Gauge fluctuation in Noncommutative Geometry and Carnot-Carathéodory distance

Pierre Martinetti

(Dipartimento di Matematica & CMTP, Università di Roma Tor Vergata)

Event: ERC Workshop on Geometric Analysis on sub-Riemannian and Metric Spaces

Date: Oct 11, 2011, time: 14:30

Place: Centro De Giorgi, Scuola Normale Superiore

Abstract. In noncommutative geometry, starting with an algebra \(A\) and an operator \(D\) which generalizes to the noncommutative framework the Dirac (or Atiyah) operator of a spin manifold, Connes defines a distance on the space of states of \(A\). We call it the "spectral distance".
In case A is chosen as the tensor product of the algebra of smooth function on a compact Riemannian manifold M with the algebra of n-square matrices, the space of state of \(A\) is a \(U(n)\) trivial bundle on \(M\).
Through the so called process of "fluctuation of the metric", one equips this bundle with a connection \(C\). This amount to turn the initial Dirac operator \(D\) into a covariant Dirac operator \(D+C\).
It was expected that the spectral distance calculated with \(D+C\) were equal to the horizontal distance defined by \(C\). We will show that the link between the two distances is more subtle, depending on the holonomy
of the connection.