Final PhD Oral Examination (Thesis Title: “Emergent Spacetime in Matrix Models”)

Abstract:
We study the noncommutative geometry associated to matrices of N quantum particles in the 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 the original matrices determine not only shape of the emergent surface, but also a unique Poisson structure. We prove that commutators of matrix operators correspond to Poisson brackets. Through our construction, we can realize arbitrary noncommutative membranes embedded in R³.
We further conjecture an embedding operator which assigns, to any (2n + 1) Hermitian matrices, a 2n-dimensional hypersurface in flat (2n + 1)-dimensional Euclidean space. This corresponds to precisely 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 noncommutative geometry. Using this correspondence, all physical properties of the emergent D(2n)-brane can be computed.
Many works have been done in exploring the geometry emerged from the matrix configuration, but they do not always produce consistent results. We apply two types of point probe methods and the supergravity charge density formula on the generalized fuzzy sphere S²_so(4). Its tangled structure challenges the applicability of the probing methods. We propose to disentangle blocks of S²_so(4) regarding the geometrical symmetry and retrieve S²_so(4) as a thick two sphere with coherent layers consistently in three methods.
The Yang-Mills matrix model with mass term representing IR cutoff on the effective radius generates remarkable spherical solutions of the emergent universe, but it is unsolvable, unlike matrix models dominated by the Gaussian potential. By coarse-graining the dimension of matrices, quantum gravity is reproduced by the Gaussian model at the fixed point of dimensional renormalization group flow. We approach the unsolvable YM model by the same dimensional renormalization and discover a non-trivial fixed point after imposing the spherical topology. The fixed point might lead to a new duality between quantum gravity and the massive YM model in the continuum limit, and its existence also sets a density condition on the generalized fuzzy sphere.