3-4 hours of lectures per week.
Lecturer
Jørgen Ellegaard Andersen
Content
We shall continue making our way through the program outlined below. 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 Kahler 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 Kähler case. Some interesting analysis due to Boutet de Monvel, Sjoestrand and Guillemin on the asymptotics of the Szeöge 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 beautiful 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 eld theory and toplogy in 3 dimensions. We begin 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. Any complex structure on the genus g surfaces induces the structure of projectiv algebraic varity on the moduli space. - 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 eld theories, which of course includes the polynomial invariants of knots introduced by Jones 15 years ago.
Prerequisites
Large parts of the spring semester will be concerned with deformation quantization and will as such not need the geometric quantization program developed in the first semester of this course. However, towards the end of the semester we will tie the two approaches to quantization together.
Literature
Please see the course homepage: http://home.imf.au.dk/andersen/gqmstqft
Evaluation
Students who do not intend to take a degree in Mathematics or Statistics from the University of Aarhus, but wish to earn credits for a 2.dels course from the Department of Mathematics, should indicate at the beginning of the course that they wish to be examined.
The form of examination for these students will be active participation together with oral or written contributions.
Credits
10 ECTS
Semester
Spring 2003