### Dijkgraaf-Witten and its Categorification by Martins and Porter

#### Posted by Urs Schreiber

I was looking at

J. F. Martins and T. Porter
*On Yetter’s invariants and an extension of the
Dijkgraaf-Witten invariant to categorical groups*

(tac)

which João Martins pointed me to in a comment to the entry BF-Theory as a higher gauge theory. (John discussed this paper here a while back).

On the train back home, this inspired me to write the following notes, which happen to be mostly about ordinary Dijkgraaf-Witten theory, but try to put the general context into perspective.

[**Update:** Typed notes on this topic are now here: On $\Sigma$-models and nonabelian differential cohomology]

The authors categorify Dijkgraaf-Witten theory, which is a sigma-model theory with target space a one-object groupoid, to a sigma-model theory with target space a strict one-object 2-groupoid.

Before talking about what these authors do, I recall and extend what we have said here before about the $n$-categorical sigma-model formulation of Dijkgraaf-Witten theory – which is not something these authors consider, but something I enjoy translating everything into, because, in particular, it explains the curious “triangulation weights” that appear in this context.

Recall (for instance from Canonical measures on configuration spaces) Dijkgraaf-Witten theory is the sigma-model type quantum field theory whose

-**par**ameter spaces
are of the form
$P_1(X)$

for $X$ some 3-manifold with chosen triangulation and $P_1(X)$ the groupoid generated from the edges of the triangulation modulo the faces (hence a finite versoin of the fundamental 1-groupoid of $X$);

- **tar**get space is the groupoid
$\mathbf{B}G := \{\bullet \stackrel{g}{\to} \bullet | g \in G\}$
whose geometric realization yields the classifying space
$|\mathbf{B}G| \simeq B G$ of some finite group $G$;

- background field
is a parallel **tra**nsport
pseudofunctor
$\mathrm{tra} : \mathbf{B}G \to \mathbf{B}^3 U(1)
\,,$
where $\mathbf{B}^3 U(1)$ is the strict 3-groupoid with
whose space of 3-morphisms is $U(1)$, which is entirely
determined by its associator, which in turn encodes
precisely a group 3-cocycle on $G$;

- **conf**iguration space is the internal hom
$\mathrm{conf} =
\mathrm{hom}_{\mathrm{Cat}}(P_1(X), \mathbf{B} G)$
wich is the groupoid whose objects are flat $G$-connections
on $X$ and whose morphisms are gauge transformations of
these.

Therefore the **action** of this theory is
the trangression of the background field
to the configuration space
$S =
\mathrm{tg} \mathrm{tra} :=
\mathrm{hom}(P_1(X),\mathrm{tra})
:
\mathrm{hom}(P_1(X),\mathbf{B}G)
\to
\mathrm{hom}(P_1(X), \mathbf{B}^3 U(1))/\sim
\,,$
where $/\sim$ means that we form the set of equivalence
classes of objects.

This is the first point where something interesting happens. Let’s figure out what $\mathrm{hom}(P_1(X), \mathbf{B}^3 U(1))/\sim$ is.

Objects of $\mathrm{hom}(P_1(X), \mathbf{B}^3 U(1))$
are pseudofunctors from
$P_1(X)$ to $\mathbf{B}^3 U(1)$ (weak 3-functors if you
wish, with $P_1(X)$ regarded as a 3-category).
The only nontrivial aspect of these pseudofunctors
are their associators and hence, I think,
they are in bijection with colorings of the tetrahedra in
$X$ by elements in $U(1)$.
Moreover, the morphisms between such pseudofunctors
are in bijection with colorings of *triangles*
in $X$ with elements in $U(1)$, where from a coloring of
tetrahedra and and one of triangles we get to a new
coloring of tetrahedra by multiplying each tetrahedron label
by the labels on its faces, with orientations taken care
of suitably.

What labels equivalence classes of objects in $P_1(X) \to \mathbf{B}^3 U(1)$? If two objects are connected by a morphism, then their “integrated $U(1)$-label” $\prod_{tetrahedra t in X} (label of tetrahedron t)$ is the same.

