On geometric quantisation of integrable systems with singularities

This thesis shows an approach to geometric quantisation of integrable systems. It extends some results by Guillemin, Kostant, Rawnsley, Sniatycki and Sternberg in geometric quantisation, considering regular fibrations as real polarisations, to the singular setting: the real polarisations concerned here are given by integrable systems with nondegenerate singularities, and the definition of geometric quantisation used is the one suggested by Kostant (via higher cohomology groups). It also presents unifying proofs for results in geometric quantisation by exploring the existence of symplectic circle actions: the tools developed here highlight and unravel the role played by circle actions in known results in geometric quantisation. The originality of this thesis relies on the following aspects. Firstly, the use of symplectic circle actions to obtain results in geometric quantisation, and secondly, the nonexistence of Poincaré lemmata for foliated cohomology when the foliation has singularities. Previous results on circle actions, due to Rawnsley, could not be used when the circle action is not free, and it is not straightforward to adapt them to accommodate fixed points. After developing these techniques, the computation of geometric quantisation is performed in a series of situations, which includes: the cotangent bundle of the circle and products of it with any quantisable manifold, and neighbourhoods of nondegenerate singularities of integrable systems (hyperbolic singularities need special treatment, since there is no natural circle action). These computations imply that the Kostant complex is a fine resolution (for the sheaf of sections of the prequantum line bundle which are flat along the polarisation) when the real polarisations are given by integrable systems with nondegenerate singularities. It is important to mention that the proofs are original, since, contrary to expectations, there is no Poincaré Lemma when singularities are allowed for the foliated cohomology associated to foliations induced by integrable systems. This nontrivial result turns out to be interesting in its own right, but only the aspects related to geometric quantisation are presented in the thesis, e.g. the need for a new proof that the Kostant complex is a fine resolution for the sheaf of flat sections. The thesis also provides a different proof of a theorem, firstly proved by Guillemin and Sternberg, that shows that the set of regular Bohr-Sommerfeld fibres is discrete -it not only bares the role played by circle actions, it also excludes the compactness assumption from the theorem. The exploitation of circle actions culminate in an alternative proof for the theorems of Sniatycki and Hamilton. It is an original and unifying proof: the argument works for both situations, Lagrangian fibre bundles and locally toric manifolds. In addition, this approach casts some light on a conjecture about the contributions coming from focus-focus type of singularities. It actually proves that, in degree zero, there is no contribution to geometric quantisation coming from focus-focus fibres for compact 4-dimensional almost toric manifolds