$\kappa$-deformation, affine group and spectral triples

TL;DR

This paper constructs a spectral triple for a 2D $$-deformation using the affine group, overcoming previous obstructions to finite summability by adjusting the spectral dimension.

Contribution

It introduces a regular spectral triple for the 2D $$-deformation based on the affine group, with a novel approach to spectral dimension.

Findings

Spectral dimension is one, metric dimension is two.

Overcomes previous obstructions to finitely-summable spectral triples.

Uses a smooth subalgebra of the affine group $C^*$-algebra.

Abstract

A regular spectral triple is proposed for a two-dimensional $\kappa$-deformation. It is based on the naturally associated affine group $G$, a smooth subalgebra of $C^*(G)$, and an operator $\caD$ defined by two derivations on this subalgebra. While $\caD$ has metric dimension two, the spectral dimension of the triple is one. This bypasses an obstruction described in \cite{IochMassSchu11a} on existence of finitely-summable spectral triples for a compactified $\kappa$-deformation.