The n-Category Café

Ah, the store of knowledge that is this blog. We already have some exposition about how

a local system on a manifold M is a representation of $\pi_1(M)$ (or perhaps on this post that should be the fundamental 1-category).

and its generalisation as constructible sheaves on stratified spaces.

And an ambitious question about

a representation of the fundamental n-category of a stratified space.

David writes:

$$\rho : \pi_1(S,*) \to GL(n,\mathbb{C})$$

So that’s about representations of the fundamental group, right?

Right! I think we can rephrase a certain version of the Riemann–Hilbert problem as asking whether, for any representation of the fundamental group of a punctured Riemann surface, we can find a linear differential equation

$$a_n(z) {d^n f\over d z^n} \; + \; \cdots \; + \; a_1(z) {d f \over d z} + a_0(z) f = 0$$

with holomorphic coefficients $a_i(z)$, such that the monodromies of the solutions realize this representation.

And the answer, by the way, is yes.

There are also versions in higher dimensions.

Will your categorified gauge theory allow you up the ladder, like to 2-representations of the fundamental 2-groupoid of some space?

Maybe so! Certainly Urs and I know what a ‘flat 2-connection’, and the holonomies of such a thing define a homomorphism

$$\rho : \Pi_2(M,*) \to G$$

from the fundamental 2-group of the base manifold $M$ to the gauge 2-group $G$.

We’re a bit less advanced when it comes to understanding ‘2-vector 2-bundles’, so we could replace $G$ by $GL(V)$ for some ‘2-vector space’ $V$. Actually this isn’t so hard formally — we can always use Baez–Crans 2-vector spaces to define 2-vector 2-bundles, and we know what it means to put 2-connections on these. But, we’d need to work with some examples to get a feel for what these things are actually like.

And, I think we reach some really unexplored territory when we ask the question analogous to the Riemann–Hilbert problem.

Then you could pose the Baez–Riemann–Hilbert problem to ask if any 2-rep comes from solutions of some, hmm what is a categorified linear differential equation?

Good question. Solve it, and we’ll be talking about the ‘Corfield–Riemann–Hilbert problem’.

Thanks for the link which describes Gelfand’s work on representation theory.

Some parts which really interested me were:

Gelfand believed that the space $\hat{G}$ of irreducible representations of (a locally compact Lie group) $G$ is a reasonable “classical space”….

… it was natural to guess that the total space $\hat{G}$ is also an algebraic variety.

Could someone could explain the basic results here to me, in layman’s terms? I’m going to expose my shocking ignorance here and admit I only know the basics about representations of compact Lie groups, and my algebraic geometry is pretty shoddy.

The reason I’d like to know is that when one works with 2-representations of Lie groups, it becomes useful to think of the space of irreducible representations as a geometric space. At least, that’s the case for finite groups… and it’s probably also the case for locally compact Lie groups; after all, this seems to be in the spirit of Yetter’s Measurable Categories (though I haven’t grasped this paper yet).

So I’m interested in this theme of thinking of $\hat{G}$ as a geometric space. I know this is really a huge theme and I have no hope of understanding the serious stuff… but just a glimpse? Does $\hat{G}$ form a nice algebraic variety? What’s a nice example?

Bruce wrote:

Could someone could explain the basic results here to me, in layman’s terms? I’m going to expose my shocking ignorance here…

I’m shockingly ignorant too. The buzzword you want is “Plancherel formula”. A very readable introduction is here.

Briefly, if the locally compact group $G$ has a Haar measure that’s invariant under both left and right translations, we say it’s unimodular.

It then makes sense to look at the space $\widehat{G}$ of all irreducible unitary representations of $G$, and ask if there’s a measure on $\widehat{G}$, the Plancherel measure $\nu$, which lets us write down this nonabelian generalization of a formula familiar from Fourier theory:

$$\int_G |f(g)|^2 d g = \int_{\widehat{G}} \| \widehat{f}(\pi) \|^2 d\nu(\pi)$$

(For details of what this formula actually means, read the reference I just gave.)

Murray and von Neumann discovered that such a measure doesn’t always exist! However, if if $G$ is ‘type I’, it does. And in this case, $\widehat{G}$ is metrizable and locally compact.

I don’t want to say what ‘type I’ means. But, you’ll be happy to know that real or complex simple Lie groups qualify, at least if they’re ‘algebraic’ — that is, defined by algebraic equations. For example, $SL(2,\mathbb{R})$ is algebraic, but its universal cover is not.

In fact we don’t need our group to be simple: any reductive algebraic group $G$ has a Plancherel measure on $\widehat{G}$. So, groups like $GL(n,\mathbb{R})$ and $GL(n,\mathbb{C})$ have a Plancherel measure.

So, take your favorite group of this sort, and try to actually get your hands on $\widehat{G}$ and its Plancherel measure. Unless your group is compact, this is usually really hard! This is why people like Harish–Chandra and Vogan are famous: they made real progress on these questions.

I certainly haven’t seen people saying that $\widehat{G}$ is an algebraic variety. Is this true?

I think we could look up stuff about the irreducible unitary representations of $G = SL(2,\mathbb{R})$ and get some sense of what’s going on. You’ve got your discrete series, your principal series, your complementary series, …

… but basically, in this case I think $\widehat{G}$ consists of a countable discrete set of points, two lines, and a closed unit interval, perhaps glued together in some way.