Gauge fluctuation in Noncommutative Geometry and Carnot-Carathéodory distance
speaker: Pierre Martinetti (Università di Napoli "Federico II")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.
timetable:
Tue 11 Oct, 14:30 - 15:20, Aula Dini
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