A new class of fuzzy spaces with classical limit

TL;DR

This paper introduces a novel matrix regularization method for fuzzy spaces based on matrix-valued functions on a cylinder, demonstrating classical limits as manifolds with applications to fuzzy space interpolation and string vertices.

Contribution

It proposes a new class of fuzzy spaces using matrix-valued functions on a cylinder, establishing conditions for classical limits and constructing specific examples like fuzzy string vertices.

Findings

Classical limits of fuzzy spaces are manifolds composed of coordinate patches.

A method for interpolating between fuzzy spaces is developed.

A fuzzy string vertex connecting three circles is constructed.

Abstract

We present a new type of matrix regularization, which is based on matrix-valued functions defined on a cylinder. If non-commutative coordinates of a fuzzy space are defined by a regularization of such functions, we show that a classical limit for the fuzzy spaces exists, when the matrix-valued functions nearly commute. In this case, the classical limit of the fuzzy space is a manifold, which is composed of coordinate patches that are defined by the diagonal entries of the diagonalized matrix-valued functions. As applications, an interpolation for direct sums of fuzzy spaces is described and a fuzzy string vertex with classical limit, i.e. a surface interconnecting three circles, is constructed.