7 research outputs found
Coverings, correspondences, and noncommutative geometry
We construct an additive category where objects are embedded graphs in the
3-sphere and morphisms are geometric correspondences given by 3-manifolds
realized in different ways as branched covers of the 3-sphere, up to branched
cover cobordisms. We consider dynamical systems obtained from associated
convolution algebras endowed with time evolutions defined in terms of the
underlying geometries. We describe the relevance of our construction to the
problem of spectral correspondences in noncommutative geometry.Comment: 31 pages, LaTe
Spin Foams and Noncommutative Geometry
We extend the formalism of embedded spin networks and spin foams to include
topological data that encode the underlying three-manifold or four-manifold as
a branched cover. These data are expressed as monodromies, in a way similar to
the encoding of the gravitational field via holonomies. We then describe
convolution algebras of spin networks and spin foams, based on the different
ways in which the same topology can be realized as a branched covering via
covering moves, and on possible composition operations on spin foams. We
illustrate the case of the groupoid algebra of the equivalence relation
determined by covering moves and a 2-semigroupoid algebra arising from a
2-category of spin foams with composition operations corresponding to a fibered
product of the branched coverings and the gluing of cobordisms. The spin foam
amplitudes then give rise to dynamical flows on these algebras, and the
existence of low temperature equilibrium states of Gibbs form is related to
questions on the existence of topological invariants of embedded graphs and
embedded two-complexes with given properties. We end by sketching a possible
approach to combining the spin network and spin foam formalism with matter
within the framework of spectral triples in noncommutative geometry.Comment: 48 pages LaTeX, 30 PDF figure