### HIM Trimester Geometry and Physics, Week 1

#### Posted by Urs Schreiber

I had thought I could get some work done here, but I hardly find the time to reply to my email.

But there are lots of things that would deserve blogging about.

The main two activities currently are one daily seminar on supergeometry in preparation for the Stolz-Teichner program next week, the other is a daily inernal seminar where we teach each other whatever comes to mind.

**Vertex operator algebras from Segal’s axioms**

In the internal seminar I very much enjoyed a talk by Liang Kong on how vertex operator algebras “arise from Segal’s CFT axioms” if you massage those a bit. This is due to theorems by Huang and Huang-and-Kong, and gives a rather nice geometric description of the otherwise rather unwieldy definition of a VOA. The following literature information thanks to Liang Kong:

The main statement is in Yi-Zhi Huang’s book:

*Two-dimensional conformal geometry and vertex operator algebras*
Progress in Mathematics, Vol. 148, Birkhauser Boston, 1997.

There is an online paper cover the main result in the book: arXiv:q-alg/9504019.

For the Theorem that proper completion of VOA gives algebra over a true operad, read the following two papers: arXiv:math/9808022, arXiv:math/0010326

For the open-string case: arXiv:math/0308248

For the open-closed case and higher genus: arXiv:math/0610293

**Supergeometry**

We are following Deligne-Morgan, *Quantum fields and strings, a course for mathematicians* (#). Today it was my turn with talking about integration over supermanifolds. While I followed it, I was a bit unsatisfied with their presentation, I must say. Here is my attempt:

*Integration over Supermanifolds*

(pdf, 6 pages)

Afterwards I had some discussion about *supergroupoids*. I was talking about integrating super $L_\infty$-algebras to super $\infty$-groupoids recently in

*On action Lie $\infty$-groupoids and action $L_\infty$-algebroids*

(blog, pdf)

but there I use a notion of superspace more general than that of supermanifolds.

The only reference mentioning the obvious idea of looking at groupoids internal to supermanifolds that we could find is

Rajan Mehta
*Supergroupoids, double structures, and equivariant cohomology*

(arXiv:math/0605356).

That thesis also consider groupoids internal to “Q-manifolds” (which I would call groupoids internal to $L_\infty$-algebroids).

Incidentally, here is a related question that we happened to come across also in discussion with Constantin Teleman over tea:

when you google “supergroupoid” you get precisely two non-identical hits: the definition by Mehta (groupoid internal to supermanifolds) and the definition by John Baez (groupoid with $\mathbb{Z}_2$-grading).

Similarly, when asked what can be said about his theorem when groups are replaced by supergroups, Constantin Teleman assumed a supergroup is a group with a $\mathbb{Z}_2$-grading. Other people would think it is a group internal to supermanifolds.

“Supergroups” in the sense of groups with a $\mathbb{Z}_2$-grading also appear in the nice proof of Doplicher-Roberts.

While it might seem like a clash of terminology, we know, on the other hand, that super-things in the first sense tend to live in supercategories in the second sense! Super D-brane categories, for instance, live in supercategories in the second sense, as I described in Lazaroiu on $G$-flows on categories.

So something interesting is going on. But I am not completely sure yet what.

## Re: HIM Trimester Geometry and Physics, Week 1

Thanks Urs this is really useful.