set of U(N ) coherent states that solves all the simplicity constraints weakly for an arbitrary Immirzi parameter.