Working Group III:

New geometric quantization schemes

Members: K.-H. Neeb, C. Meusburger

Geometric quantisation is a powerful technique for constrained mechanical systems. Recently, the underlying mathematical framework (a Hamiltonian action of a Lie group on a Poisson manifold with a momentum map) has been generalized to the concept of a quasi-Hamiltonian action, where the Lie group is replaced by a pair of groups corresponding to the two Lie algebras in a Manin pair and the momentum map takes values in a Lie group. A key motivation for this generalisation was the possibility of obtaining moduli spaces of flat connections on Riemann surfaces as quasi-Hamiltonian quotients of direct products of Lie groups. Moduli spaces of flat connections have attracted extensive interest in both mathematics and physics and play a fundamental role in Chern-Simons gauge theory and three-dimensional gravity. From a physics viewpoint, the moduli space of flat connections on a surface is the gauge invariant (or reduced) phase space of Chern-Simons gauge theory for three-dimensional gravity. The quantisation of moduli spaces associated to certain Lie groups is thus equivalent to the task of constructing a diffeomorphism invariant quantum theory of three-dimensional gravity.

Classically, (quasi-)Hamiltonian actions and momentum maps have been studied intensively for compact groups. However, there are many instances, where quasi-Hamiltonian actions show up naturally for non-compact or even infinite-dimensional groups. In particular, this is the case for the moduli spaces of flat connections arising as the phase spaces of three-dimensional gravity. In this situation, the reduction procedures that implement the constraints and the corresponding quantisation procedures are still very poorly understood. Here, specific examples should be understood more systematically to extract the natural structural framework. This would provide a new quantisation formalism applicable to gravity in three dimensions and would allow one to gain important insights into the role of contraints and diffeomorphism invariance in quantum gravity in general.

Particular aspects to be explored in this context are quasi-Hamiltonian actions of doubles of Lie groups, which arise in three-dimensional gravity and correspond to Hamiltonian actions of loop groups. More generally, there is a prospect of obtaining (projective) unitary representations of loop groups from suitable quantisation procedures or from more advanced quantisation formalisms such as combinatorial quantisation. Moreover, the mathematical structures arising in quantum gravity suggest that one should also undertake a closer study of a class of infinite-dimensional Lie groups that generalise loop groups to groups involving maps on a graph.

Impressum