## Final Report Summary - FGQ (The future of geometric quantisation)

The topic of this project was geometric quantisation. In physics, this is a way to construct the quantum mechanical description of a physical system from its classical mechanical description. For a system with symmetries, the way that geometric quantisation relates its classical and quantum symmetries to each other is particularly relevant. The central principle here is that quantisation commutes with reduction, or symbolically [Q,R]=0. This means that using symmetries to simplify the classical or quantum description of a system is compatible with geometric quantisation. This principle has also led to new insights in representation theory, the mathematics of quantum symmetries.

The [Q,R]=0 principle was proved in the 1990s for systems with compact phase spaces and compact groups of symmetries. An example of such a system is a spinning charged particle in a constant magnetic field. Then the phase space is the two-sphere (representing the possible directions of the angular momentum of the particle), and the symmetry group is the circle group of rotations around the axis through the particle, in the direction of the magnetic field. For most systems, however, the phase space is noncompact. For example, the phase space of a pendulum is a cylinder: the circle on which the end of the pendulum can move combined with a line representing its possible velocities. This cylinder is unbounded, and hence noncompact. Also, symmetry groups may be noncompact, as for example in the case of a free particle in three-dimensional space, where the translation symmetries make up a noncompact group.

The overall goal of this project was to generalise [Q,R]=0 to systems with noncompact phase spaces and symmetry groups. This generalisation is relevant to physics, because most phase spaces are noncompact, as was just pointed out. Furthermore, it is relevant to mathematics, because it provides new information about representation theory of noncompact groups. The overall goal was divided into three specific aims: to prove the [Q,R]=0 principle

1. for a large class of representations;

2. in as generally noncompact settings as possible;

3. in cases where classical reduction leads to singularities.

All of these aims were more than achieved.

We now give a more technical description of the results obtained in each of these three directions.

1. Previous results for K-theory classes of discrete series representations of semisimple groups were extended to K-theory classes of more general representations. These include principal series representations of complex semisimple groups. Also, the formal geometric quantisation of Weitsman and Paradan was generalised to noncompact groups, in a K-theoretic setting. Very explicit [Q,R]=0 results, not involving K-theory, were obtained for discrete series representations, in cases where the orbit space of the group action is not compact, and for irreducible representations of compact groups acting on noncompact symplectic and Spin-c manifolds. Finally, a fixed point theorem was proved that relates geometric quantisation of elliptic coadjoint orbits of semisimple groups to the Harish-Chandra character formula for discrete series representations.

2. Early in this project, the [Q,R]=0 principle was generalised to actions by noncompact groups on symplectic manifolds, with noncompact orbit spaces. An example is the action by a noncompact Lie group on its cotangent bundle induced by left multiplication. This very general noncompact version of [Q,R]=0 was later generalised to Spinc-manifolds, and equivariant index theory was developed for general Dirac-type operators.

3. Singular reduced spaces were allowed in most results obtained, so that the third aim was achieved as well.

As in any research project, it would have been disappointing if only the aims that could have been imagined beforehand had been achieved. An important set of results of this project are about the unexpected generalisation of the [Q,R]=0 principle from symplectic manifolds to the much more general Spin-c manifolds. This was proved in the compact case by Paradan and Vergne in 2014, and generalised to various noncompact settings in this project.

The training objectives of this project were aimed developing the fellow into an independent researcher with an international reputation in the field. To meet these objectives, the fellow spoke at various conferences and seminars, organised two conferences himself, obtained funding for these conferences, started and reinforced collaborations, participated in outreach activities, and managed the successful execution of the project. His status as an independent researcher was confirmed when the University of Adelaide offered him a permanent position, which he took up after completing this project.

Contact details

Peter Hochs

School of Mathematical Sciences

North Terrace Campus

The University of Adelaide

SA 5005 Australia

peter.hochs@adelaide.edu.au

http://maths.adelaide.edu.au/peter.hochs/

The [Q,R]=0 principle was proved in the 1990s for systems with compact phase spaces and compact groups of symmetries. An example of such a system is a spinning charged particle in a constant magnetic field. Then the phase space is the two-sphere (representing the possible directions of the angular momentum of the particle), and the symmetry group is the circle group of rotations around the axis through the particle, in the direction of the magnetic field. For most systems, however, the phase space is noncompact. For example, the phase space of a pendulum is a cylinder: the circle on which the end of the pendulum can move combined with a line representing its possible velocities. This cylinder is unbounded, and hence noncompact. Also, symmetry groups may be noncompact, as for example in the case of a free particle in three-dimensional space, where the translation symmetries make up a noncompact group.

The overall goal of this project was to generalise [Q,R]=0 to systems with noncompact phase spaces and symmetry groups. This generalisation is relevant to physics, because most phase spaces are noncompact, as was just pointed out. Furthermore, it is relevant to mathematics, because it provides new information about representation theory of noncompact groups. The overall goal was divided into three specific aims: to prove the [Q,R]=0 principle

1. for a large class of representations;

2. in as generally noncompact settings as possible;

3. in cases where classical reduction leads to singularities.

All of these aims were more than achieved.

We now give a more technical description of the results obtained in each of these three directions.

1. Previous results for K-theory classes of discrete series representations of semisimple groups were extended to K-theory classes of more general representations. These include principal series representations of complex semisimple groups. Also, the formal geometric quantisation of Weitsman and Paradan was generalised to noncompact groups, in a K-theoretic setting. Very explicit [Q,R]=0 results, not involving K-theory, were obtained for discrete series representations, in cases where the orbit space of the group action is not compact, and for irreducible representations of compact groups acting on noncompact symplectic and Spin-c manifolds. Finally, a fixed point theorem was proved that relates geometric quantisation of elliptic coadjoint orbits of semisimple groups to the Harish-Chandra character formula for discrete series representations.

2. Early in this project, the [Q,R]=0 principle was generalised to actions by noncompact groups on symplectic manifolds, with noncompact orbit spaces. An example is the action by a noncompact Lie group on its cotangent bundle induced by left multiplication. This very general noncompact version of [Q,R]=0 was later generalised to Spinc-manifolds, and equivariant index theory was developed for general Dirac-type operators.

3. Singular reduced spaces were allowed in most results obtained, so that the third aim was achieved as well.

As in any research project, it would have been disappointing if only the aims that could have been imagined beforehand had been achieved. An important set of results of this project are about the unexpected generalisation of the [Q,R]=0 principle from symplectic manifolds to the much more general Spin-c manifolds. This was proved in the compact case by Paradan and Vergne in 2014, and generalised to various noncompact settings in this project.

The training objectives of this project were aimed developing the fellow into an independent researcher with an international reputation in the field. To meet these objectives, the fellow spoke at various conferences and seminars, organised two conferences himself, obtained funding for these conferences, started and reinforced collaborations, participated in outreach activities, and managed the successful execution of the project. His status as an independent researcher was confirmed when the University of Adelaide offered him a permanent position, which he took up after completing this project.

Contact details

Peter Hochs

School of Mathematical Sciences

North Terrace Campus

The University of Adelaide

SA 5005 Australia

peter.hochs@adelaide.edu.au

http://maths.adelaide.edu.au/peter.hochs/