Spectral triples of holonomy loops

TL;DR

This paper applies noncommutative geometry to the space of connections via spectral triples constructed from holonomy loops, revealing limitations in the Dirac operator's axiomatic compliance.

Contribution

It introduces a spectral triple framework over the space of connections using holonomy loops, advancing the application of noncommutative geometry in gauge theories.

Findings

Spectral triple constructed from holonomy loops over the space of connections.

Dirac operator does not fully satisfy spectral triple axioms.

Identifies challenges in applying noncommutative geometry to infinite-dimensional spaces.

Abstract

The machinery of noncommutative geometry is applied to a space of connections. A noncommutative function algebra of loops closely related to holonomy loops is investigated. The space of connections is identified as a projective limit of Lie-groups composed of copies of the gauge group. A spectral triple over the space of connections is obtained by factoring out the diffeomorphism group. The triple consist of equivalence classes of loops acting on a separable hilbert space of sections in an infinite dimensional Clifford bundle. We find that the Dirac operator acting on this hilbert space does not fully comply with the axioms of a spectral triple.