Supercoherent states,super-Kähler geometry and geometric quantization

Generalized coherent states provide a means of connecting square integrable representations of a semi-simple Lie group with the symplectic geometry of some of its homogeneous spaces. In the first part of the present work this point of view is extended to the supersymmetric context through the study of the OSp(2/2) coherent states. These are explicitly constructed starting from the known abstract typical and atypical representations of osp(2/2). Their underlying geometries turn out to be those of supersymplectic OSp(2/2)-homogeneous spaces. Moment maps identifying the latter with coadjoint orbits of OSp(2/2) are exhibited via Berezin's symbols. When considered within Rothstein's general paradigm, these results lead to a natural general definition of a super-Kähler supermanifold, the supergeometry of which is determined in terms of the usual geometry of holomorphic Hermitian vector bundles over Kähler manifolds. In particular, the supergeometry of the above orbits is interpreted in terms of the geometry of Einstein-Hermitian vector bundles. In the second part, an extension of the full geometric quantization procedure is applied to the same coadjoint orbits. Thanks to the super-Kähler character of the latter, this procedure leads to explicit super-unitary irreducible representations of osp(2/2) in super-Hilbert spaces of superholomorphic square-integrable sections of prequantum bundles of the Kostant type. This work lays the foundations of a program aimed at classifying Lie supergroups' coadjoint orbits and their associated irreducible representations, ultimately leading to harmonic superanalysis. For this purpose a set of consistent conventions is exhibited.Address after October 1, 1995: Faculté des sciences Jean Perrin, Université d'Artois, rue Jean Souvraz, S.P. 18, F-62307 Lens, France. E-mail: amine@gat.univ-lillel.fr.     

Geometric Quantization of Vector Bundles and the Correspondence with Deformation Quantization

I repeat my definition for quantization of a vector bundle. For the cases of the Toeplitz and geometric quantizations of a compact K?hler manifold, I give a construction for quantizing any smooth vector bundle, which depends functorially on a choice of connection on the bundle. Using this, the classification of formal deformation quantizations, and the formal, algebraic index theorem, I give a simple proof as to which formal deformation quantization (modulo isomorphism) is derived from a given geometric quantization. Received: 16 November 1998 / Accepted: 29 June 2000     

Contractions of group representations via geometric quantization

Letters in Mathematical Physics - We propose a general framework to contract unitary dual of Lie groups via holomorphic quantization of their coadjoint orbits, using geometric quantization. The...     

Quantum orbits of theR-matrix type

Given a simple Lie algebra g, we consider the orbits in g^* which are of theR-matrix type, i.e., which possess a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the so-calledR-matrix bracket. We call an algebra quantizing the latter bracket a quantum orbit of theR-matrix type. We describe some orbits of this type explicitly and we construct a quantization of the whole Poisson pencil on these orbits in a similar way. The notions ofq-deformed Lie brackets, braided coadjoint vector fields, and tangent vector fields are discussed as well.     

Quantization of symplectic orbits of compact Lie groups by means of the functional integral

The functional integral for the quantization of the coadjoint orbits of the unitary and orthogonal groups is given by means of an explicit construction of the corresponding ≪Darboux≫ variables.     

Upper Bounds for Regularized Determinants

We conjecture that the zeta-regularized determinant of the Laplace operator with coefficients in a holomorphic vector bundle on a compact K?hler manifold remains bounded when the metric on the bundle varies. This conjecture is shown to be true for certain classes of line bundles on Riemann surfaces. Received: 7 July 1997 / Accepted: 20 April 1998     

Dynamical Yang-Baxter Equation and Quantum Vector Bundles

We develop a categorical approach to the dynamical Yang-Baxter equation (DYBE) for arbitrary Hopf algebras. In particular, we introduce the notion of a dynamical extension of a monoidal category, which provides a natural environment for quantum dynamical R-matrices, dynamical twists, etc. In this context, we define dynamical associative algebras and show that such algebras give quantizations of vector bundles on coadjoint orbits. We build a dynamical twist for any pair of a reductive Lie algebra and its Levi subalgebra. Using this twist, we obtain an equivariant star product quantization of vector bundles on semisimple coadjoint orbits of reductive Lie groups.The research is supported in part by the Israel Academy of Sciences grant no. 8007/99-03, the Emmy Noether Research Institute for Mathematics, the Minerva Foundation of Germany, the Excellency Center Group Theoretic Methods in the study of Algebraic Varieties of the Israel Science foundation, and by Russian Foundation for Basic Research grant no. 03-01-00593.Deceased January 2004Acknowledgement We are grateful to J. Bernstein, V. Ostapenko, and S. Shnider for stimulating discussions within the Quantum groups seminar at the Department of Mathematics, Bar Ilan University. We appreciate useful remarks by M. Gorelik, V. Hinich, and A. Joseph during a talk at the Weizmann Institute. Our special thanks to P. Etingof for his comments on various aspects of the subject.     

