Carnot-Carathéodory Metric and Gauge Fluctuation in Noncommutative Geometry

Gauge fields have a natural metric interpretation in terms of horizontal distance. The latest, also called Carnot-Carathéodory or subriemannian distance, is by definition the length of the shortest horizontal path between points, that is to say the shortest path whose tangent vector is everywhere horizontal with respect to the gauge connection. In noncommutative geometry all the metric information is encoded within the Dirac operator D. In the classical case, i.e. commutative, Connes’s distance formula allows to extract from D the geodesic distance on a riemannian spin manifold. In the case of a gauge theory with a gauge field A, the geometry of the associated U(n)-vector bundle is described by the covariant Dirac operator D+A. What is the distance encoded within this operator? It was expected that the noncommutative geometry distance d defined by a covariant Dirac operator was intimately linked to the Carnot-Carathéodory distance dh defined by A. In this paper we make precise this link, showing that the equality of d and dH strongly depends on the holonomy of the connection. Quite interestingly we exhibit an elementary example, based on a 2 torus, in which the noncommutative distance has a very simple expression and simultaneously avoids the main drawbacks of the riemannian metric (no discontinuity of the derivative of the distance function at the cut-locus) and of the subriemannian one (memory of the structure of the fiber).