On globally non-trivial almost-commutative manifolds

Abstract

Within the framework of Connes' noncommutative geometry, we define and study globally non-trivial (or topologically non-trivial) almost-commutative manifolds. In particular, we focus on those almostcommutative manifolds that lead to a description of a (classical) gauge theory on the underlying base manifold. Such an almost-commutative manifold is described in terms of a 'principal module', which we build from a principal fibre bundle and a finite spectral triple. We also define the purely algebraic notion of 'gauge modules', and show that this yields a proper subclass of the principal modules. We describe how a principal module leads to the description of a gauge theory, and we provide two basic yet illustrative examples.

Krein Spectral Triples and the Fermionic Action

“…As in [BD14], we construct almost-commutative manifolds as F × M instead of M × F (though the latter is more common in the literature). The reason is that the order F × M is more natural for the generalisation to the globally non-trivial case and its description as a Kasparov product (see [BD14, §III.C]).…”

Spin geometry of the rational noncommutative torus

“…that has the usual product form (7) though with a vanishing second term and acting not on the full tensor product of Hilbert spaces but only on its subspace.…”

“…So far we established isomorphisms between spectral triples by selecting Dirac operators, both for the first and second case, which act trivially as the identity on the finite part of the respective Hilbert spaces. In order to push further the analogy with almost commutative manifolds, hence with standard model of particles ([1], [7]), it would be interesting to get a Dirac operator whose action on the "internal" degrees of freedom is non trivial. In quantum field theory, internal degrees of freedom of a single, isolated fermion change whenever it moves in a space-time region with a gauge field (e.g.…”

“…The gauge theories mentioned above are, by construction, topologically trivial, in the sense that the corresponding principal bundles are globally trivial bundles. A generalization to nontrivial bundles has been obtained in ( [7]) for the special case of Yang-Mills theory.…”

“…It will be interesting to investigate in future works possible relations of our spectral geometry approach with the approach of [7] which make use of connections on principal bundles, and also with [13], where spectral triples on the noncommutative torus have been employed to study topological insulators. Another direction to be explored regards the noncommutative coverings and in particular spectral triples on self-coverings of the rational noncommutative torus in [14], and the quantum coverings studied in [15] via Hopf algebroids as the noncommutative analogue of groupoids.…”