Designing the sound of a cut-off drum

TL;DR

This paper explores how the spectral action in noncommutative geometry can be used to design the shape of a drum by interpreting the eigenvalues of a Dirac operator as musical notes, linking spectral data to geometric form.

Contribution

It introduces a novel approach to designing geometric shapes from spectral data using noncommutative geometry, connecting spectral cut-offs to physical space metrics.

Findings

Spectral data can determine the shape of a drum.

The Dirac operator's eigenvalues relate to the geometry of space.

A new method links high-momentum cut-offs to short-distance geometry.

Abstract

The spectral action in noncommutative geometry naturally implements an ultraviolet cut-off, by counting the eigenvalues of a (generalized) Dirac operator lower than an energy of unification. Inverting the well known question "how to hear the shape of a drum ?", we ask what drum can be designed by hearing the truncated music of the spectral action ? This makes sense because the same Dirac operator also determines the metric, via Connes distance. The latter thus offers an original way to implement the high-momentum cut-off of the spectral action as a short distance cut-off on space. This is a non-technical presentation of the results of arXiv:1305.2605