# Manchester Geometry Seminar 2009/2010

**Thursday 18 February 2010. ** *The Frank Adams Room 1 (Room 1.212), the Alan Turing Building. 4.15pm*
## Quantization and Conformal Geometries of the Supercotangent and Spinor Bundles

Jean-Philippe Michel (Université Claude Bernard Lyon 1)

`jpmichel@math.univ-lyon1.fr`

We start by an introduction to spinor and super geometries.

The geometric quantization links the supercotangent bundle of
a pseudo-riemannian manifold (M,g) and its spinor bundle, as well as the respective actions of the conformal vector fields. From their explicit expressions, we deduce, on a conformally flat manifold (M,g), the classification of the conformally covariant spinor differential operators (chirality, Dirac operator,...) and of their symbols, which are functions on the supercotangent of M.
We prove then the existence and uniqueness of a conformally equivariant quantization between symbols and operators (if (M,g) is conformally flat), which leads to a correspondence between the conformally covariant symbols and operators.