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Atle Hahn, Group of Mathematical Physics of the University of Lisbon

From the Chern-Simons path integral to the Reshetikhin-Turaev invariant

Topological Quantum Field Theory Seminar

Atle Hahn, Group of Mathematical Physics of the University of Lisbon

From the Chern-Simons path integral to the Reshetikhin-Turaev invariant

The study of the heuristic Chern-Simons path integral by E. Witten inspired (at least) two general approaches to quantum topology. Firstly, the perturbative approach based on the CS path integral in the Lorentz gauge and, secondly, the "quantum group approach" by Reshetikhin/Turaev. While for the first approach the relation to the CS path integral is obvious for the second approach it is not. In particular, it is not clear if/how one can derive the relevant R-matrices or quantum 6j-symbols directly from the CS path integral. In my talk, which summarizes the results of a recent preprint, I will sketch a strategy that should lead to a clarification of this issue in the special case where the base manifold is of product form. This strategy is based on the "torus gauge fixing" procedure introduced by Blau/Thompson for the study of the partition function of CS models. I will show that the formulas of Blau/Thompson can be generalized to Wilson lines and that the evaluation of the expectation values of these Wilson lines leads to the same state sum expressions in terms of which Turaev's shadow invariant is defined. Finally, I will sketch how one can obtain a rigorous realization of the path integral expressions appearing in this treatment.

George Zoupanos, Physics Department, National Technical University, Athens

Dimensional Reduction of Gauge Theories over continuous and fuzzy coset spaces

We review the dimensional reduction of N=1 higher dimensional Gauge Theories over Coset Spaces with emphasis on the possibility to obtain four-dimensional GUTs with chiral fermions and softly broken supersymmetry. Next we consider gauge theories defined in higher dimensions, where the extra dimensions form a fuzzy space (a finite matrix manifold). We emphasize some striking features emerging such as (i) the appearance of non-abelian gauge theories in four dimensions starting from an abelian gauge theory in higher dimensions, (ii) the fact that the spontaneous symmetry breaking of the theory takes place entirely in the extra dimensions and (iii) the renormalizability of the theory.

Louis Kauffman, Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago

Virtual Knot Theory

New developments in Virtual Knot Theory.

John Baez, Department of Mathematics, University of California at Riverside

Topological 2-Groups and Their Classifying Spaces

Categorifying the concept of topological group, one obtains the notion of a topological 2-group. This in turn allows a theory of "principal 2-bundles" generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group $G$ and a space $M$, principal $G$-bundles over $M$ are classified by either the Cech cohomology ${H}^{1}(M,G)$ or the set of homotopy classes $[M,BG]$, where $BG$ is the classifying space of $G$. Here we review work by Bartels, Jurco, Baas-Bökstedt-Kro, and others generalizing this result to topological 2-groups. We explain various viewpoints on topological 2-groups and the Cech cohomology ${H}^{1}(M,G)$ with coefficients in a topological 2-group $G$, also known as "nonabelian cohomology". Then we sketch a proof that under mild conditions on $M$ and $G$ there is a bijection between ${H}^{1}(M,G)$ and $[M,B\mid G\mid ]$, where $B\mid G\mid $ is the classifying space of the geometric realization of the nerve of $G$. Applying this result to the ''string 2-group" $\mathrm{String}(G)$ of a simply-connected compact simple Lie group $G$, we obtain a theory of characteristic classes for principal $\mathrm{String}(G)$-2-bundles.

Tim Porter, University of Wales, Bangor, UK

The “Crossed” technology and its applications

In this short series of lectures I will introduce crossed modules and crossed complexes, both algebraically and topologically, will look at their relevance for combinatorial group theory, the theory of syzygies and group cohomology, and then will head for higher order objects namely 2-crossed modules, and related complexes. In the final parts I will introduce some of the constructions of non-Abelian cohomology, sheaves, torsors and Bitorsors and consider the interaction between the crossed gadgetry of the earlier lectures and this area. ##### References

Schedule:

Lecture 1: Wednesday, 3rd December, 2008, 11h30 - 13

Lecture 2: Wednesday, 3rd December, 2008, 16h30 - 18

Lecture 3: Thursday, 4th December, 2008, 14h30 - 16

Lecture 4: Thursday, 4th December, 2008, 16h30 - 18

- The pdf file at the start of the following page, which includes discussions: n-Category Café
- Slightly updated version of the crossed menagerie file: Crossed Menagerie

Schedule:

Lecture 1: Wednesday, 3rd December, 2008, 11h30 - 13

Lecture 2: Wednesday, 3rd December, 2008, 16h30 - 18

Lecture 3: Thursday, 4th December, 2008, 14h30 - 16

Lecture 4: Thursday, 4th December, 2008, 16h30 - 18

Tim Porter, University of Wales, Bangor, UK

The “Crossed” technology and its applications - II

(see Lecture 1)

Tim Porter, University of Wales, Bangor, UK

The "Crossed" technology and its applications - III

(see Lecture 1)

Tim Porter, University of Wales, Bangor, UK

The "Crossed" technology and its applications - IV

(see Lecture 1)