The converse does not quite hold in general, unless $X$ is sufficiently nice, I think. But if $X$ is the 3-sphere, we can essentially model it by $\mathbf{B}^3 \mathbb{Z}$ and then we’d have $\mathrm{hom}(\mathbf{B}^3\mathbb{Z}, \mathbf{B}^3 U(1))/\sim = U(1) \,.$

So let me assume that $X$ is the 3-sphere from now on. Then
we have found that the *action* of Dijkgraaf-Witten theory,
which was the transgression of the background field on
**tar**get space $\mathbf{B}G$ to
**conf**iguration space, is the map
$S :=
\mathrm{hom}(P_1(X),\mathrm{tra})
:
\mathrm{hom}(P_1(X),\mathbf{B}G)
\to
\mathrm{hom}(P_1(X), \mathbf{B}^3 U(1))/\sim$
which sends each flat $G$-connection $A$ on $X$ to the
number
$S : A \mapsto
\prod_{tetrahedra t in X}
\mathrm{tra}(A(t))
\in U(1)
\,.$

This action we want to integrate over all of configuration space. To do so we should regard it as a functor on configuration space and somehow for its colimit over all of configuration space. The best current idea we have for what that actually means is:

we sum the action over all objects of configuration space
and weigh each contribution by the Leinster measure
$d\mu$
of configuration space, which here amounts just to weighing
it by the *groupoid cardinality*

$d\mu : A \mapsto (number of morphisms starting at A) \,.$

We get one morphism starting at any flat $G$-connection on $X$ for every choice of element in $G$ per vertex of $X$. If there are $v$ many vertices, this means that $d\mu(A) = \frac{1}{|G|^v} \,.$

So, finally, the **partition function**
of Dijkgraaf-Witten theory for given parameter space $X$,
which we want to think of as
$Z(X) = colim_{hom(P_1(X),\mathbf{B}G)}
\mathrm{tg}(\mathrm{tra})$

is

$Z(X) = \sum_{flat G-connections A} \prod_{tetrahedra t in X} \mathrm{tra}(A(t)) \; \frac{1}{|G|^v} \,.$

Unfortunately, despite lots of thinking and
conjecturing and
toy-example-checking
here on the $n$-Café
it is still not *quite* clear how the *morally* right
way to think of a colimit here makes strictly sense as a
colimit, and so for the time being we have to invoke
by hand the rule that “path integral means sum of transgressed
background field weighted by the Leinster measure”.
But still, it’s pretty cool.

(I should add that the picture which I am drawing here has been painted a lot by Simon Willerton and Bruce Bartlett, based on old work by Quillen. What I am adding are some, I think, refinements concerning aspects like the background field, parallel transport and transgression.)

Okay, that sets the stage. This is Dijkgraaf-Witten theory. One shows that this $Z$ is a homotopy invariant of 3-manifolds (and hence in particular independent of the chosen triangulation). In order for this to be true, the precise nature of the Leinster measure $d\mu(A) = \frac{1}{|G|^v}$ is crucial.

Now we want to know: what happens to this setup
if **tar**get space is not necessarily
a (finite) 1-group
$\mathbf{B}G$
but an arbitrary strict (finite) 2-group
$\mathbf{B}G_{(2)}$
where $G_{(2)}$ comes from the crossed module
$(H \stackrel{t}{\to} G \stackrel{\alpha}{\to} Aut(G))
\,.$

This is best approached in two steps:

- first using just the *trivial* background field,
concentrating on what happens to the path integral
domain and Leinster measure;

- then introducing also nontrivial background fields.

The first step was done by D. Yetter in his TQFTs from homotopy 2-types. There he finds what we’d call the “Leinster 2-measure” for the path integral, namely the combinatorial weight $d\mu$ that depends on the chosen triangulation, by fixing some weighting, and then demonstrating that it yields a partition function $Z$ which is a homotopy invariant.

Martins and Porter then also include a nontrivial background field in this context.

They also generalize the proof that the partition
function $Z$ is a homotopy invariant to higher dimensional
**par**ameter spaces. For $n=4$ and trivial
background field, this is also done by
Girelli-Pfeiffer-Popescu.

