Noncommutative spaces and matrix embeddings on flat R^{2n+1}

TL;DR

This paper proposes a conjecture for embedding 2n+1 hermitian matrices into a hypersurface in flat (2n+1)-dimensional space, enabling the study of emergent noncommutative geometries and D-branes.

Contribution

It introduces a new embedding operator conjecture that links hermitian matrices to hypersurfaces, advancing the understanding of fuzzy D-branes and noncommutative spaces.

Findings

Defined a hypersurface embedding for hermitian matrices

Computed physical properties of emergent D(2n)-branes

Constructed a rotationally symmetric flat noncommutative space in 4D

Abstract

We 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 emergent 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. We apply our conjecture to noncommutative flat and spherical spaces. As a by-product, we obtain a construction of a rotationally symmetric flat noncommutative space in 4 dimensions.