Matrix embeddings on flat R³ and the geometry of membranes

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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 R³ using an index function defined for points in R³ that is constructed from the three matrices and the point. The set of surfaces is covariant under rotations, dilatations and translation operations on R³, 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.