# Gauge fluctuation in Noncommutative Geometry and Carnot-Carathéodory distance

## Pierre Martinetti

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

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.