Moduli spaces of discrete gravity I: A few points

TL;DR

This paper explores the quantization of spectral triples in finite-dimensional commutative algebras, constructing moduli spaces of Dirac operators, and investigates effects like fermion coupling and spontaneous symmetry breaking.

Contribution

It introduces a method to construct moduli spaces of Dirac operators in finite spectral triples and analyzes their properties under quantization and symmetry transformations.

Findings

Construction of moduli space of Dirac operators

Coupling to fermions regularizes infinities

Observation of spontaneous spectral symmetry breaking

Abstract

Spectral triples describe and generalize Riemannian spin geometries by converting the geometrical information into algebraic data, which consist of an algebra $A$, a Hilbert space $H$ carrying a representation of $A$ and the Dirac operator $D$ (a selfadjoint operator acting on $H$). The gravitational action is described by the trace of a suitable function of $D$. In this paper we examine the (path-integral-) quantization of such a system given by a finite dimensional commutative algebra. It is then (in concrete examples) possible to construct the moduli space of the theory, i.e. to divide the space of all Dirac operators by the action of the diffeomorphism group, and to construct an invariant measure on this space. We discuss expectation values of various observables and demonstrate some interesting effects such as the effect of coupling the system to Fermions (which renders finite quantities in cases, where the Bosons alone would give infinite quantities) or the striking effect of spontaneous breaking of spectral invariance.