Departmental Oral Examination (Thesis Title: “Emergent Spacetime in Matrix Models")

Abstract:
We study the noncommutative geometry associated to matrices of N quantum dots in the matrix models. The earlier work established a surface embedded in flat R^3 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 R^3.

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 S^2(so(4)) . Its tangled structure challenges the applicability of the probing methods. We propose to disentangle blocks of S^2(so(4)) regarding the geometrical symmetry and retrieve S^2(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. After imposing the spherical topology, we approach the model approximately by coarse-graining the dimension of matrices and find the free energy satisfies the Callan-Symanzik like renormalization group equation and reproduces 2D quantum gravity near the fixed point of the running mass. Among classical solutions of generalized fuzzy spheres, the critical sphere has its mass at the fixed point within a range of matrix dimensions for a given effective radius cutoff.