(Re)constructing Dimensions

TL;DR

This paper explores how to infer the size, shape, and properties of extra-dimensional manifolds in string theory from the observed mass spectrum of Kaluza-Klein modes, including examples of isospectral manifolds and the limits of reconstruction accuracy.

Contribution

It introduces methods to deduce geometric features of compact manifolds from Kaluza-Klein spectra and provides examples of isospectral manifolds relevant for string theory backgrounds.

Findings

Examples of isospectral manifolds with different geometries.

Analysis of the accuracy of reconstructing manifold properties from finite spectra.

Discussion on the limitations of spectral data in determining manifold topology and metric.

Abstract

Compactifying a higher-dimensional theory defined in R^{1,3+n} on an n-dimensional manifold {\cal M} results in a spectrum of four-dimensional (bosonic) fields with masses m^2_i = \lambda_i, where - \lambda_i are the eigenvalues of the Laplacian on the compact manifold. The question we address in this paper is the inverse: given the masses of the Kaluza-Klein fields in four dimensions, what can we say about the size and shape (i.e. the topology and the metric) of the compact manifold? We present some examples of isospectral manifolds (i.e., different manifolds which give rise to the same Kaluza-Klein mass spectrum). Some of these examples are Ricci-flat, complex and K\"{a}hler and so they are isospectral backgrounds for string theory. Utilizing results from finite spectral geometry, we also discuss the accuracy of reconstructing the properties of the compact manifold (e.g., its dimension, volume, and curvature etc) from measuring the masses of only a finite number of Kaluza-Klein modes.