Membrane topology and matrix regularization

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.