The String Coffee Table

[Update 1. Feb 2006: I have corrected a couple of typos, improved the discussion of Morita equivalence and added a crucial reference to a a review paper. ]

Here is a transcript of the first of the talks mentioned in the previous entry.

The talk pretty closely followed the review paper

I. Moerdijk
Orbifolds as Groupoids: an Introduction
math.DG/0203100 .

Suggested background reading is the book

I. Moerdijk & J. Mrčun
Introduction to Foliations and Lie Groupoids
Cambridge studies in advanced mathematics 91 (2003).

In that book orbifolds are first described (sections 1 and 2) in terms of local charts, i.e. as spaces locally diffeomorphic to $\mathbb{R}^n/G$ for some $G$ which may change from point to point. In section 5 Lie groupoids are introduced and the relation to orbifolds is briefly discussed in 5.6.

Suggested further reading is the latest paper by Lupercio and Uribe (whose work I mentioned recently)

Ernesto Lupercio, Bernardo Uribe, Miguel A. Xicotencatl
Orbifold String Topology
math.AT/0512658

as well as a an upcoming book

Leida, Ruan, Adém
Orbifolds and String Topology
(to appear).

These deal with loop spaces of orbifolds. This will, hopefully, be the topic of tomorrow’s second lecture.

Ieke Moerdijk is giving a series of talks in Hamburg on the general topic of orbifolds.

The first lecture today was mainly concerned with highlighting the right way to think about orbifolds, that which allows to neatly talk about maps between them and about extra structure on them.

If I may rephrase this point of view in my words, I would state it in terms of a slogan as

Orbifolds are to be thought of as decategorified groupoids.

Apart from technical issues related to the fact that one wants everything to be smooth in a suitable sense, the simple (and well known) idea is the following.

To every topological space $M$ on which some group $G$ acts (or rather, on which several such groups act locally) , we may associate the action groupoid

whose objects are the points of $x$ and which has a morphism between $x$ and $y$ if and only if there is a $g\in G$ such that acting with $g$ on $x$ yields $y$.

To divide out by the action of $G$, i.e. to identify points on a commong $G$-orbit amounts to nothing but passing to the isomorphism classes $|\mathbf{G}_{M,G}|$ of the groupoid $\mathbf{G}_{M,G}$.

Passing to isomorphism classes is known as ‘decategorification’. Hence, up to some technical fine print, orbifolds are decategorified groupoids.

As always when some decategorified structure is encountered, one obtains a deeper understanding of its nature by undoing the decategorification and studying the original category it came from. And that’s what Ieke Moerdijk is emphasizing is the right point of view also in the case of orbifolds.

I plan to report on the details of his talks as soon as possible.

(This entry has been momentarily removed. I need to correct something.)

Motivated by some private correspondence on the content of the previous entry, (String) Physics from (Higher) Algebra, I would like to improve some of the statements made there.

1) First of all, the phrasing in the previous entry suggested that the 2-category of bimodules internal to $\mathrm{BiMod}(R_2)$ is equivalent to the 2-category of left $R_2$-modules when Ostrik’s theorem applies. Actually, what is pretty obvious is only that the former sits inside the latter in a certain sense. That’s good enough for the general argument I was interested in. Still, I suspect that the two really are equivalent, but that’s not so obvious. I have tried to cleanly describe the situation and the corresponding conjecture in these notes:

$\;\;\;\;$Module Categories and internal Bimodules

If anyone feels like helping to prove or disprove the conjecture stated at the very end of these notes, please let me know.

2) Second, the formulations in the previous entry didn’t make it clear which (2-)ring action the morphism in the various (2-)categories were supposed to respect. I give a new, somewhat more accurate re-formulation of the general idea below. Further refinements are obviously still necessary.

Finally, I should mention that Jeffrey Morton has a new preprint concerned with categorifying quantum mechanics:

Jeffrey Morton
Categorified Algebra and Quantum Mechanics
math.QA/0601458 .

Superficially this looks very different from what I am talking about here. But I think it might be related. However, I have not fully digested that paper yet.

I should finally begin to learn a little bit about what goes under the curious name topological conformal field theory (TCFT). Kevin Costello had already pointed me to his work on TCFT last year, and, more recently, Aaron Bergman has emphasized the relevance of this work here.

As long-time readers of this blog know, I have the strange idée fixe of realizing string physics as a categorification of point particle quantum mechanics.

Last week I talked about a theorem by Viktor Ostrik which stated roughly that more or less every 2-module of a 2-ring $R_2$ is equivalent to a category of ordinary modules of some algebra object internal to $R_2$. I note that this provides one more puzzle piece in a picture of strings in terms of categorified quantum mechanics.

Today the third session of Nikita Nekrasov’s lecture was made available online. It contains some new things. At least I had not seen them before.

A while ago we had talked about (I, II, III) how gerbes of chiral differential operators make an appearance in Berkovits’ formulation of the superstring in terms of a curved $\beta-\gamma$-system of pure spinors. As Peter Woit has already noted on his blog, Nikita Nekrasov is currently lecturing on his latest preprint

Nikita Nekrasov
Lectures on curved beta-gamma systems, pure spinors, and anomalies
hep-th/0511008

in Jerusalem, and a quicktime video of his lectures is available online.

I have watched the first two parts, hoping to see some hints concerning open questions like those mentioned here. But no luck so far…

Today appeared a preprint

Andreas Thom
A Remark about the Connes Fusion Tensor Product
math.OA/0601045

which reviews technicalities in the definition of a certain ‘fusion’ operation on bimodules over von Neumann algebras. This operation is due to Alain Connes and is called Connes Fusion at least since Antony Wassermann’s article

Antony Wassermann
Operator Algebras and Conformal Field Theory III
math.OA/9806031.

For von Neumann algebras coming from positive energy representations of loop groups Connes Fusion is the rigorous version of the fusion operation of primary fields in conformal field theory. It plays a crucial role in geometric approaches to elliptic cohomology.

I continue where I left off yesterday.

As far as I am aware, there are two rigorous formulations of 2D conformal field theory.

1) Segal said CFT is a functor from a suitable category of 2D cobordisms to $\mathrm{Vect}$. Stolz and Teichner refined this by, essentially, decomposing vector spaces into bimodules. To them, a CFT is a 2-functor from conformal surface elements to a weak 2-category (bicategory) of bimodules.

2) Fuchs, Runkel, Schweigert et al. on the other hand realize CFT in terms decorations of dual triangulations of surfaces by means of modular tensor categories.

Is there a conceptual link between these two approaches? What do 2-functors to bimodules have to do with Wilson graphs/Feynman diagrams with labels in tensor categories?

I am getting the impression that 2) is what you get by locally trivializing 1) in a suitable sense. At the heart of this mechanism seems to be a deep relation between the keywords in the title of this entry, namely between algebra bimodules, adjunctions and a construction known as the internal $\mathrm{Hom}$. I have ever so briefly mentioned this in the context of Aaron Lauda’s recent preprint on Frobenius algebras and ambidextrous adjunctions. Here I would like to talk about more details that I have learned about, meanwhile.

Many puzzle pieces are beginning to fall into place, but I know I am only scratching the surface. For instance, below I make a conjecture, albeit a rather obvious one, about the relation between the internal $\mathrm{Hom}$ and Eilenberg-Moore objects. If anybody knows more about this, please drop me a note!

A while ago I had discussed a preprint by Andreas Gustavsson dealing with ‘nonabelian strings’. Now he has a followup:

Andreas Gustavsson
The non-Abelian tensor multiplet in loop space
hep-th/0512341 .