Abstract
In work [1], a surface embedded in flat ℝ 3 is associated to any three hermitian matrices. We study this emergent surface when the matrices are large, by constructing coherent states corresponding to points in the emergent geometry. We 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: for example, we examine a round sphere with a non-spherically symmetric Poisson structure. We also give a natural construction for a noncommutative torus embedded in ℝ 3. Finally, we make remarks about area and find matrix equations for minimal area surfaces.
Highlights
Such as planes, tori and spheres.1 In their paper, [1], take this one step further: using the BFSS model they found a geometric interpretation of three matrix coordinates as a co-dimension one surface embedded in three dimension
We study this emergent surface when the matrices are large, by constructing coherent states corresponding to points in the emergent geometry
We find the original matrices determine shape of the emergent surface, and a unique Poisson structure
Summary
As for the first expression in equation (2.10), it will turn out that if we take the normal vector to point along the x3 direction, we have α|(X1 − x1)2|α ≈ α|(X2 − x2)2|α α|(X3 − x3)2|α , so the coherent state is ‘flattened’ to lie predominantly in the 1-2-plane and balanced (‘round’). To flesh out these ideas, we will examine a series of increasingly complex examples. We will construct the approximate eigenvector |αp and study corrections to the large N limit described above
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