Symmetries, Matrices, and de Sitter Gravity

Using simple algebraic methods along with an analogy to the BFSS model, we explore the possible (target) spacetime symmetries that may appear in a matrix description of de Sitter gravity. Such symmetry groups could arise in two ways, one from an ``IMF'' like construction and the other from a ``DLCQ'' like construction. In contrast to the flat space case, we show that the two constructions will lead to different groups, i.e. the Newton-Hooke group and the inhomogeneous Euclidean group (or its algebraic cousins). It is argued that matrix quantum mechanics based on the former symmetries look more plausible. Then, after giving a detailed description of the relevant one particle dynamics, a concrete Newton-Hooke matrix model is proposed. The model naturally incorporates issues such as holography, UV-IR relations, and fuzziness, for gravity in $dS_{4}$. We also provide evidence to support a possible phase transition. The lower temperature phase, which corresponds to gravity in the perturbative regime, has a Hilbert space of infinite dimension. In the higher temperature phase where the perturbation theory breaks down, the dimension of the Hilbert space may become finite.