Semidirect products and the Pukanszky condition

We study the general geometrical structure of the coadjoint orbits of a semidirect product formed by a Lie group and a representation of this group on a vector space. The use of symplectic induction methods gives new insight into the structure of these orbits. In fact, each coadjoint orbit of such a group is obtained by symplectic induction on some coadjoint orbit of a “smaller” Lie group. We study also a special class of polarizations related to a semidirect product and the validity of Pukanszky's condition for these polarizations. Some examples of physical interest are discussed using the previous methods     

Quantum cosmology and conformal invariance

According to Belinsky, Khalatnikov, and Lifshitz, gravity near a spacelike singularity reduces to a set of decoupled one-dimensional mechanical models at each point in space. We point out that these models fall into a class of conformal mechanical models first introduced by de Alfaro, Fubini, and Furlan (DFF). The deformation used by DFF to render the spectrum discrete corresponds to a negative cosmological constant. The wave function of the Universe is the zero-energy eigenmode of the Hamiltonian, or the spherical vector of the representation of the conformal group SO(1,2). A new class of conformal quantum mechanical models with enhanced ADE symmetry is constructed, based on the quantization of nilpotent coadjoint orbits.     

On Algebraic Supergroups,Coadjoint Orbits and Their Deformations

In this paper we study algebraic supergroups and their coadjoint orbits as affine algebraic supervarieties. We find an algebraic deformation quantization of them that can be related to the fuzzy spaces of non-commutative geometry.Investigation supported by the University of Bologna, funds for selected research topics.     

Dynamical algebras in classical mechanics

For a spectrum-generating algebra of classical observables, it is proven that the phase space dynamics simplifies to a Hamiltonian system on submanifolds of the algebra's dual. These submanifolds are coadjoint orbits if the algebra arises from a symplectic group action. If the Hamiltonian splits into the sum of a function of the algebra generators plus a commuting part, then the dynamics transfers to the dual space and an explicit formula is given for the flow vector field on the coadjoint orbits. A unique feature of the presentation is that all constructions are at the Lie algebra level.     

Central extensions of current groups in two dimensions

The paper is devoted to generalization of the theory of loop groups to the two-dimensional case. To every complex Riemann surface we assign a central extension of the group of smooth maps from this surface to a simple complex Lie group G by the Jacobian of this surface. This extension is topologically nontrivial, as in the loop group case. Orbits of coadjoint representation of this extension correspond to equivalence classes of holomorphic principalG-bundles over the surface. When the surface is the torus (elliptic curve), classification of coadjoint orbits is related to linear difference equations in one variable, and to classification of conjugacy classes in the loop group. We study integral orbits, for which the Kirillov-Kostant form is a curvature form for some principal torus bundle. The number of such orbits for a given level is finite, as in the loop group case; conjecturedly, they correspond to analogues of integrable modules occurring in conformal field theory.     

Fiber Bundles in Non-relativistic Quantum Mechanics

The problem of describing a quantum mechanical system with symmetry by a fiber bundle is considered. The quantization of a fiber bundle is introduced. Fiber bundles for the Kepler problem and the rotator are constructed. The fiber bundle concept provides a new model for a physical system: it provides us with a model for an elementary particle with extension having integral values of spin.     

All local classical symmetries in Hamiltonian mechanics

Using the formalism of symplectic group actions and coadjoint orbits, we give a complete list of all classical simple Lie algebras which are local symmetries for a given Hamiltonian vector field.     

Quantum coadjoint orbits in gl(n)*

We present a doubleU _h(gl(n, ℂ))-equivariant quantization on semisimple coadjoint orbits of the group GL(n, ℂ) as a quotient of the extended reflection equation algebra by relations which are given explicitly. Such a quantization is a two-parameter family including an explicit GL(n)-equivariant quantization of the Kirillov-Kostant-Souriau Poisson bracket. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Symplectic structures associated to Lie-Poisson groups

The Lie-Poisson analogues of the cotangent bundle and coadjoint orbits of a Lie group are considered. For the natural Poisson brackets the symplectic leaves in these manifolds are classified, and the corresponding symplectic forms are described. Thus the construction of the Kirillov symplectic form is generalized for Lie-Poisson groups.Supported in part by a Soros Foundation Grant awarded by the American Physical SocietyUnité Associée au C.N.R.S., URA 280     

On geometric quantization of the Dirac magnetic monopole

We give a simple derivation of the spectrum of the Dirac magnetic monopole on a unit sphere S² based on geometric quantization and the Frobenius reciprocity formula. The starting point is the calculation by Novikov and Schmelzer of the canonical symplectic structure on the coadjoint orbits of the isometry group of 3-dimensional Euclidean space E(3), which showed the appearance of the Dirac magnetic term.