The String Coffee Table

In case there is anyone out there who knows what a category is but not what the Doplicher-Roberts theorem says, here is a brief outline.

Given a compact group $G$, its category $\mathrm{Rep}_f(G)$ of unitary finite dimensional representation over $\mathbb{C}$ (no guarantee that I give exactly all the right qulifications in the following!) is well known to be a symmetric tensor-$*$-category.

The question is, conversely, given any symmetric tensor $*$-category, is it equivalent to $\mathrm{Rep}_f(G)$ for some compact $G$?

This turns out to be almost true.

Call a concrete symmetric tensor-$*$-category one that is a subcategory (in general not a full subcategory, of course) of that of finite dimensional Hilbert spaces $\mathrm{Hilb}_f$. For this case Tannaka proved in 1939 (in an article in German language but published in a Japanese journal), that, yes, such a $G$ exists and is unique up to isomorphism.

More generally, though, we may have any (‘abstract’) symmetric tensor-$*$-category which does not arise as a subcategory of $\mathrm{Hilb}$. In this case there is more freedom. Essentially (and unsurprisingly for the physicists Doplicher and Roberts but apparently more suprisingly for instance for the mathematician Deligne) this is because of the existence of fermionic symmetries. So this means that we really have to work not just with Hilbert spaces but with graded/super Hilbert spaces.

Given an abstract tensor $*$-category with $\mathrm{End}(1) = \mathbb{C}$ and assuming there is at least one faithful symmetric $*$-preserving tensor functor from this to $\mathrm{Hilb}$ then the category of such functors is a connected groupoid and the symmetric tensor-$*$-category is equivalent to the representation category of the vertex group of this groupoid. This is also due to Tannaka.

Such a functor to $\mathrm{Hilb}$ need not exist. But if we super everything, then it does.

More precisely, let’s call a group $G$ with a specified element $\sigma \in G$ of order 2, $\sigma^2 = 1$, a $\mathbb{Z}_2$-graded group $(G,\sigma)$. Then $\mathrm{Rep}_f(G,\sigma)$ is defined to be equal to $\mathrm{Rep}_f(G)$ as tensor-$*$-categories but with the usual signs introduced when graded Hilbert spaces are commuted.

Combining results by Doplicher-Roberts and Deligne we then find that for every symmetric tensor-$*$-category $C$ with $\mathrm{End}(1) = \mathbb{C}$

1) there does exist a symmetric $*$-preserving functor

2) $E$ is unique up to isomorphism

3) the group of natural automorphisms of $E$ is a compact graded group - that’s the group we are after