Membrane topology and matrix regularization

TL;DR

This paper investigates how membrane topology in M-theory's matrix model can be understood through eigenvalue distributions and introduces a new matrix-function correspondence rule that captures topological information.

Contribution

It reveals the role of eigenvalue sequences in encoding membrane topology and proposes a semi-local matrix-function correspondence rule applicable to all membrane topologies.

Findings

Eigenvalue distributions encode topology information.

Eigenvalue sequences exhibit Morse-theoretic branching.

New correspondence rule relates matrix elements to Fourier components.

Abstract

The problem of membrane topology in the matrix model of M-theory is considered. The matrix regularization procedure, which makes a correspondence between finite-sized matrices and functions defined on a two-dimensional base space, is reexamined. It is found that the information of topology of the base space manifests itself in the eigenvalue distribution of a single matrix. The precise manner of the manifestation is described. The set of all eigenvalues can be decomposed into subsets whose members increase smoothly, provided that the fundamental approximations in matrix regularization hold well. Those subsets are termed as eigenvalue sequences. The eigenvalue sequences exhibit a branching phenomenon which reflects Morse-theoretic information of topology. Furthermore, exploiting the notion of eigenvalue sequences, a new correspondence rule between matrices and functions is constructed. The new rule identifies the matrix elements directly with Fourier components of the corresponding function, evaluated along certain orbits. The rule has semi-locality in the base space, so that it can be used for all membrane topologies in a unified way. A few numerical examples are studied, and consistency with previously known correspondence rules is discussed.