The n-Category Café

Yep, it’s just Dijkgraaf-Witten theory. The talk was just an exposition of Dan Freed’s ideas.

So, by “derive”, we mean that the path-integral gluing and sewing relations tell us that the category associated to $S^1$ is the category of representations of the quantum double of functions on a finite group, in the same way that the path integral gluing and sewing in the case of 2d topological field theory tells us that the algebra associated to $S^1$ is a Frobenius algebra.

This is all in Dan Freed’s notes. I’ve added, at most, errors in my own notes.

Urs wrote:

Probably a “finite quantum group” is nothing but a finite group with a cocycle on it? (We are talking about Dijkgraaf–Witten theory, I suppose?)

Yes! I can’t resist saying some well-known fun stuff about this subject.

A “finite quantum group” is the name some people use for a finite-dimensional Hopf algebra. An example of this is the group algebra $\mathbb{C}[G]$ of a finite group. Another easy example is the algebra $Fun(G)$ of complex-valued functions on $G$, with pointwise multiplication.

As you probably know, $\mathbb{C}[G]$ and $Fun(G)$ are dual Hopf algebras. The category of representations of $\mathbb{C}[G]$ is the category of representations of $G$. The category of representations of $Fun(G)$ is the category of $G$-graded vector spaces — or if you prefer, vector bundles over $G$.

So, $G$-representation and $G$-grading are dual concepts in a certain sense.

Indeed, if we have an finite abelian group $A$, it has a ‘Pontryagin dual’ — the finite abelian group

$$A^* = hom(A,\mathrm{U}(1))$$

which is equipped with natural isomorphisms

$$Fun(A^*) \cong \mathbb{C}[A]$$ $$\mathbb{C}[A] \cong Fun(A^*)$$

as Hopf algebras. We’ve been talking about this recently elsewhere, so I couldn’t resist pointing out how it fits in here.

More interesting finite quantum groups arise when we take a 3-cocycle

$$\alpha: G \times G \times G \to \mathbb{C}^*$$

and use this to ‘twist’ the associator in the category of $G$-graded vector spaces. The resulting monoidal category turns out to be the category of representations of some more interesting finite-dimensional Hopf algebra.

Even more interesting stuff happens when we form the ‘quantum double’ of any of the Hopf algebras mentioned so far!

The best way to understand the quantum double goes via its category of representations. Just as any monoid has a ‘center’ which is a commutative monoid, any monoidal category $M$ has a ‘center’ $Z M$ which is a braided monoidal category. This is part of the general philosophy of centers!

If we take the center of

$$Rep(\mathbb{C}[G]) \simeq [vector bundles over G]$$

we get the category of equivariant vector bundles over $G$ — that is, vector bundles equipped with a lift of the adjoint action of $G$ on itself. This is the category of representations of a certain Hopf algebra, the ‘quantum double’ of $\mathbb{C}[G]$. Amusingly, this is also the quantum double of $Fun(G)$.

Finally, the most fun occurs when we take the quantum double of a ‘twisted’ group algebra.

But still, from which starting point do you mean to derive that?

As you probably know, we can derive all these Hopf algebras starting from an extended TQFT, the Dijkgraaf–Witten model, and considering its vector spaces and 2-vector spaces of states on various manifolds.

Jeffrey Morton is handing in his thesis tomorrow! It shows how to define the (untwisted) Dijkgraaf–Witten model as an extended TQFT in all dimensions.

It sounds like Jeffrey Morton, A. J. Tolland and Bruce Bartlett have a common interest in this Dijkgraaf–Witten stuff. Maybe they should collaborate!

I love it too, as a simple example of lots of fun ideas.