Spectral Quadruples

A set of data supposed to give possible axioms for spacetimes. It is hoped that such a proposal can serve to become a testing ground on the way to a general formulation. At the moment, the axioms are known to be sufficient for cases with a sufficient number of symmetries, in particular for 1+1 de Sitter spacetime.Comment: AMS-LaTeX, 7 pages, presented at the Euroconference "BRANE NEW WORLD and Noncommutative Geometry", Torino, Villa Gualino (Italy), October 2 - 7, 200

Quantum Gravity coupled to Matter via Noncommutative Geometry

We show that the principal part of the Dirac Hamiltonian in 3+1 dimensions emerges in a semi-classical approximation from a construction which encodes the kinematics of quantum gravity. The construction is a spectral triple over a configuration space of connections. It involves an algebra of holonomy loops represented as bounded operators on a separable Hilbert space and a Dirac type operator. Semi-classical states, which involve an averaging over points at which the product between loops is defined, are constructed and it is shown that the Dirac Hamiltonian emerges as the expectation value of the Dirac type operator on these states in a semi-classical approximation.Comment: 15 pages, 1 figur

Moduli spaces of discrete gravity I: A few points

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.Comment: 35 pages, Latex, uses xy-package

Multiple noncommutative tori and Hopf algebras

We derive the Kac-Paljutkin finite-dimensional Hopf algebras as finite fibrations of the quantum double torus and generalize the construction for quantum multiple tori.Comment: 18 pages; AMSLaTeX (major revision, the construction of dual rewritten using approach of multiplier Hopf algebras, references added

The Spectral Geometry of the Equatorial Podles Sphere

We propose a slight modification of the properties of a spectral geometry a la Connes, which allows for some of the algebraic relations to be satisfied only modulo compact operators. On the equatorial Podles sphere we construct suq2-equivariant Dirac operator and real structure which satisfy these modified properties.Comment: 6 pages. Latex. V2: Minor changes; to appear in Comptes Rendus Mathematiqu