Dynamical symmetries, coherent states and nonlinear realizations: Towards noncommutative gravity

Based on the gravity model as a gauge theory previously by the authors, the introduction of noncommutative structures in the very geometry is discussed and developed in a new different way. Taking into account a linear Lagrangian in curvature and considering a pair of coherent vectors responsible, among other things, for the symmetry breaking, the introduction of noncommutativity in the very structure of the geometry is achieved. The fact that the vectors are coherent states ensure not only the natural and total quantization of the model, but also the formulation of the noncommutative structure, in particular by the introduction of a star product from the convolutory properties thereof. This new star product, in a sharp contrast with other proposals, is independent of the Bargmann index or other parameters depending on the dimension of the representation of the structure group of the noncommutativity. We also explain in the same way that a matrix model, through the properties of these coherent vectors, can be easily formulated. The pros and cons of the implementation of star products versus a theory of matrices based on coset coherent states are briefly discussed.