Papers by John Rawnsley
- Harmonic Maps Bibliography
[PS]
F.E. Burstall, L. Lemaire, J. Rawnsley
The Harmonic Maps Bibliography is an ongoing project to provide a
comprehensive and up-to-date biliography in the area of harmonic maps.
It is the joint work of many volunteers.
- Quantization of Kähler Manifolds. III
[PS]
[PDF]
Quantization of Kähler Manifolds. IV
[PS]
[PDF]
Michel Cahen, Simone Gutt, John Rawnsley
We use Berezin's dequantization procedure to define a formal *-product
on the algebra of smooth functions on bounded homogeneous complex domains.
We prove that this formal *-product is convergent on a dense subalgebra of
the algebra of smooth functions.
- Some Remarks on the Classification of Poisson Lie Groups
[PS]
[PDF]
Michel Cahen, Simone Gutt and John Rawnsley
We describe some results in the problem of classifying the bialgebra
structures on a given finite dimensional Lie algebra.
We consider two aspects of this problem.
One is to see which Lie algebras arise (up to isomorphism) as the big
algebra in a Manin triple,
and the other is to try and determine all the exact Poisson structures for a
given semisimple Lie algebra.
We follow here the presentation of the talk that one of us gave at the
Yokohama Symposium; in particular, we recall many well known properties so
that it is essentially self-contained.
- On tangential star products for the coadjoint Poisson structure
[PS]
[PDF]
M. Cahen, S. Gutt, J. Rawnsley
We derive necessary conditions on a Lie algebra from the existence of a
star product on a neighbourhood of the origin in the dual of the Lie
angebra for the coadjoint Poisson structure which is both differential
and tangential to all the coadjoint orbits. In particular we show that
when the Lie algebra is semisimple there are no differential star products
on any neighbourhood of the origin in the dual of its Lie algebra.
- Hamiltonian circle actions on symplectic manifolds and the signature
[PS]
John Jones and John Rawnsley
Let M be a symplectic manifold with a Hamiltonian circle action
with isolated fixed points. We prove that the signature of M is the
alternating sum of its even Betti numbers.
- Equivalence of star products on symplectic manifolds
[PS]
S. Gutt and J. Rawnsley
Expanded notes from the Quantisation Seminar on Deligne's approach to
the equivalence of star products, and of the lecture of the
first author in the Workshop on Quantization and Momentum Maps
at the University of Warwick in December 1997.
- Preferred invariant symplectic connections on compact coadjoint orbits
[PS]
M. Cahen, S. Gutt and J. Rawnsley
We prove the existence of at least one $G$-invariant preferred
symplectic connection on any coadjoint orbit of a compact semisimple Lie
group $G$. We look at the case of the orbits of $SU(3)$ and show that
in this case the invariant preferred connection is unique.
- Symmetric symplectic spaces with Ricci-type curvature
[PS]
M. Cahen, S. Gutt and J. Rawnsley
We determine the isomorphism classes of symmetric symplectic manifolds
of dimension at least 4 which are connected, simply-connected
and have a curvature tensor which has only one non-vanishing
irreducible component -- the Ricci tensor.
- Homogeneous symplectic manifolds with Ricci-type curvature
[PS]
[PDF]
M. Cahen, S. Gutt, J. Horowitz and J. Rawnsley
We consider invariant symplectic connections $\nabla$ on homogeneous
symplectic manifolds $(M,\omega)$ with curvature of Ricci type. Such
connections are solutions of a variational problem studied by Bourgeois
and Cahen, and provide an integrable almost complex structure on the
bundle of almost complex structures compatible with the symplectic
structure. If $M$ is compact with finite fundamental group then
$(M,\omega)$ is symplectomorphic to $\P_n(\C)$ with a multiple of its
K\"ahler form and $\nabla$ is affinely equivalent to the Levi-Civita
connection.
- Symplectic connections with parallel Ricci tensor
[PDF]
M. Cahen, S. Gutt and J. Rawnsley
A variational principle introduced to select some symplectic connections
leads to field equations which, in the case of the Levi Civita
connection of K\"ahler manifolds, are equivalent to the condition that
the Ricci tensor is parallel. This condition, which is stronger than the
field equations, is studied in a purely symplectic framework.
- Moduli space of symplectic connections of Ricci type on $T^{2n}$ -
a formal approach
[PDF]
M. Cahen, S. Gutt, J. Horowitz and J. Rawnsley
We consider analytic curves $\nabla^t$ of symplectic connections of Ricci
type on the torus $T^{2n}$ with $\nabla^0$ the standard connection. We show,
by a recursion argument, that if $\nabla^t$ is a formal curve of such
connections then there exists a formal curve of symplectomorphisms
$\psi_t$ such that $\psi_t\cdot\nabla^t$ is a formal curve of flat
$T^{2n}$invariant symplectic connections and so $\nabla^t$ is flat for all $t$.
Applying this result to the Taylor series of the analytic curve, it
means that analytic curves of symplectic connections of Ricci type
starting at $\nabla^0$ are also flat.
The group $G$ of symplectomorphisms of the torus $(T^{2n},\omega)$ acts
on the space $\E$ of symplectic connections which are of Ricci type. As
a preliminary to studying the moduli space $\E/G$ we study the moduli of
formal curves of connections under the action of formal curves of
symplectomorphisms.
- Natural star products on symplectic manifolds and
quantum moment maps
[PDF]
S. Gutt and J. Rawnsley
We define a natural class of star products: those which are given by a
series of bidifferential operators which at order $k$ in the deformation
parameter have at most $k$ 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.
- Compact Coadjoint Orbits
[PDF]
J. Rawnsley
I give an answer to the question ``Which groups have compact coadjoint
orbits?''. Whilst I thought that the answer, which is straightforward,
must be in the literature, I was unable to find it. This note aims to
rectify this.
It is also a plea: If the result is already published then I would like
to know the reference.