117 research outputs found
Natural star products on symplectic manifolds and quantum moment maps
We define a natural class of star products: those which are given by a series
of bidifferential operators which at order in the deformation parameter
have at most derivatives in each argument. We show that any such star
product on a symplectic manifold defines a unique symplectic connection. We
parametrise such star products, study their invariance and give necessary and
sufficient conditions for them to yield a quantum moment map. We show that
Kravchenko's sufficient condition for a moment map for a Fedosov star product
is also necessary.Comment: Expanded bibliograph
Construction of Ricci-type connections by reduction and induction
Given the Euclidean space endowed with a constant symplectic
structure and the standard flat connection, and given a polynomial of degree 2
on that space, Baguis and Cahen have defined a reduction procedure which yields
a symplectic manifold endowed with a Ricci-type connection. We observe that any
symplectic manifold of dimension greater than 2 endowed with a symplectic
connection of Ricci-type is locally given by a local version of such a
reduction.
We also consider the reverse of this reduction procedure, an induction
procedure: we construct globally on a symplectic manifold endowed with a
connection of Ricci-type a circle or a line bundle which
embeds in a flat symplectic manifold as the zero set of a
function whose third covariant derivative vanishes, in such a way that
is obtained by reduction from .
We further develop the particular case of symmetric symplectic manifolds with
Ricci-type connections
Traces for star products on symplectic manifolds
We give a direct elementary proof of the existence of traces for arbitrary
star products on a symplectic manifold. We follow the approach we used in
\cite{refs:GuttRaw}, solving first the local problem. A normalisation
introduced by Karabegov \cite{refs:Karabegov} makes the local solutions unique
and allows them to be pieced together to solve the global problem
On Mpc-structures and Symplectic Dirac Operators
We prove that the kernels of the restrictions of symplectic Dirac or
symplectic Dirac-Dolbeault operators on natural subspaces of polynomial valued
spinor fields are finite dimensional on a compact symplectic manifold. We
compute those kernels for the complex projective spaces. We construct
injections of subgroups of the symplectic group (the pseudo-unitary group and
the stabilizer of a Lagrangian subspace) in the group Mpc and classify
G-invariant Mpc-structures on symplectic spaces with a G-action. We prove a
variant of Parthasarathy's formula for the commutator of two symplectic
Dirac-type operators on a symmetric symplectic space
Extrinsic symplectic symmetric spaces
We define the notion of extrinsic symplectic symmetric spaces and exhibit
some of their properties. We construct large families of examples and show how
they fit in the perspective of a complete classification of these manifolds. We
also build a natural star-quantization on a class of examples
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