### Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

#### Posted by Urs Schreiber

*guest post by Bruce Bartlett*

Recently, Richard Hepworth gave a seminar at Sheffield:

*2-Vector Bundles and the Volume of a Differentiable Stack*,
(pdf, 9 pages).

**Abstract:** This seminar is an account of Alan Weinstein’s recent
paper The Volume of a
Differentiable Stack. I’ll explain that differentiable stacks
are a generalization of smooth manifolds and that they crop up in many
interesting situations, like the study of orbifolds or the study of
flat connections. Just as every manifold has a tangent bundle, every
stack has a tangent something, and I’ll explain that the something in
question is a bundle of Baez–Crans 2-vector spaces. These 2-vector
bundles are often horrible compared with vector bundles, but they
still admit a ‘top exterior power’. We’ll see that sections of this
top exterior power can be treated just like volume forms on a
manifold, and in particular can be integrated to define the volume of
a stack.

In other words, Richard’s talk is all about the stuff that we were discussing a few weeks ago here at the n-category cafe when Weinstein’s paper came out. Richard kicked off that discussion by saying that Weinstein’s construction for defining the volume of a stack could be placed into a more conceptual framework:

1) Define the tangent bundle of the stack. It is not a vector bundle over the original stack, but rather a form of ‘Baez-Crans 2-vector bundle’. For more details about tangent stacks, see Richard’s recent paper, Vector Fields and Flows on Differentiable Stacks.

2) Take the ‘top exterior power’ of the tangent 2-vector bundle to obtain a line bundle over the stack. Weinstein considered a line bundle $Q_A = \Lambda^{top}(T X)^*$, where $A$ denotes a Lie algebroid. Richard’s whole point is that

this line bundle can perhaps be thought of more conceptually as the top exterior power of the tangent 2-vector bundle!3) A volume form on the stack is a nowhere-vanishing section of this top exterior power line bundle.

4) Integrate!

So, the game of categorification is alive and well in the world of differentiable stacks.

One thing that interests me: do the vector fields on a differentiable stack form a Lie 2-algebra, and can we gain conceptual capital this way?

## Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Neat stuff! Everything is starting to gradually fit together!

By the way, someone wrote:

It would be nice to actually see the link.