paper

Spectral distance on the circle

arXiv:math/0703586 · doi:10.1016/j.jfa.2008.07.018

Abstract

A building block of noncommutative geometry is the observation that most of the geometric information of a compact riemannian spin manifold M is encoded within its Dirac operator D. Especially via Connes' distance formula one is able to extract from the spectral properties of D the geodesic distance on M. In this paper we investigate the distance d encoded within a covariant Dirac operator on a trivial U(n)-fiber bundle over the circle with arbitrary connection. It turns out that the connected components of d are tori whose dimension is given by the holonomy of the connection. For n=2 we explicitly compute d on all the connected components. For n>2 we restrict to a given fiber and find that the distance is given by the trace of the module of a matrix. The latest is defined by the holonomy and the coordinates of the points under consideration. This paper extends to arbitrary n and arbitrary connection the results obtained in hep-th/0506147 for U(2)-bundle with constant connection. It confirms interesting properties of the spectral distance with respect to another distance naturally associated to connection, namely the horizontal or Carnot-Caratheodory distance d_H. Especially in case the connection has irrational components, the connected components for d are the closure of the connected components of d_H within the euclidean topology on the torus.

Published version. Conclusions unchanged but paper considerably enhanced to make it more readable. One section ("Why the circle ?") added, together with simple examples of horizontal distance. Several figures added as well

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