### Categorical Trace and Sections of 2-Transport

#### Posted by Urs Schreiber

*What is a trace?*

In quantum field theory, the answer is: let $\mathrm{QFT} : n\mathrm{Cob} \to \mathrm{Vect}$ be 1-functor describing an $n$-dimensional QFT. Then the trace of

is

*What is a 2-trace?*

In an *extended* QFT we want to refine $\mathrm{QFT}$ to an $n$-functor #. Do we get an $n$-trace this way?

Curiously, the right way to think of the extended $\mathrm{QFT}$ is, apparently, to think in terms of *spaces of sections* of $n$-bundles #.

I am trying to understand the implications of a general abstract proposal # for what that might actually mean.

As a consistency check, I would like to understand if in the case where the QFT’s target space is a 2-group and the 2-bundle is a 2-representation: does the canonical space of states

know about the notion of **2-character of a 2-representation** as discussed by Kapranov and Ganter ?

Here $\mathrm{conf}$ denotes the configuration space of the QFT, $\mathrm{phas}$ the space of phases, $\mathrm{par}$ the parameter space and $\mathrm{sect}$ the space of states - as described here.

Clearly, I am taking the liberty here to think more about arrow theory than about the technicalities to be dealt with in concrete implementations. If you are rather interested in the latter, you might enjoy the most recent work by Graeme Segal, the originator of the functorial description of CFT.

What I instead would like to understand here is this:

**a general idea for the canonical extended QFT associated to a charged $n$-particle**

To a given $n$-transport # over target space $P$, coupled to an $n$-particle represented by a parameter space $\mathrm{par}$, we can, in a natural way, associate the space $\mathrm{sect}$ of sections of the $n$-transport “transgressed” to the $n$-particle’s configuration space.

But Freed et al. tell us (and moreover Bruce Bartlett and Simon Willerton # tell me, who is greatly indebted to them for lots of good conversation) that we should, roughly, be looking for a map from parameter space $\mathrm{par}$ to a space of sections.

I notice that, quite abstractly, there is a canonical candidate, namely the image of $H$ under what ought to be a canonical equivalence in the world of 2-categories:

To flesh out this abstract nonsense, we need

**an application**

Take target space to be a strict 2-group

Take parameter space to be the open 2-particle

or the closed one,

coupled to a 2-bundle with 2-connection on target space

which is nothing but a 2-representation of $G_2$.

For the different but similar setup of lax 2-representations of discrete 2-groups, Kapranov and Ganter have proposed a notion of **2-character**. The canonical “space of states”

ought to know about such 2-characters.

*Does it?*

To decide this, we need just follow that tao of abstract nonsense, work out $i(H)$ and check.

I found this to be harder that I had expected. But now I do have something like a partial result, constructing not the full $i(H)$ in the above example but just some restriction of it.

I am not sure if anything but the introduction and the remark at the end is actually readable, but in case anyone cares, some details can be found in these notes:

$\;\;\;$ sections of 2-reps

The result of these considerations (in as far as they happen to be correct) is:

*
In the above example (open string $a \to b$ propagating on a 2-group and coupled to a 2-rep) the “canonical space of states” $i(H)$ associates collections of 2-morphisms
$\array{
& \nearrow \searrow^{\mathrm{Id}}
\\
\mathbb{C}
&e \Downarrow \;&
\mathbb{C}
\\
&
\searrow \nearrow_{\rho(g)}
}$
for all $g \in \mathrm{Mor}_1(\Sigma(G_2))$ to $a$, similar collections to $b$ and
associates to the morphism $a \to b$ a map from one to the other that comes from
conjugation with represented 2-group elements.
*

Actually, one gets something more general than this. But in this special case this structure is indeed rather similar in nature to the proposal by Kapranov and Ganter.

**reminder: the 2-character by Kapranov and Ganter #**

Kapranov and Ganter considered 2-reps $\rho$ of 2-groups $G_2$ of the form

which have an ordinary group $G$ as objects and only identity morphisms. But that restriction has no bearing on the notion of 2-trace they invent in order to define 2-characters of such reps.

Let

be the lax 2-functor representing the 2-group, where $T$ is some 2-category of 2-vector spaces. So

Then, by definition, the Kapranov-Ganter 2-trace of the image under $\rho$ of some $g \in G$ is the Hom-space

An element in that space $\mathrm{Tr}(\rho(g))$ is hence a 2-morphism $e$ of the form

Moreover, for any element $h \in G$ we may map that morphism to the one obtained by conjugating with $\rho(h)$ and inserting the unit $\mathrm{Id}_A \Rightarrow \stackrel{\rho(h)}{\to}\stackrel{\rho(h^{-1})}{\to}$

This way, the categorical trace actually becomes a functor on the action groupoid of $G$ acting on itself.

