### Dijkgraaf-Witten Theory and its Structure 3-Group

#### Posted by Urs Schreiber

Chern-Simons theory, for every choice of compact Lie group $G$ and class $\tau \in H^4(B G,\mathbb{Z})$, is a theory of volume holonomies. Therefore one might want to understand it in terms of parallel 3-transport # with respect to a suitable structure 3-group.

A toy example for Chern-Simons theory is Dijkgraaf-Witten theory. This instead depends on a group 3-cocycle $\alpha$ of a finite group $G$.

As recently mentioned here, there are attempts to categorify Dijkgraaf-Witten theory by suitably replacing $G$ by some $n$-group.

But do we even understand the ordinary theory in natural terms?

In particular, since Dijkgraaf-Witten theory, too, is a theory of parallel 3-transport, it should really come from a 3-group itself already. A 3-group, that is, which is naturally obtained from an ordinary finite group $G$ and a group 3-cocycle. And preferably in such a way that it illuminates the structure of Chern-Simons theory itself.

Which 3-group is that?

After briefly recalling the idea of Dijkgraaf-Witten theory, I shall argue that the 3-group in question is the inner automorphism 3-group

where $G_\alpha$ is the skeletal weak 2-group whose group of objects is $G$, whose group of morphisms is $U(1)$ and whose associator is determined by the given group 3-cocycle $\alpha \in H^3(G,U(1))$.

This gives a concise way to say what Dijkgraaf-Witten theory is. It also fits in nicely with the claim # that the 3-group $\mathrm{INN}(\mathrm{String}(G))$ governs Chern-Simons theory - since $\mathrm{String}(G)$ # is essentially the Lie analog of $G_\alpha$.

The setup we are dealing with in Dijkgraaf-Witten theory was introduced on pp. 42-46 of

Robbert Dijkgraaf & Edward Witten
*Topological Gauge Theories and Group Cohomology*

Commun. Math. Phys. 129 (1990), 393.

(pdf).

In its simplest form, it defines a map from 3-manifolds to the abelian group $U(1)$, induced by any $U(1)$-valued group 3-cocycle $\alpha \in H^3(G,U(1))$.

Recall that this simply means that $\alpha$ is a map

which satisfies the cocycle condition

meaning that

for all $g,h,k,l \in G$.

Consider now a 3-manifold $X$. Choose any oriented triangulation $T$, decomposing $X$ into tetrahedra.

A (necessarily flat, since $G$ is finite) $G$-connection on $X$ is now represented by an assignment of elements $g(v_1,v_2) \in G$ to edges $v_1 \to v_2$ of the triangulation, such that all triangles commute.

The flatness condition ensures that the coloring of all tetrahedra by group elements is fixed already if three of the tetrahedron’s edges are labelled.

Given any such flat connection, we can hence assign to each tetrahedron an element in $U(1)$, by evaluating our group cocycle $\alpha$ on three such group elements.

Dijkgraaf-Witten theory is the study of the functional obtained by sending each such colored triangulated X to the product of these elements of $U(1)$ over all tetrahedra of the triangulation.

Now I reformulate this procedure as the computation of the 3-transport in a principal 3-bundle with 3-connection on $X$.

It is a theorem (see corollary 44 of HDA V for details) that every 2-group is equivalent to a 2-group $G_\alpha$

- whose group of objects is given by an ordinary group $G$

- which is skeletal as category (all isomorphic objects are in fact equal)

- whose automorphism group of every object is an abelian group $A$

- whose nonvanishing associator (a map sending three objects to a morphism) is a group 3-cocycle $\alpha \in H^3(G,A)$.

Hence the basic data on which Dijkgraaf-Witten theory is built is precisely that classifying 2-groups (up to equivalence).

We are therefore very tempted to try to see if the 3-group that we expect is canonically associated to the Dijkgraaf-Witten model is obtained from $G_\alpha$.

I think to make contact with the more general case of Chern-Simons theory, we want to look at $\mathrm{INN}(G_\alpha)$, the 3-group of inner automorphisms of $G_\alpha$.

But for the present purpose it is helpful to first restrict attention to the much “smaller” and more accessible 3-group

which has objects and 1-morphisms those of $G_\alpha$, and which has precisely one 2-morphism between any ordered pair of parallel 1-morphisms.

A principal 3-transport with values in this 3-group is a lax functor

Here I write $P_1(X) = (X\times X \stackrel{\to}{\to} X)$ for the pair groupoid of $X$, and $\Sigma(G'_3)$ for the 3-category which has a single object “$\bullet$” such that $\mathrm{Hom}(\bullet,\bullet) = G'_3$.

