# Gauge fluctuation in Noncommutative Geometry and Carnot-Carathéodory distance

### (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.