Geometric quantisation, Hitchin connection, and all that.
Geometric quantisation is a tentative approach to extending the mathematical model of ordinary quantum mechanics to an arbitrary symplectic manifold. There are, however, heavy limitations to the extent to which this can be done in general, as well as several different ways to apply geometric quantisation, which is ultimately less of a fixed construction and more of a general recipe to be adapted to the specific cases. In particular, the entire process depends on the choice of an auxiliary piece of structure on the symplectic manifold, called a polarisation, which are not guaranteed to be unique when they exist. The study dependence of the outcome on this piece of data is one of the most important problems in this theory.
Nonetheless, geometric quantisation has been successfully carried out in a number of examples, famously on the moduli spaces of flat G-connections on smooth surfaces for G = SU(n) (Hitchin, Axelrod-Della-Pietra-Witten) and SL(n,C) (Witten). In both cases, polarisations can be obtained from the choice of a complex structure on the surface, resulting in a family of Hilbert spaces parametrised by the Teichmüller space, and the problem of the dependence on this additional structure is addressed by regarding these families as vector bundles and constructing connections on them, called the Hitchin and Hitchin-Witten connections.
In my talk, I wish to follow the flow of questions and discussion to choose exactly what materials to cover. I plan to start from the basic ideas of geometric quantisation leading to the construction of the Hitchin and Hitchin-Witten connections, focusing on more specific aspects depending on the audience’s interest.
