Question

What are some examples of theorems about topology or differential geometry that have been proven using topological/differentiable stacks, or, some examples of proofs made easier by them? I'm well aware of several statements made more beautiful in the LANGUAGE of stacks, but, I'm looking for a concrete application.

Answer 1

I would like to point out that stacks are "just" higher analogues of sheaves - a very basic tool to arrange structure. The same is true for topological or differentiable stacks. So I think everybody who expects amazing applications of stacks should be able to name an equally amazing application of a sheaf. (I am not saying that those don't exist!)

That said, let me mention an application. In view of the fact that you didn't get any answers so far (apart from your own), I hope it's not too inappropriate to take one from my own research. It applies abelian gerbes with connection to lifting problems for principal bundles.

I hope the following specifications qualify the theorem below as application: its statement does not involve any stacks or gerbes, just "basic" differential geometry. Its proof, however, is a simple composition of two gerbe-theoretical theorems.

Theorem. Let $M$ be a connected smooth manifold, let $P$ be a principal $G$-bundle with connection over $M$, let $\hat G$ be a central extension of $G$ by an abelian Lie group $A$, and let $\rho \in \Omega^2(M,\mathfrak{a})$ be a 2-form. Then, there exists a principal $A$-bundle $\mathcal{L}_P$ over $LM$ with a connection and with a fusion product, and a bijection between

1. isomorphism classes of lifts of the structure group of $P$ from $G$ to $\hat G$ with compatible connection of scalar curvature $\rho$, and

2. smooth sections of $\mathcal{L}_P$ that preserve the fusion product and pull back the connection to the transgressed 1-form $L\rho \in \Omega^1(LM,\mathfrak{a})$.

Of course some concepts that appear here would need some more explanation - but that's not the point. Let me better point out how gerbes with connection come into the picture. We employ two results from gerbe theory:

1. Associated to every lifting problem posed by a bundle $P$ is an $A$-gerbe over $M$, called the "lifting gerbe" and denoted $\mathcal{G}_P$. This gerbe represents geometrically the obstruction against lifts. Moreover, the actual lifts are in equivalence with trivializations of $\mathcal{G}_P$. The same works if one wants to include connections into the lifting problem. These are results of Murray and Gomi.

2. The category of $A$-gerbes with connection over $M$ is equivalent to a certain category of principal $A$-bundles with connection over $LM$ which are additionally equipped with "fusion products". The equivalence is established by a transgression functor, which has been introduced by Brylinski and McLaughlin. It takes trivializations of gerbes to sections of bundles.

Now, define $\mathcal{L}_P$ as the transgression of $\mathcal{G}_P$. Since transgression is an equivalence of categories, it is a bijections on Hom-sets, and this bijection is exactly the statement of the theorem.

Ok, in order to complete my claim that this is an application, I should probably mention an example where the theorem is useful. That's the case for $spin$ and $spin^c$ structures on manifolds, and I have learned about it from Stephan Stolz and Peter Teichner. In the case of $spin$ structures, $\mathcal{L}_P$ is a $\mathbb{Z}_2$-bundle over $LM$ and plays the role of the orientation bundle of $LM$. Since $\mathbb{Z}_2$ is discrete, all the connections disappear and forms are identically zero. So, the theorem says that isomorphism classes of $spin$ structures on $M$ are in bijection to "fusion-preserving orientations" of $LM$. In the $spin^c$ case, a similar statement follows that additionally includes the scalar curvature of the $spin^c$-structures.

Answer 2

Whilst asking this, I nearly forgot that one application does come to mind:

http://www.math.fsu.edu/~aluffi/archive/paper325.pdf

In this paper Behrang Noohi shows how to use topological stacks to calculate the fundamental group of the quotient of a topological space by a group(oid) action by using fixed-point data.