Examples of non integer dimensional geometries

TL;DR

This paper explores spectral triples with non-integer dimension spectra, including examples with complex and real non-integer dimensions, revealing their geometric and algebraic properties.

Contribution

It provides explicit examples of non-integer dimensional spectral triples involving commutative C*-algebras, highlighting their differential algebra and spectral properties.

Findings

One example has a complex dimension spectrum and trivial differential algebra.

Another example features a real non-integer dimension spectrum with non-trivial differential algebra.

Distance depends non-trivially on the deformation parameter.

Abstract

Two examples of spectral triples with non-integer dimension spectrum are considered. These triples involve commutative C*-algebras. The first example has complex dimension spectrum and trivial differential algebra. The other is a parameter dependent deformation of the canonical spectral triple over S1. Its dimension spectrum includes real non-integer values. It has a non-trivial differential algebra and in contrast with the one dimensional case there are no junk forms for a non-vanishing deformation parameter. The distance on this space depends non-trivially on this parameter.