The String Coffee Table

Next February there is an entire winter school all about Stolz & Teichner’s work on elliptic objects.

From Field Theories to Elliptic Objects
Schloss Mickeln, Düsseldorf (Germany)
February 28 - March 4, 2006
Graduiertenkolleg 1150: Homotopy and Cohomology
Prof. Dr. G. Laures, Universität Bochum
Dr. E. Markert, Universität Bonn
Details : http://www.math.uni-bonn.de/people/GRK1150 $\to$ study program
Program
Announcement and Application

Unfortunately, application deadline was already Dec. 15. Too bad. If anyone knows anything about this, please let me know.

Yesterday Michael Müger gave two very nice talks on the (various flavors of the) Doplicher-Roberts reconstruction theorem, on occasion of his new, drastically simplified proof of this classical result:

Michael Müger
Abstract Duality Theory for Symmetric Tensor $*$-categories
available here.

The original proof by Doplicher and Roberts was spread over several papers and had around 200 pages. The new one fits, self-containedly with an introduction to the category theoretic language included, snugly into 40 pages.

As far as I understood, the main point is that once you use nowadays obvious category-theoretic reasoning and building on ideas by Deligne concerning this problem, the problem becomes pretty easy. Once you know how to do it, that is.

If you wonder why the above paper seems to start with an appendix, note that it is an appendix, namely to

Hans Halvoren
Quantum Field Theory: Algebraic
to appear in
J. Butterfield & J. Earman (eds.)
Handbook of Philosophy and Physics

So, yes, the ‘physical’ motivation for this is algebraic QFT, which - unfortunately - only describes physically uninteresting field theories so far. Therefore, no one of our string theory group bothered to attend the talk.

But, as Müger emphasized, already these physically trivial QFTs give rise to lots of very interesting mathematics, in particular to something that Michael Müger calls a Galois theory of local quantum fields. That was mainly the topic of his second talk, which I am not going to go into right now.

In the last entry I announced our new paper on gerbe holonomy for unoriented surfaces, referring to this as an instance of a (generalized ) notion of equivariant structure on a gerbe. There will be more to say about this, but lest the impression arises that my statement suggests that equivariant structures on gerbes haven’t been considered before, I hasten to emphasize that quite the opposite is true. Here is some selected literature.

We have a new preprint:

K. Waldorf & C. Schweigert & U. S.
Unoriented WZW Models and Holonomy of Bundle Gerbes
hep-th/0512283

Abstract:

The Wess-Zumino term in two-dimensional conformal field theory is best understood as a surface holonomy of a bundle gerbe. We define additional structure for a bundle gerbe that allows to extend the notion of surface holonomy to unoriented surfaces. This provides a candidate for the Wess-Zumino term for WZW models on unoriented surfaces. Our ansatz reproduces some results known from the algebraic approach to WZW models.

As a motivation, recall that an ordinary bundle $E \to M$ on a base space $M$ on which a finite orbifold group $K$ acts by diffeomorphisms $K \ni k : M \to M$ is called equivariant if there are isomorphisms $\phi^k : k^* E \to E$ relating the bundle to any of its images under the pullback induced by the actions of the elements of the orbifold group. These isomorphism have to satisfy a certain compatibility condition. There may be different choices of such isomorphisms and hence different choices of equivariant structures on bundles.

This is relatively straightforwardly catgorified to the context of (bundle) gerbes. A 2-equivariant structure on a (bundle) gerbe $G$ over a base space $M$ is a choice of (bundle) gerbe isomorphisms $\phi^k : k^* G \to G$ (known as ‘stable isomorphisms’ in the case of bundle gerbes) that satisfy the above compatibility condition up to coherent 2-isomorphism.

In the above paper this is not discussed in generality, but for the case where $K = \mathbb{Z}_2$ acts by (possibly orientation-reversing) diffeomorphisms and where the gerbe isomorphism for the nontrivial element $\sigma \in \mathbb{Z}_2$ relates, in the language of bundle gerbes, not $\sigma^* G$ with $G$, but $\sigma^* G$ with $G^*$.

Here $G^*$ is the dual bundle gerbe of $G$, obtained by replacing the line bundle appearing in the definition of $G$ by its dual line bundle. Passing to the dual gerbe essentially corresponds to what in the physics literature is called the orientation involution on the worldsheet. Hence an equivariant structure

for $\sigma \in K = \mathbb{Z}_2$ describes a gerbe on an orientifold, i.e. on an orbifold with ‘additional twist’.

There is a unified 2-categorical picture behind this, which will be discussed in a sequel, but the above paper just postulates such orientifold structures on gerbes and shows that this has the right properties.

An ‘orientifold structure’ on a bundle gerbe in the above sense is called a Jandl structure on a gerbe in that paper. This term was already used for certain involutive structures on Frobenius algebras in hep-th/0306164 which describe CFTs on unoriented (and possibly unorientable) worldsheets. A Jandl structure on a bundle gerbe is a geometric realization of (aspects of) an abstract Jandl structure on an algebra object in a modular tensor category.

