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Emergent spacetime in matrix models Yeh, Ken Huai-Che


We study the non-commutative geometry associated with matrices of N quantum particles in matrix models. The earlier work established a surface embedded in flat ℝ³ from three Hermitian matrices. We construct coherent states corresponding to points in the emergent geometry and find that the original matrices determine not only the shape of the emergent surface, but also a unique Poisson structure. Through our construction, we can realize arbitrary non-commutative membranes embedded in ℝ³. We further conjecture an embedding operator that assigns, to any 2n+1 N-dimensional Hermitian matrices, a 2n-dimensional hypersurface in flat (2n+1)-dimensional Euclidean space. This corresponds to defining a fuzzy D(2n)-brane corresponding to N D0-branes. Points on the hypersurface correspond to zero eigenstates of the embedding operator, which have an interpretation as coherent states underlying the emergent non-commutative geometry. Using this correspondence, all physical properties of the emergent D(2n)-brane can be computed. Many studies have been carried out exploring the geometry emerging from the matrix configuration, but they have not always produced consistent results. We apply two types of point-probe methods, as well as the supergravity charge density formula to the generalized fuzzy sphere. Its tangled structure challenges the applicability of these probing methods. We propose to disentangle blocks of the generalized fuzzy sphere regarding the geometrical symmetry and retrieve the generalized fuzzy sphere as a thick two sphere with coherent layers consistently in three methods. The Yang-Mills (YM) matrix model with mass term representing a cutoff radius generates remarkable spherical solutions of the emergent universe, but it is unsolvable, unlike for matrix models dominated by a Gaussian potential. By coarse-graining the dimension of matrices, quantum gravity is reproduced by the Gaussian model at the fixed point of the dimensional-renormalization group flow. We approach the unsolvable YM model using the same dimensional-renormalization and discover a non-trivial fixed point after imposing spherical topology. This fixed point might lead to a new duality between quantum gravity and the massive YM model, and its existence also sets a density condition on the generalized fuzzy sphere.

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