Geometric Quantization, Moduli Spaces and TQFT III
3-4 hours of lectures per week.
Lecturer
Jørgen Ellegaard Andersen
Content
In this course I will discuss in detail the gauge theory construction of the quantum representations of the mapping class groups, which are part of the Reshetihin-Tureav TQFT.
This means we will study Hitchin's construction of a projectively flat connection over Teichm{\"u}ller space in the bundle, whose fiber over a given Riemann surface is the geometric quantization of the moduli spaces of semi-stable bundles on that Riemann Surface.
In the second half of the course, I will go through the proof of my Asymptotic Faithfulness result, which states that any non-trivial mapping class is detected by these quantum representations.
This uses the theory of Toeplitz operators and the relation between geometric quantization and deformation quantization via Berezin-Toeplitz quantization.
Prerequisites
The courses "Geometric quantization, moduli spaces and TQFT I & II" and "Moduli space of vector bundles".
Literature
N. Hitchin,
Flat connections and geometric quantization, Comm.Math.Phys., 131 (1990) 347-380.
M. Schlichenmaier,
Deformation quantization of compact Kähler manifolds by Berezin-Toeplitz quantization, Conference Moshé Flato 1999 (September 1999, Dijon, France) (eds. G. Dito, and D. Sternheimer), Kluwer, 2000, Vol. 2, pp. 289-306, math.QA/9910137.
J. E. Andersen,
Asymptotic faithfulness of the quantum SU(n) representations of the mapping class groups. MPS-Preprint 2002-32. math.QA/0204084.
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
Quarter
Spring 2004