John Rawnsley's Personal Pages :: Research

Downloadable Papers

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Quantization of Kähler Manifolds. III
Letters in Math. Phys. 30 (1994) 291-305.
Quantization of Kähler Manifolds. IV
Letters in Math. Phys. 34 (1995) 159-168.
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
Contemp. Math. 179 (1994) 1-16.
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
Commun. Math. Phys. 180 (1996) 99-108.
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
J. Geom. Phys. 23 (1997) 301-307.
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
J. Geom. Phys. 29 (1999) 347-392.
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
Letters in Math. Phys. 48 (1999) 353-364.
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
Conférence Moshé Flato. Dijon, July, 1999. 81-91, Math. Phys. Stud., 22, Kluwer Acad. Publ., Dordrecht, 2000, math.SG/9912181
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
J. Geom. Phys. 38 (2001) 140-151, math.DG/0006212
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 $\mathbb{P}_n(\mathbb{C})$ with a multiple of its K\"ahler form and $\nabla$ is affinely equivalent to the Levi-Civita connection.

Traces for star products on symplectic manifolds
J. Geom. Phys. 42 (2002) 12-18, math.QA/0105089
S. Gutt and J. Rawnsley
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 our paper above, solving first the local problem. A normalisation introduced by Karabegov makes the local solutions unique and allows them to be pieced together to solve the global problem.

Moduli space of symplectic connections of Ricci type on $T^{2n}$ - a formal approach
J. Geom. Phys. 46 (2003) 174-192, math.SG/0201167
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 $\mathcal{E}$ of symplectic connections which are of Ricci type. As a preliminary to studying the moduli space $\mathcal{E}/G$ we study the moduli of formal curves of connections under the action of formal curves of symplectomorphisms.

Natural star products
Lett. Math. Phys. 66 (2003) 123-139, math.SG/0304498
S. Gutt & 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
math.RT/0306336
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.

Symmetries of star products

J. Rawnsley
Lectures given at the UK-Japan Winter School, University of Durham, 6-9 January, 2004. Seminar on Mathematical Sciences, Keio University 30 (2004) 37-49.

Symplectic Connections
Int. J. Geom. Methods Mod. Phys. 3 (2006) 375-420, math.SG/0511194v2,
P. Bieliavsky, M. Cahen, S. Gutt, J. Rawnsley and L. Schwachhöfer
Expanded version of lectures given at the XXth International Workshop on Differential Geometrical Methods in Theoretical Mechanics, University of Ghent, August 22-26, 2005.