A unitary invariant in Riemannian geometry

TL;DR

This paper introduces a new invariant in Riemannian geometry that, combined with the Dirac operator spectrum, fully characterizes the geometry, analogous to the role of the CKM matrix in particle physics.

Contribution

It proposes a novel invariant measuring the relative position of von Neumann algebras, enhancing the spectral approach to Riemannian geometry.

Findings

The invariant captures geometric information beyond the spectrum.

It completes the spectral characterization of Riemannian manifolds.

Analogous to the CKM matrix, it encodes additional geometric data.

Abstract

We introduce an invariant of Riemannian geometry which measures the relative position of two von Neumann algebras in Hilbert space, and which, when combined with the spectrum of the Dirac operator, gives a complete invariant of Riemannian geometry. We show that the new invariant plays the same role with respect to the spectral invariant as the Cabibbo--Kobayashi--Maskawa mixing matrix in the Standard Model plays with respect to the list of masses of the quarks.