Smoother than a circle, or How non commutative geometry provides the torus with an egocentred metric

Details Smoother than a circle, or How non commutative geometry provides the torus with an egocentred metric Smoother than a circle, or How non commutative geometry provides the torus with an egocentred metric

2006-03-07

arxiv.org/abs/hep-th/0603051v2

Proceedings of the international conference on "differential geometry and its application", Deva, October 2005. Cluj universit

Physics

High Energy Physics

High Energy Physics - Theory

Short and non technical version of hep-th/0506147. Proceedings of the international conference on "differential geometry and i

Scientific paper

We give an overview on the metric aspect of noncommutative geometry, especially the metric interpretation of gauge fields via the process of "fluctuation of the metric". Connes' distance formula associates to a gauge field on a bundle P equipped with a connection H a metric. When the holonomy is trivial, this distance coincides with the horizontal distance defined by the connection. When the holonomy is non trivial, the noncommutative distance has rather surprising properties. Specifically we exhibit an elementary example on a 2-torus in which the noncommutative metric d is somehow more interesting than the horizontal one since d preserves the S^1-structure of the fiber and also guarantees the smoothness of the length function at the cut-locus. In this sense the fiber appears as an object "smoother than a circle". As a consequence, from a intrinsic metric point of view developed here, any observer whatever his position on the fiber can equally pretend to be "the center of the world".

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