The String Coffee Table

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.