Manchester Geometry Seminar 2011/2012

Quantization is a concept which comes from quantum mechanics. It consists in associating with a classical observable a quantum observable. Geometrically, the space of quantum observables can be identified with a space of differential operators on a manifold $M$ whereas the space of classical observables can be identified with the corresponding space of symbols. Actually, the space of symbols is the graded space corresponding to the filtered space of differential operators. The space of symbols can be viewed too as a space of polynomial functions on the cotangent bundle of the manifold $M$.

There is no natural quantization on a manifold $M$: namely, there is no quantization which commutes with the action of all local diffeomorphisms of $M$. At the infinitesimal level, there is no quantization which commutes with the Lie derivative in the direction of all vector fields. However, if $M$ is the Euclidean space, one can wonder if there exists a quantization which exchanges the actions of maximal Lie subalgebras in the Lie algebra of vector fields. This leads to the concept of equivariant quantization. P. Lecomte and V. Ovsienko showed the existence of quantizations equivariant with respect to the action of a Lie subalgebra isomorphic to the projective algebra, $sl(m+1,\mathbb{R})$. After that, C. Duval, P. Lecomte and V. Ovsienko proved the existence of $so(p+1,q+1)$-equivariant quantizations, i.e. quantizations equivariant with respect to the action of a Lie subalgebra isomorphic to the conformal algebra.

The problem of equivariant quantization has a counterpart on an arbitrary manifold $M$. It consists in the projective (resp. conformal) setting in finding a natural and projectively (resp. conformally) invariant quantization, i.e. a quantization which depends on a connection (resp. a pseudo-Riemannian metric) but only on its projective class (resp. conformal class), the quantization being natural in all of its arguments. The problem of natural and invariant quantization has been recently completely solved.

I will speak in my talk about the generalization of the problem of equivariant quantization in the context of supergeometry. The works in this direction have been recently performed by T. Leuther, P. Mathonet and myself. In order to prove the existence of a natural and projectively invariant quantization on an arbitrary supermanifold, we use the Thomas fiber bundle described in J. George's thesis.