Matrix embeddings on flat $R^3$ and the geometry of membranes

TL;DR

This paper introduces a method to derive oriented Riemann surfaces in three-dimensional space from three hermitian matrices, linking matrix models to membrane geometry and topological features relevant to holography and black holes.

Contribution

It provides a covariant procedure to construct embedded surfaces from matrices, revealing geometric and topological structures and their relation to physical effects like the Hanany-Witten effect.

Findings

Defined a set of oriented Riemann surfaces from matrices

Established covariance under geometric transformations

Connected surfaces to line bundles with connections

Abstract

We show that given three hermitian matrices, what one could call a fuzzy representation of a membrane, there is a well defined procedure to define a set of oriented Riemann surfaces embedded in $R^3$ using an index function defined for points in $R^3$ that is constructed from the three matrices and the point. The set of surfaces is covariant under rotations, dilatations and translation operations on $R^3$, it is additive on direct sums and the orientation of the surfaces is reversed by complex conjugation of the matrices. The index we build is closely related to the Hanany-Witten effect. We also show that the surfaces carry information of a line bundle with connection on them. We discuss applications of these ideas to the study of holographic matrix models and black hole dynamics.