The term derives from a rather famous example of German-language experimental poetry by the Austrian poet Ernst Jandl, who in 1995 added the following insight to mankind’s pool of wisdom:

manche meinen
lechts und rinks
kann man nicht velwechsern
werch ein illtum

I haven’t ever before tried to translate poetry, much less experimental poetry, but if the above doesn’t enlighten you the following should give you the idea:

some peopre think
light und reft
cannot be muddred up
what an ellol

Bettel use a Jandr stluctule on youl gelbe to avoid that ellol!

As you may have noticed, I am beginning to learn about the 3D TFT approach towards 2D CFT. Here is a collection of some useful background literature which happens to be available online (thus circumventing the more ambitious task of listing the standard literature available in print).

Here are some notes on the preprint

A. Lauda
Frobenius algebras and planar open string topological field theories
math.QA/0508349

The author

1) points out a relation between adjunctions and Frobenius algebras.

and

2) uses categorified adjunctions in order to describe categorified Frobenius algebras and topological membranes.

Turns out that point 1) relies on the same mechanism which is also responsible for the phenomenon that 2-trivializations of 2-functors give rise to Frobenius algebras.

Thanks to John Baez for making me aware of this paper by means of his TWF.

(I had a pretty long review of this paper almost done yesterday when a computer crash erased it all. Here is a second but inevitably shorter version. )

Here is a post which I have just submitted to sci.math.research, as a followup to John Baez’s latest TWF.

I would be grateful for comments on the following notes.

2D TFT from 2-Transport

Abstract:

It is shown that the parallel surface transport given by certain locally trivialized 2-functors from 2-paths to $\mathbf{Vect}$ reproduces the class of 2-dimensional topological field theories introduced by Fukuma, Hosono and Kawai. In general, every full 2-trivialization of a transport 2-functor gives rise to a Frobenius algebra bundle.

Eric Sharpe watched my feeble attempts (I,II) to say something about derived categories and he decided to help out. Here are his remarks. Also check out his article

E. Sharpe
Derived Categories and D-Branes
Encyclopedia of mathematical physics

Wolfang Lerche today talked about his work on topological strings derived from the so-called Landau-Ginzburg model, their relation to twisted complexes, derived categories, mirror symmetry and all that.

That’s a lot of ground to be covered and I am really supposed to be doing something that I am actually being paid for. But I would like to just quickly mention some key ideas underlying all this.

I’d be grateful for comments on the following notes.

Nonabelian Bundle Gerbes from 2-Transport, Part I: Synthetic Bibundles

Abstract:

It is shown that 1-morphisms between 2-functors from 2-paths to the category $\mathbf{BiTor}(H)$ define bibundles with bibundle connection the way they appear in nonabelian bundle gerbes. An arrow theoretic description of bibundle connections is given using synthetic differential geometry. In a sequel to this paper this will be used to show that pre-trivializations of 2-functors to $\mathbf{BiTor}(H)$ are in bijection with fake flat nonabelian bundle gerbes.

A similar result for abelian gerbes was discussed previously here.

We have a new, prettified and more concise version of our work on 2-connections in 2-bundles, prepared for the proceedings of the Streetfest conference last summer.

J. Baez & U. S.
Higher Gauge Theory
math.DG/0511710

Compared to our previous preprint the discussion has been made more transparent. The interpretation of 2-transitions of 2-connections as 2-functor $p$-morphisms is now included (as discussed here).

For a nice overview of the key ideas see the transparencies of the talk that John is giving next Sunday at the Union College Math Conference.

Update (May, 28, 2006): We have now received the (anonymous) referee report on that paper. Here it is

Report on the paper “Higher Gauge Theory” by John Baez and Urs Schreiber.

This paper is an introduction to ongoing research by the authors of higher dimensional generalization of gauge theory (see [BS]). If to think of gauge theory as the study of parallel transport of 0- dimensional objects (points) the higher gauge theory will deal with parallel transport of 1-dimensional gadgets (strings, paths). The authors “categorify” such standard ingredients of the gauge theory as structure group G, principal G-bundle, connection on it. For some of them category analogs are well know, for others the authors work out what they should be to fit nicely in the gauge picture. The testing ground is the “holonomy functor” which is the assignment to a path in the base of a principal G-bundle with connection the holonomy along it. The higher version of this is the holonomy of paths between paths taking values in the structure 2-group. It should be noted that the whole picture is “smooth” meaning that everything is “internalized” in the category of smooth spaces.

The paper is very well written. The definitions and constructions are carefully motivated and explained. The (mostly omitted) proofs can be found in the preprint [BS]. Aimed at “categorically oriented” mathematical public the paper will fit perfectly into the volume and should be published in its present form.