The n-Category Café

I would like to understand 2-dimensional field theories describing charged 2-particles (strings) arrow-theoretically, from the point of view of 2-transport #.

In a previous posting # I had begun formulating the functorial notion of a section of a 2-bundle with 2-connection. I used this to discuss the arrow theory of the disk diagrams that describe the parallel transport of such sections over topologically disk-shaped surfes.

Notice that, for a charged $n$-particle, a section of the bundle it is charged under is essentially what in physics you would call a quantum state of the $n$-particle.

The point of expressing this entirely arrow-theoretically is that I want the formalism to tell me how the notion of section generalizes as we move from $n$ to $n+1$-particles. I don’t want to think. I want to follow the Tao.

So, in particular, the hope was that with using the right general notion of section, we would automatically be lead to find that a section for an open 2-particle, i.e. for something that looks like an interval

encodes something like an ordinary section of some bundle over path space of target space, together with certain boundary data associated to the endpoints. This boundary data should essentially be a gerbe module #, otherwise known as a Chan-Paton bundle.

With the right notion of 2-vector space used, it is easy to see that a 2-section associates some kind of module to endpoints of the open string. This is what I described in my previous posting.

Namely, if we use the canonical inclusion

and take our vector 2-bundle with connection on target space, $\mathrm{tar}$, to be a 2-functor

which transgresses # to configuration space

as

then, clearly, sections

will come, over each path $x \stackrel{\gamma}{\to} y$ in target space, from squares

in $\mathrm{Bim}$. But this means that the naively expected section $e(\gamma)$ of the fiber over $\gamma$ is accompanied by modules $e(x)$ and $e(y)$.

All I want to do in this posting here is to present a brief argument, suggested by considerations discussed recently #, showing that, in a suitable setup, a 2-section $e : 1_* \to \mathrm{tra}_*$ associates precisely a gerbe module to each endpoint of the string.

$\;\;\;$ gerbe modules from 2-sections

The trick is to use a particularly well-behaved incarnation of the 2-functor that encodes the gerbe and its connection.

If we choose a good covering $U \to X$ of target space $X$, we may consider the bundle gerbe with connection as a 2-functor from the 2-category of paths in the transition groupoid #

to 1-dimensional vector spaces #.

Notice the canonical inclusion

Configuration space, then, as always, is that sub-2-category of maps

whose morphisms don’t physically move the string, but just gauge transform its configuration.

For $P_2(U^\bullet)$, this involves in particular morphisms that come from 2-cells

which encode how a path $\gamma$ may “jump” from being regarded as sitting in $U_i$ to being regarded as sitting in $U_j$.

One then has to unwrap the definition of a morphism $e : 1_* \to \mathrm{tra}_*$ to see what it means. This cannot well be described with words, but requires drawing the relevant diagrams.

It turns out, that precisely the required compatibility of $e$ with morphisms $(\gamma,i) \to (\gamma,j)$ forces the boundary part of the 2-section to be a module for the gerbe in question.