Constructing spectral triples over holonomy-diffeomorphisms and the problem of reconciling general relativity with quantum field theory

TL;DR

This paper constructs a spectral triple over gauge connection spaces, linking non-commutative geometry with quantum field theory and general relativity, and demonstrates how a dynamical metric naturally emerges within this framework.

Contribution

It introduces a spectral triple on gauge connection quotient spaces, incorporating a dynamical metric and connecting to quantum Yang-Mills theory, advancing the unification of gravity and quantum fields.

Findings

A Dirac-type operator on gauge connection space is rigorously constructed.

The metric on the configuration space is shown to be dynamical and regularized physically.

A Bott-Dirac operator links the framework to quantum Yang-Mills theory.

Abstract

In this paper we construct a candidate for a spectral triple on a quotient space of gauge connections modulo gauge transformations and show that it is related to a Kasparov type bi-module over two canonical algebras: the HD-algebra, which is a non-commutative C*-algebra generated by parallel transports along flows of vector fields, and an exterior algebra on a space of gauge transformations. The latter algebra is related to the ghost sector in a BRST quantisation scheme. Previously we have shown that key elements of bosonic and fermionic quantum field theory on a curved background emerge from a spectral triple of this type. In this paper we show that a dynamical metric on the underlying manifold also emerges from the construction. We first rigorously construct a Dirac type operator on the a quotient space of gauge connections modulo gauge transformations, and discuss the commutator between this Dirac type operator and the HD-algebra. To do this we first construct a gauge-covariant metric on the configuration space and use it to construct the triple. The key step in this construction is to require the volume of the quotient space to be finite, which amounts to an ultra-violet regularisation. Since the metric on the configuration space is dynamical with respect to the time-evolution generated by the Dirac type operator in the triple, it is possible to interpret the regularisation as a physical feature (as opposed to static regularisations, which are always computational artefacts). Finally, we construct a Bott-Dirac operator that connects our construction with quantum Yang-Mills theory.