## Re: Dijkgraaf-Witten and its Categorification by Martins and Porter

Proposition.The combinatorial weighting for the “categorified DW” theory introduced by Yetter (and studied by Girelli, Pfeiffer, Popescu, Porter, Martins etc.) is indeed the Leinster measure of the corresponding configuration space.Recall that for ordinary DW theory for finite group $G$ config space is the groupoid $\mathrm{hom}_{Cat}(P_1(X), \mathbf{B}G)$ and that since there are $|G|^\nu$ many ways to assign group element to the $\nu$ vertices of $X$, there are $|G|^\nu$ many natural transformations starting at any functor $P_1(X) \to \mathbf{B}G$ and therefore the Leinster measure of the hom-groupoid is $d\mu = \frac{1}{|G|^\nu} \,.$

Now, Yetter et al find that when target space is replaced by $\mathbf{B}G_{(2)}$ for $G_{(2)}$ the finite strict 2-group coming from the strict crossed module $H \stackrel{t}{\to} G \stackrel{\alpha}{\to} Aut(H) \,,$ then the corresponding combinatorial weight becomes $|G|^{- \nu} |H|^{-\nu + \lambda} \,,$ where $\nu$ is the number of vertices, as before, and where $\lambda$ is the number of edges in $X$.

(See for instance theorem 2.13 on p. 127 of Martins-Porter’s On Yetter’s invariants or the discussion below theorem III.2 on p. 6 of Girelli-Pfeiffer-Popescu’s BF to BFCG (while the factor is not manifest in their expression (15))).

I am claiming that this factor is precisely the Leinster measure on the configuration space

$conf = \mathrm{hom}_{2Cat}(P_1(X), \mathbf{B}G_{(2)})/\sim$

Here I am taking, to replace awkward pseudofunctors with nicer strict 2-functors, $P_1(X)$ now to be the strict 2-groupoid generated from the triangles in $X$ (which is a manifold with chosen triangulation, recall).

Then the above hom-groupoid is supposed to be the 1-groupoid obtained by passing to equivalence classes of 1-morphisms in the 2-groupoid whose objects are strict 2-functors $P_1(X) \to \mathbf{B}G_{(2)}$, whose morphisms are pseudonatural transformations of these, and whose 2-morphisms are modifications of those.

To prove this, we need to compute how many 1-morphisms in $conf$ there start at any given object.

But that’s easy: a 1-morphism $g : A \to A'$ in the 2-groupoid of strict 2-functors $P_1(X) \to \mathbf{B}G_{(2)}$ is a transformation whose component is an assignment of squares in $\mathbf{B}G_{(2)}$ to edges in $X$:

$g : (x \stackrel{\gamma}{\to} y) \;\;\;\; \mapsto \;\;\;\; \array{ \bullet &\stackrel{A(\gamma)}{\to}& \bullet \\ \downarrow^{g(x)} & \Downarrow^{g(\gamma)} & \downarrow^{g(y)} \\ \bullet &\stackrel{A'(\gamma)}{\to}& \bullet } \,.$

By the general Yoga of strict 2-groups, we know that the square on the right is entirely and precisely fixed by

- the source 2-functor value $A(\gamma) \in G$ on the top;

- the two transformation elements $g(x), g(y) \in G$ on the two sides;

- and the label $g(\gamma) \in H$ of the 2-morphism filling the square.

This uniquely determines the value $A'(\gamma)$ of the pseudofunctor which is the target of the transformation whose component $g$ is.

Therefore, clearly, there are precisely

$|G|^\nu |H|^\lambda$

many 1-morphisms in $\mathrm{hom}_{2Cat}(P_1(X), \mathbf{B}G_{(2)})$

starting at any given object.

But some of them are related by 2-morphisms

To figure out many, we need to compute how many modifications $\eta : g \to g'$ there are starting at any transformation $g$.

But again the same logic applies: such a transformations is given by a component map which sends vertices in $X$ to 2-cells in $\mathbf{B}G_{(2)}$:

$\eta : x \mapsto \array{ & \nearrow \searrow^{g(x)} \\ \bullet & \Downarrow^{\eta(x)}& \bullet \\ & \searrow \nearrow_{g'(x)} }$ for $\eta(x) \in H$.

Again one sees, by drawing the relevant naturality tin-can diagram which I won’t bother to do here in MathML, that for each such choice of assignments of elements in $H$ to vertices in $X$ there is precisely one modification $\eta : g \to g'$.

This means that we have

- $|G|^\nu |H|^\lambda$ 1-morphisms emanating at each object

and

- $|H|^\nu$ 2-morphisms emanating at every morphism.

The Leinster measure in this situation is $d\mu(A) = |G|^{-\nu} |H|^{\nu - \lambda} \,,$ precisely the combinatorial factor which people found makes the Yetter state sum model a topological invariant.

[

Update: A more detailed discussion of this is now in On $\Sigma$-models and nonabelian differential cohomology.]