The n-Category Café

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

-parameter 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 $X$);

- target 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 transport 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 $G$;

- configuration 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 target space $\mathbf{B}G$ to configuration 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 target 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 parameter spaces. For $n=4$ and trivial background field, this is also done by Girelli-Pfeiffer-Popescu.