Emergent geometry of membranes

TL;DR

This paper explores how large hermitian matrices can encode emergent geometries of membranes, establishing a link between matrix commutators and classical Poisson brackets, and demonstrating realizations of noncommutative membranes.

Contribution

It introduces a method to derive emergent membrane geometries and Poisson structures from large matrices, extending the understanding of noncommutative surfaces.

Findings

Matrix commutators correspond to Poisson brackets.

Constructed noncommutative spheres and tori in R^3.

Provided matrix equations for minimal area surfaces.

Abstract

In work arXiv:1204.2788, 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.