Geometric quantization, moduli spaces and TQFT
3-4 hours of lectures per week.
Lecturer Joergen Ellegaard Andersen Content After a short introduction to symplectic manifolds, we will discuss the general program of geometric quantization. We shall consider both real and complex polarizations, but concentrate on the Kãhler case. After that we shall discuss deformation quatization and just touch on Kontsevich main theorem, which states such always exists for general Poisson manifolds. Continuing with the symplectic case, we will following ideas which goes back to Berezin, see how the geometric quantization via the use of Toeplitz operators, gives a particular deformation quantization in the Kãhler case. Some interesting analysis due to Boutet de Monvel, Sjoestrand and Guillemin on the asymptotics of the Szeõ kernel will be used at this point.
Following this we shall see how this program can be applied in the gaugetheoretic approach to quantum invariants of 3 dimensional manifolds.
We will start with a baby example, where we study the abelian moduli spaces. With the use of Theta-functions, we will "discover" a beautifull relations between the Moyal star product, the heat equation and Berezin-Toeplitz quantization of Abelian varieties, which has only recently been understood.
The non-abelian analog of this is far more involved, but offer considerable insight into Topological quantum field theory and toplogy in 3 dimensions. We beging by studying the moduli space at G-connections on a genus g surface from scratch. We will see how this space is a singular symplectic manifold. Kãhler surfaces of the same genus. - Having understood that we shall then move on to apply the geometric quantization program to these moduli spaces and study the projective connection in the resulting Verlinde bundle over Teichmuller space introduced some 10 years ago by Axelrod, Della Pietra and Witten, Hitchin and Faltings. - The course will end at frontline research by discussing how the Berezin-Toeplitz quantization procedure can be made compatible with this projectively at connection so as to induce topological conclusions about the associated Topological quantum field theories, which of course includes the polynomial invariants of knots introduced by Jones 15 years ago.
Prerequisites The starting level of the course will be set at the minimum of that of the participants and that of the lecturer. - Depending on that, we may have to continue the course beyond the fall semester.