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November 8, 2006

Flat Sections and Twisted Groupoid Reps

Posted by Urs Schreiber

Tomorrow, Simon Willerton will be visiting Hamburg, giving a talk on Topological Field Theory and Gerbes. I have long meant to say something about one of his last papers on this subject, namely

Simon Willerton
The twisted Drinfeld double of a finite group via gerbes and finite groupoids

While this paper is motivated by the desire to understand a certain groupoid algebra, the “Drinfeld double”, it actually does so by embedding the problem into a much larger context, namely the categorical description of topological field theory, and in particular of Dijkgraaf-Witten theory.

After sketching a general picture of topological field theory, which is a finite analog (finite in the sense of finite groups instead of Lie groups and finite groupoids instead of topological spaces) of the big picture that Michael Hopkins sketched for Chern-Simons theory, the paper demonstrates a couple of interesting cross-relations between apparently different topics that are obtained this way.

In particular, and that shall be the aspect which I will concentrate on here, Simon Willerton makes the point that one should think of representations of a finite groupoid which are twisted by a groupoid 2-cocycle, as flat sections of a gerbe with flat connection on that groupoid.

Below, I will briefly review some relevant aspects from the paper. Then I would like to propose a way to understand these spaces of sections in an arrow-theoretic manner, along the general lines that I talked about recently # in the context of categorified quantum mechanics.

My observation is that

the space of flat sections “over the pp-point” of an nn-bundle ((n1)(n-1)-gerbe) with connection on a space XX is the space of natural transformations ee

(1) Id 1* [d p,P .(X)] e [d p,nVect] tra *, \array{ &\nearrow \searrow^{\mathrm{Id}_{1*}} \\ [d^p, P_.(X)] &e \Downarrow \;& [d^p, n\mathrm{Vect}] \\ & \searrow \nearrow_{\mathrm{tra}_*} } \,,

where d pd^p is the pp-particle and tra\mathrm{tra} the transport functor # of the nn-bundle, as described here.

I claim that for the special cases of flat (n=1)(n=1)- and (n=2)(n=2)-bundles with connection this reproduces the twisted groupoid representations discussed by Simon Willerton in section 2.2 and 2.3 of the above paper.

For the present purpose we are interested in nn-connections on trivial nn-bundles. This means that we may choose our domain path category to be

(1)P .(X):=Π(X) P_.(X) := \Pi(X)

the fundamental groupoid of XX, based on a couple of chosen points in XX, at least one for each connected component. (All choices will lead to equivalent groupoids.)

An nn-connection on a flat nn-bundle over XX is then a pseudofunctor

(2)tra:P .(X)T, \mathrm{tra} : P_.(X) \to T \,,

where TT is some codomain nn-category. (Equivalently, we could turn P .(X)P_.(X) into an nn-category by filling in higher simplices and look at proper nn-functors.)

In the present case the codomain TT will be nothing but

(3)T=Σ n(U(1)), T = \Sigma^n(U(1)) \,,

the nn-fold suspension of the group U(1)U(1). This has U(1)U(1)-worth of nn-morphisms with everything else being trivial. So this describes flat principal Σ n(U(1))\Sigma^n(U(1))-nn-bundles ((n1)(n-1)-gerbes) on XX.

Using the nature of pseudofunctors, it is now easy to see that such nn-functor tra\mathrm{tra} is nothing but a groupoid nn-cocycle on Π(X)\Pi(X). We have talked about that rather recently here. It is a special case of how pp-functors with values in Σ p(U(1))\Sigma^p(U(1)) are a generalization of differential pp-forms #.

A groupoid cocycle is defined in straightforward analogy to a group cocycle. For instance a group 3-cocycle on a finite group GG governs Dijkgraaf-Witten theory. Since G=Π 1(BG)G= \Pi_1(B G) we may, from the above point of view, regard such a 3-cocycle on GG as a flat 2-gerbe with connection on BGB G. This is precisely the 2-gerbe on BGB G which Carey-Johnson-Murray-Stevenson-Wang construct in terms of transition bundle gerbes, as discussed here.

Let’s try to say what a section and in particular a flat section of an nn-bundle with connection is in this language.

First, to get a feeling for what is going on, consider the familiar case n=1n=1.

Since our (n=1)(n=1)-U(1)U(1)-bundle is flat, it is trivializable and we may assume it to be in fact trivial. Then, if we consider the associated complex line bundle, every fiber is precisely the standard copy of \mathbb{C}.

A section is hence a map

(4)e:Obj(P .(X)) e : \mathrm{Obj}(P_.(X)) \to \mathbb{C}

which assigns a complex number to every point in P .(X)P_.(X).

The section is flat (covariantly constant) if for any two points xx and yy connected by a morphism xγyx \stackrel{\gamma}{\to} y the value over yy is the same as that over xx parallel transported along γ\gamma:

(5)e(y)=tra(γ)(e x). e(y) = \mathrm{tra}(\gamma)(e_x) \,.

(For convenience I am not distinguishing notationally between the principal transport tra\mathrm{tra} and its associated vector transport ρ *tra\rho_* \mathrm{tra} induced by the standard rep ρ:Σ(U(1))Vect\rho : \Sigma(U(1)) \to \mathrm{Vect}).