The functor $\mathrm{tra}$ associates elements $g(v_1,v_2)$ of $G$ to edges $v_1 \to v_2$ such that all triangles

commute.

This commutativity is a consequence of the fact that $G_\alpha$ is skeletal. Being lax, the functor $\mathrm{tra}$ also associates an element in $U(1)$ to each triangle, representing a 2-morphism (the compositor) from $g(v_1,v_2)g(v_2,v_3)$ to $g(v_1,v_3)$ - but since the source and target of that 2-morphism must coincide, it requires $g(v_1,v_2)g(v_2,v_3) = g(v_1,v_3)$.

Furthermore, $\mathrm{tra}$ associates a unique element in $U(1)$ to every tetrahdron,

representing the unique 3-morphism filling it.

The crucial point is that the image of this tetrahedron involves the nontrivial associator of $G_\alpha$, which relates

the boundary of the left hand side to the boundary of the right hand side.

Both 1-morphisms here are actually equal. Still, by the general rules, they are related by a nontrivial associator.

If we assume for the moment that $\mathrm{tra}$ is such that the coloring of all faces is by the neutral element in $U(1)$, then we find that the 3-morphism assigned by our 3-transport to any tetrahedron is precisely the appropriate Dijkgraaf-Witten amplitude.

But even is the faces are colored nontrivially we get the Dijkgraaf-Witten volume holonomy: the contribution of the faces cancels out if the boundary of the 3-manifold we do the parallel transport over vanishes. Only the contribution of the associator remains. That’s the Dijkgraaf-Witten amplitude of a closed manifold.

So we can concisely define the Dijkgraaf-Witten amplitude of a flat connection on $X$ as the volume holonomy of a transport 3-functor with values in $\Sigma(G'_3)$.

And the same argument applies, I think, also if we use the full 3-group $G_3 = \mathrm{INN}(G_\alpha)$ (which should come from something like a weak 2-crossed module of groups, which looks like $(U(1) \to U(1)\times G \to G)$ (two copies of $U(1)\to G$, one shifted in degree)).

Finally, notice that the 2-group $\mathrm{String}_G$ is, for (simply connected, compact, simple) Lie groups $G$ essentially what $G_\alpha$ is for finite groups. This is most manifest when one looks at the skeletal version of the Lie 2-algebra of $\mathrm{String}_G$: that’s $\mathrm{Lie}(G)$ of objects, $\mathrm{Lie}(U(1))$ of morphisms and the only nontrivial structure is the associator given by a Lie algebra cocycle $H^3(\mathrm{Lie}(G),\mathbb{R})$.

## Re: Dijkgraaf-Witten Theory and its Structure 3-Group

Please excuse my stepping back right to the beginning, but I haven’t thought much about 3-transport in a 3-connection except in broad outlines… Thanks for describing it.

So I guess I really have a question for John about the background to this. I’ve been wanting for a while to understand better how the associator for this 2-group $G_{\alpha}$ is related to the 3-cocycle in $H^3(G,U(1))$. This gave me an incentive to go back and think about it.

In the 2004/5 sessions of John’s QG seminar, we discussed the Dijkgraaf-Witten model as related to an extended TQFT coming from categorification of the Fukuma-Hosono-Kawai construction. In that case, it’s clear that to the interior of a tetrahedron has to be assigned the associator for the two composition operations on the front and back faces. So when I went to look up Dijkgraaf and Witten’s paper, I was confused to find it presented in terms of this cocycle.

There are a few differences in presentation - one is straightforward, namely that in the seminar we were thinking of the extended TQFT as giving an operator for a cobordism, as opposed to giving amplitudes for manifolds. This is okay, since for closed manifolds, the operator (from $\mathbf{C}$ to $\mathbf{C}$) can be interpreted as a complex number, and indeed it’ll be an amplitude.

The other is the one that I’m less sure of. It has something to do with the fact that in the seminar, we were talking about edges labelled by elements of the 2-algebra $Vect[G]$, and it’s clear what this is. The product in this 2-algebra is a convolution product, and the standard associator is inherited from the associator for tensor products of vector spaces. One can twist this associator using a group cocycle, multiplying the maps of vector spaces by a constant - the cocycle condition guarantees the result still satisfies the pentagon identity and is an associator.

So if we’re given:

then we get a new associator for $Vect[G]$. The Dijkgraaf-Witten amplitude for a given relates the two different 2-morphisms we get by using the two different associators.

I’m not entirely clear on the connection between the 2-algebra and the 2-group point of view, beyond what I said above. There’s some kind of action of $G_{\alpha}$ on $Vect[G]$ - is there an illuminating way to put this?