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Topological Quantum Field Theory and Related Algebraic Structures

(Spring 2008)

The master course "Topological Quantum Field Theory and Related Algebraic Structures" (course description) is part of the master class 2007-2008 Quantum groups, affine Lie algebras and their applications.

Schedule: Monday, 14:15-17:00, wk 6-wk 12, wk 14-wk 18, wk 21-wk 22
Location:Universiteit van Amsterdam, Faculty of Science, Korteweg-de Vries Institute for Mathematics, Plantage Muidergracht 24, 1018 TV Amsterdam, room P-0.18 (ground floor)


This course is a continuation of "Quantum groups and knot theory" which was taught in the Fall 2007 by Eric Opdam and Jasper Stokman. The lecture notes for this course can be found here. If you did not take this course it is OK, but you should look at the lecture notes and work more on this subject.
Here is the list of references. It will grow. Here is the list of homeworks:

HW1-2.

HW3
The homeworks are due to the Monday after they are assigned. You should choose three from the suggested list. Of course you are welcome to do more then this.



Lecture 1 was a reminder of basic facts about bialgebras, Hopf algebras, and rigid monoidal categories. Here is the draft of the lecture. More information on bialgebras and Hopf algebras can be found in the lecture notes from the Fall class. The notes on monoidal categories can be found here . The notes on rigidity in monoidal categories is here .


Lecture 2. The subject was: rigid, braided monoidal categories, ribbon categories and examples. The construction of the double of Hopf algebras was explained with some basic examples related to quantized sl_2. Lecture notes on braided categories can be found here . Here you can also find the draft of the lecture.


Lecture 3. Categories of tangles and diagrams were introduced. There was a brief discussion of the Knizhnik-Zamolodchikov equation. The draft of the lecture is here .


Lecture 4. The functor \tilde{F} from diagrams decorated by a braided monoidal rigid category C to C was discussed. The trace of an endomorphisms of an objects in a ribbon category was defined. Then quantized universal enveloping algebras of sl_2 and their representations were discussed. The homework is here. The draft of the lecture is here . Here are notes on here distributions and Fourier transform by Roland van der Veen.


Lecture 5 has some basic facts about irreducible representations of quantized universal enveloping algebras for sl-2. It has also the construction of invariants of framed tangles. The draft is here . The homework is here.


Lecture 6 was given by Roland van der Veen. The representation of 3-manifolds as a surgery on a framed link is the main topic of this lecture. The lecture notes and the homework can be found here .

Lecture 7 was a short overview of basic structures of a quantum field theory. The draft of the lecture is here .


Lecture 8 did not take place due my absence. Solutions to some home-works can be found here here .


In Lecture 9 I introduce modular categories and review their basic properties. The draft of this lecture is here .The homework is here


Lecture 10 . The draft of this lecture is here .The homework for this lecture is given in Lecture 12.


Lecture 11. The draft of this lecture is here .
The homework for this lecture is given in Lecture 12.

Lecture 12. The draft of this lecture is here . The notes on connections are here . The homework is here


Lecture 13. The draft of this lecture is here .The homework is here


Last modified: Wednesday, 23-Jan-2008 42:42:42 MET