# Trimester Seminar

Venue: HIM lecture hall, Poppelsdorfer Allee 45 (if not stated otherwise)

## Monday, May 19

16:30 Igor Bakovic: Bigroupoid 2-Bundles

## Friday, May 23

14:00 Adrian Tanasa (MPI): Non-commutative quantum field theory and renormalization

Abstract: We consider two different types of scalar quantum field theoretical models on the non-commutative Moyal space. Both of them are perturbatively renormalizable. For the first of these models, the amplitudes of the associated Feynman graphs are thoroughly looked at. Different representations (the parametric and the Mellin one) of these Feynman amplitudes are presented. The latter further allows the proof of meromorphy of such an amplitude in the space-time dimension. This paves the road for the dimensional renormalization of these theories, which will be overviewed. In the last part of this talk, the second type of scalar model will be presented.

## Friday, May 30

12:30 Danny Stevenson: The classifying space of a topological 2-group

Abstract: Suppose that K is a non-commutative topological group. Then we can form the set H^{1}(M,K), the non-abelian cohomology of M with coefficients in K, or in other words the set of isomorphism classes of principal K-bundles on M. It is well known that under mild hypotheses on M and K there is a space BK, called the classifying space of K, which has the property that there is a bijection between H^{1}(M,K) and the set [M,BK] of homotopy classes of maps from M into BK. All of this can be generalized if we replace the topological group K with a topological 2-group G, i.e a group object in the category of topological groups. Thus we can form the set H^{1}(M,G), the non-abelian cohomology of M with coefficients in the topological 2-group G. It is known from work of Baas, Bokstedt and Kro that there is a topological space BG, called the classifying space of G, with the property that there is a bijection between H^{1}(M,G) and the set [M,BG]. In this talk we will describe some joint work with John Baez giving a simple proof of this fact. We will comment on the relation of the set H^{1}(M,G) to the set of equivalence classes of "G-gerbes" on M, and also comment on an interpretation of String bundles in this context.

14:00 Adrian Tanasa (MPI): Hopf algebra of scalar field theory on the nocommutative Moyal space

Abstract: Perturbative quantum field theory on the noncommutative Moyal space requires ribbon graph representation. The generalization of the Connes-Kreimer Hopf algebra structure of Feynman graphs to the Hopf algebra of these ribbon graphs is presented.

## Monday, June 2

11:00 Evgeny Feigin: Affine algebras and coset models

## Thursday, June 12

11:00, HIM seminar room (Pop. Allee 45, basement): Tilmann Wurzbacher (University of Metz, France): On the (hypothetical) Dirac operator on loop spaces

## Tuesday, June 17

11:00 Constantin Teleman: Towards a topological construction of Chern-Simons theory

## Tuesday, July 15

14:15 Henri Moscovici: Relative Connes-Chern character for manifolds with boundary

## Wednesday, July 16

14:00 Qingtao Chen: Representation of Quantum Groups and new invariants of links

Abstract: The colored HOMFLY polynomial is a quantum invariant of oriented links in S³ associated with a collection of irreducible representations of each quantum group U_q(sl_N) for each component of the link. We will discuss in detail how to construct these polynomials and their general structure, which is the part of Labastida-Marino-Ooguri-Vafa conjecture. The new integer invariants are also predicted by the LMOV conjecture and recently has been proved. LMOV also give the application of Licherish-Millet type formula for links. The corresponding theory of colored Kauffman polynomial could also be developed in a same fashion by using more complicated algebra method.