### Talks in Hamburg, Lectures in Bad Honnef

#### Posted by urs

I had to interrupt my virtual Streetfest at home for a moment. Thursday I visited the Quantum Physics and Geometry Seminar in Hamburg. After that I travelled to Bad Honnef where today I gave the first of two parts of a mini-lecture on ‘nonabelian strings’ aka nonabelian gerbe holonomy. Here are some random notes.

In Hamburg I learned some elements of ‘HQFT’ (homotopy quantum field theory) from Jens Fjelstad. That’s a fun add-on to topological field theory studied in particular by Turaev, Moore and Segal.

Very roughly, consider the domain category of a TFT with

- objects based intervals and based oriented circles equipped with homotopy classes of maps to some pointed topological ‘target’ space

- and morphisms 2D cobordisms that know about these basepoints and which are, too, equipped with homotopy classes of maps to the target.

The base point always gets sent to the distingished point in the target.

Now consider the special case that the target is a $K(G\mathrm{,1})$, i.e. a topological space which has fundamental group $G$ and all other homotopy groups trivial.

In that case circle objects in the above category are nothing but circles colored by an element of $G$.

Now consider the pair-of pants cobordism on which the basepoint runs along a V-shaped diagram. Call the two group elements associated to the two ends of the V $g$ and $h$.

It is then easy to see that the group element at the remaining end of the cobordism has to be the path $\mathrm{gh}$ in $K(G\mathrm{,1})$ , where the order of the factors is determined by the orientation of the circles.

Now move the $h$-end around the $g$-end to the other side. Take care of what happens at the basepoint. Note that $h$ picks up a ‘twist’. It gets conjugated by $g$ and becomes ${\mathrm{ghg}}^{-1}$. This way the product remains inavriant.

There would be much more to say, but I realize that without drawing diagrams this is not that much fun after all. I can’t do that here in the public computer pool at Bad Honnef Physik Zentrum. Maybe I’ll supply some diagrams and a little bit more of information and background later. (Or maybe Jens does, if he reads this.) See for instance

V. Turaev

**Homotopy Field Theory in Dimension 2 and Group-Algebras**

math.QA/9910010.

Sorry for this incomplete account, but I am getting tired. There is lots of interesting literature on this stuff that I should eventually read and which I note here just so that I don’t forget about these links myself.

For instance there is

U. Bunke, P. Turner & S. Willerton

**Gerbes and Homotopy Quantum Field Theories**

Algebraic & Geometric Topology **4** (2004) 407

**Abstract:**

For smooth finite dimensional manifolds, we characterise gerbes with connection as functors on a certain surface cobordism category. This allows us to relate gerbes with connection to Turaev’s 1+1-dimensional homotopy quantum field theories, and we show that flat gerbes are related to a specific class of rank one homotopy quantum field theories.

Now this sounds interesting! Too bad that the printer here has run out of paper…

While I was on the train to Hamburg I was reading the old

F. Gabbiani & J. Fröhlich

**Operator Algebras and Conformal Field Theory**

CMP **155** (1993) 569

in order to learn how type ${\mathrm{III}}_{1}$ factors of von Neumann algebras arise in the representation theory of loop groups (and hence in order to better understand Stolz & Teichner).

This was coincidentally precisely the right introduction to Brunetti’s talk, which was a review of

R. Longo & K.-H. Rehren

**Local Fields in Boundary Conformal Field Theory**

RMP **16** (2004) 909

where this machinery is adapted to CFT with boundary. After the talk I was friendly scolded by R. Brunetti that I should also have read his et al.’s

R. Brunetti, D. Guido & R. Longo

**Modular Structure and Duality in Conformal Quantum Field Theory**

CMP **156** (1993) 201

Oh dear, I have to quit to catch some sleep. Tomorrow morning I am supposed to continue my mini-lecture.

## Re: Talks in Hamburg, Lectures in Bad Honnef

Urs,

are you a slide person, or a board person. If the former - why not post your slides?

We just had two interesting talks here at the Streetfest (the first one I think I slept through) - one on Temperly-Lieb (spelling?) algebras and the other on Segal categories, simplicial monoids and algebraic theories. A review will be forthcoming…

Yrs,

D.