## transgression

I was asked to say something about how the way I turn functors on target space into functors on configuration space above is related to the ordinary notion of

transgression.I’ll illustrate this with the example that shows how an abelian gerbe on $X$ transgresses to a line bundle on $L X$.

Let $X$ be a space and let $P_2(X)$ be 2-paths in $X$, i.e. objects are points in $X$, morphisms are paths between points and 2-morphisms are surfaces between paths.

Let $\mathrm{par}$ be some model for the circle. Let’s take

the category with a single object and $\mathbb{Z}$ worth of morphism, with composition corresponding to addition of natural numbers.

Regard this as a 2-category with only identity 2-morphisms.

Now, one finds that the functor 2-category

has

objects being closed paths in $X$, to be thought of as loops with basepoint

morphisms being cylinders between such based loops, containing the information of the path traced out by thre basepoint

2-morphisms are deformations of the path of this basepoint.

So $[\mathrm{par},P_2(X)]$ is much like the 1-category

whose objects are (based) loops in $X$ and whose morphisms are (based) cylinders in $X$, only that $[\mathrm{par},P_2(X)]$ also knows how to deal with transformations that move the basepoint.

Now consider on $P_2(X)$ a 2-functor that sends paths to 1-dimensional vector spaces:

such that this guy is smooth and locally trivializable in a suitable sense.

I claim that such 2-functors are equivalent to bundle gerbes # with connection #.

(I have shown how every such functor gives rise to a bundle gerbe with connection here. For the converse statement you will have to trust me for the moment.)

So $\mathrm{tra}$ sends each path to a 1-dimensional vector space, sends each surface to a morphism between such vector spaces, and sends composition of paths to tensor product of vector spaces.

Notice that in the case that this 2-functor is trivial in the sense that it sends every path to the canonical 1-d vector space $V_\gamma = \mathbb{C}$ we can regard it as a functor

If, moreover, that transport were

flat, we could think of this as a pseudofunctor on the fundamental groupoid of $X$, or equivalently as a groupoid 2-cocycle.But we need not. Here I want to consider the most general line bundle gerbe with connection and hence an arbitrary functor $P_2(X) \to \Sigma(1d\mathrm{Vect}_\mathbb{C})$.

The standard construction now turns $\mathrm{tra}$ into a 2-functor $\mathrm{tra}_*$ on our “extended” loop space 2-category $[\mathrm{par},P_2(X)]$:

Here $\mathrm{tra}_*$ simply takes any object $\ell : \Sigma(\mathbb{Z}) \to P_2(X)$ and sends it to the object

in $[\Sigma(\mathbb{Z}),\Sigma(1d\mathrm{Vect})]$. Similarly for cylinders and basepoint trajectory deformations.

We can easily read off what $\mathrm{tra}_*$ does in detail:

it sends every based loop to the 1-dimensional vector space associated with it (to become the fiber of a line bundle on loop space)

it sends every cylinder between loops with given basepoint trajectory to an isomorphism between the vector spaces associated to the loops (to become a parallel transport on the bundle over loop space), but twisted by the vector space associated to the base point trajectory

but furthermore, it associates a morphism between the vector spaces of the basepoint trajectory and a deformed trajectory to every such deformation.

This last bit is a certain gauge transformation that we don’t usually consider when talking about line bundles on loop space.

And in fact, it is only apparently present here. If we have a closer look, these “twists” are actually not there.

It’s probably quite obvious to everybody reading this, but since I already typed that much, I can just as well also spell this out in detail.

So let $\ell_1$ be an incoming based loop and let $\ell_2$ be the outgoing loop of a cylinder $\Sigma$, whose basepoint runs along the path $\gamma$. Then $\mathrm{tra}_*(\Sigma)$ is a 2-morphism in $\Sigma(1d\mathrm{Vect})$ of the form

i.e. a linear map

But all these space are 1-dimensional! Hence this is

canonicallyisomorphic to a plain mapMoreover, the nature of the base point deformations (coming from a modification of pseudonatural transformations) tells us precisely that this morphism is independent of which base line trajectory we choose.

In other words: $\mathrm{tra}_*$ is well defined on isomorphism classes of 1-morphisms in $[\Sigma(\mathbb{Z}),P_2(X)]$. Hence it descends to a 1-functor

That’s the line bundle with connection on loop space obtained from the line bundle gerbe with connection on target space.