Spectral Torsion

TL;DR

This paper introduces a spectral torsion functional for spectral triples, demonstrating its ability to recover classical torsion and analyzing its behavior across various noncommutative geometries.

Contribution

It defines a new spectral torsion functional for spectral triples and explores its properties in both classical and noncommutative settings.

Findings

Recovers classical torsion for spin manifolds

Shows nonvanishing torsion in coupled noncommutative geometries

Analyzes spectral triples including quantum groups

Abstract

We introduce a trilinear functional of differential one-forms for a finitely summable regular spectral triple with a noncommutative residue. We demonstrate that for a canonical spectral triple over a closed spin manifold it recovers the torsion of the linear connection. We examine several spectral triples, including Hodge-de\,Rham, Einstein-Yang-Mills, almost-commutative two-sheeted space, conformally rescaled noncommutative tori, and quantum $SU(2)$ group, showing that the third one has a nonvanishing torsion if nontrivially coupled.