We can say this in a more natural way: denote by

(6)Id :P .(X)Vect \mathrm{Id}_\mathbb{C} : P_.(X) \to Vect

the constant functor which sends sends every object to \mathbb{C} and every morphism to the identity morphism. Then a flat section is precisely a natural transformation

(7) Id 1 P .(X) e Vect tra, \array{ &\nearrow \searrow^{\mathrm{Id}_{1}} \\ P_.(X) &e \Downarrow \;& \mathrm{Vect} \\ & \searrow \nearrow_{\mathrm{tra}} } \,,

That’s because the corresponding naturality squares read

(8) Id e x e y tra(γ) \array{ \mathbb{C} &\stackrel{\mathrm{Id}}{\to}& \mathbb{C} \\ e_x\downarrow\;\; && \;\,\downarrow e_y \\ \mathbb{C} &\stackrel{\mathrm{tra}(\gamma)}{\to}& \mathbb{C} }

for all γMor(P .(X))\gamma \in \mathrm{Mor}(P_.(X)).

(More generally we also want to understand non-flat sections this way. This then requires that we move from codomain TT to codomain inn(T)\mathrm{inn}(T) such that these naturality squares can be filled with something - namely with the covariant derivative of the section #.)

A little reflection # shows that we actually want to think of P .(X)P_.(X) not as what physicist call target space, but as what they call configuration space. In the case p=1p=1 it is the configuration space of the 1-particle

(9)d 1:={} d^1 := \{\bullet\}

for which, by coincidence, target space P .(X)P_.(X) and configuration space Hom(d 1,P .(X))\mathrm{Hom}(d^1,P_.(X)) happen to coincide:

(10)Hom(d 1,P .(X))P .(X). \mathrm{Hom}(d^1,P_.(X)) \simeq P_.(X) \,.

But this is no longer true for p>1p \gt 1. So we might want to equivalently reformulate the above by saying that a flat section is a natural transformation

(11) Id 1* [d p,P .(X)] e [d p,Vect] tra *, \array{ &\nearrow \searrow^{\mathrm{Id}_{1*}} \\ [d^p,P_.(X)] &e \Downarrow \;& [d^p,\mathrm{Vect}] \\ & \searrow \nearrow_{\mathrm{tra}_*} } \,,

Now move on to n=2n=2. Simon Willerton gives a couple of definitions of what he wants to understand under a section of a flat U(1)U(1)-gerbe defined in terms of a groupoid 2-cocycle on P .(X)P_.(X) on p.17 and following of his paper, showing that they are all equivalent.

One definition goes like this: a flat section associated to the 2-cocycle τ\tau on the groupoid P .(X)P_.(X) is an assignment of vector spaces to objects of P .(X)P_.(X) and of linear maps

(12)F(x)F(γ)F(y) F(x) \stackrel{F(\gamma)}{\to} F(y)

between these, such that for all composable γ\gamma and γ\gamma' we have

(13)F(γ)F(γ)=τ(γ,γ)F(γγ). F(\gamma) \circ F(\gamma') = \tau(\gamma,\gamma')F(\gamma\circ \gamma') \,.

Whenever you see an apparently twisted functor like that, chances are you are actually looking at a pseudonatural transformation between 2-functors.

Same here:

By the argument given here, a connection on a line 2-bundle (line bundle gerbe) is a 2-functor locally with values in bimodules which sends all points to the \mathbb{C}-algebra \mathbb{C} (representing the 2-vector space Mod =Vect\mathrm{Mod}_\mathbb{C} = \mathrm{Vect}), sends all morphisms to \mathbb{C}, regarded as a \mathbb{C}-\mathbb{C}-bimodule, and finally sends 2-morphisms (here: triangles spanned by pairs of composable morphisms) to elements of U(1)U(1) (regarded as bimodule homomorphisms).

In the present context this means that we should think of the groupoid 2-cocycle τ\tau on P .(X)P_.(X) as defining a pseudofunctor whose nontrivial compositor is

(14)tra: y γ γ x γγ z τ(γ,γ) . \mathrm{tra} \;\; : \;\; \array{ & y & \\ {}^\gamma\nearrow &\Downarrow& \searrow^{\gamma'} \\ x & \stackrel{\gamma'\circ \gamma}{\to} & z } \;\;\; \mapsto \;\;\; \array{ & \mathbb{C} & \\ {}^\mathbb{C}\nearrow &\;\;\Downarrow\tau(\gamma,\gamma')& \searrow^{\mathbb{C}} \\ \mathbb{C} & \stackrel{\mathbb{C}}{\to} & \mathbb{C} } \,.

Even though the flatness condition on our transport makes most everything rather trivial, it is crucial that the above is really considered as taking values in bimodules

(15)T=Bim(Vect) VectMod, T = \mathrm{Bim}(\mathrm{Vect}) \subset {}_\mathrm{Vect}\mathrm{Mod} \,,

because this affects the properties of pseudonatural transformations

(16) Id * [d 1,P .(X)] e [d 1,Bim] tra *. \array{ &\nearrow \searrow^{\mathrm{Id}_{\mathbb{C}*}} \\ [d^1,P_.(X)] &e \Downarrow \;& [d^1,\mathrm{Bim}] \\ & \searrow \nearrow_{\mathrm{tra}_*} } \,.

Here d 2d^2 is the 2-particle (the “open string”), i.e. the category

(17)d 2={ab} d^2 = \{a \to b\}

with two objects and a single nontrivial morphism. Accordingly, [d 2,P .(X)][d^2, P_.(X)] is the configuration space of this 2-particle and [d 2,Bim][d^2, \mathrm{Bim}] its space of phases.

By the general reasoning #, we would like to address such an ee as a covariantly constant section of our flat 2-bundle with connection tra\mathrm{tra}.

Does that notion coincide with Simon Willerton’s definition?

To answer that, we need to write down the naturality tin can diagram in Bim\mathrm{Bim} which defines the above pseudonatural transformation. It reads

(18) 1 e x e(γγ) e z = e y e x e(γ) e(γ)e z τ(γ,γ) , \array{ & \mathbb{C} & \\ {}^\mathbb{C}\nearrow &\;\;\Downarrow 1& \searrow^{\mathbb{C}} \\ \mathbb{C} & \stackrel{\mathbb{C}}{\to} & \mathbb{C} \\ e_x\downarrow\;\; &e(\gamma'\circ \gamma)& \;\,\downarrow e_z \\ \mathbb{C} &\stackrel{\mathbb{C}}{\to}& \mathbb{C} } \;\; = \;\; \array{ & \mathbb{C} \\ {}^{\mathbb{C}}\nearrow & e_y\downarrow\;& \searrow^{\mathbb{C}} \\ e_x \downarrow\;\;{}^{\Downarrow e(\gamma)} & \mathbb{C} &{}^{\Downarrow e(\gamma')} \;\;\downarrow e_z \\ {}^\mathbb{C}\nearrow &\;\;\Downarrow \tau(\gamma,\gamma')& \searrow^{\mathbb{C}} \\ \mathbb{C} & \stackrel{\mathbb{C}}{\to} & \mathbb{C} } \,,

for all composable γ\gamma, γ\gamma'.

Notice that, since this diagram lives in Bim\mathrm{Bim}, all the e xe_x, e ye_y, etc must be \mathbb{C}-bimodules - hence vector spaces - and all the e(γ)e(\gamma), e(γ)e(\gamma') must be morphisms of \mathbb{C}-bimodules - hence linear maps between vector spaces.

So we find that a pseudonatural transformation ee like this is precisely the same as the twisted groupoid representation from Simon Willerton’s paper.

Connaisseurs of gerbes will notice that the above is pretty similar to the situation encountered in the study of modules and trivializations of gerbes. A gerbe with connection may be trivialized by a vector bundle with connection \nabla, such that the curving BB of the gerbe equals the curvature F F_\nabla of the bundle as

(19)B=F , B = F_\nabla \,,

roughly. This equation is nothing but the infinitesimal version of the twisted groupoid rep condition

(20)F(γ)F(γ)=τ(γ,γ)F(γγ). F(\gamma) \circ F(\gamma') = \tau(\gamma,\gamma')F(\gamma\circ \gamma') \,.

Of course all this completely harmonizes with the general picture advertized in the paper. All I have would have to add here to that is the observation that the general way to say what a (flat) section of an nn-bundle with connection is seems to be:

a flat section over the pp-point of an nn-bundle with connection is a 2-morphism

(21) Id 1* [d p,P .(X)] e [d p,nVect] tra *, \array{ &\nearrow \searrow^{\mathrm{Id}_{1*}} \\ [d^p, P_.(X)] &e \Downarrow \;& [d^p, n\mathrm{Vect}] \\ & \searrow \nearrow_{\mathrm{tra}_*} } \,,
Posted at November 8, 2006 9:53 PM UTC

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I had some nice discussion about this behind the scenes with Simon Willerton and Bruce Bartlett.

One insight I gained from these exchanges is that the following might be a way to make the concept of section of an nn-vector bundle with connection that I am proposing here look more natural.

So let PP be some domain category and let nVectn\mathrm{Vect} be your favorite version of the nn-category of nn-vector spaces.

That should in any case have a monoidal structure on it.

But this then implies that also the (nn-)functor (nn-)category

(1)[P,nVect] [P,n\mathrm{Vect}]

is monoidal.

In particular, the tensor unit

(2)1[P,nVect] 1 \in [P,n\mathrm{Vect}]

is that constant functor I was talking about, which sends everything to the identity on the tensor unit in nVectn\mathrm{Vect}.

But then what I am saying is nothing but that a (flat) section of

(3)tra[P,nVect] \mathrm{tra} \in [P,n\mathrm{Vect}]

is a morphism

(4)1tra. 1 \to \mathrm{tra} \,.

That’s quite natural.

Accordingly the “space of (flat) sections” is

(5)[1,tra]. [1,\mathrm{tra}] \,.
Posted by: urs on November 9, 2006 5:59 PM | Permalink | Reply to this
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