DBranes from Tin Cans: Arrow Theory of Disks
Posted by Urs Schreiber
Participants of the seminar Quantization and Cohomology are being asked to think about categorification of quantum mechanics. Being the remote participant that I am, I follow the seminar by gnawing on
Exercise C: Categorify the quantum theory of the charged particle.
Hints:

Find the right arrow theory.
You know that it is right when
 it categorifies smoothly;

the result reproduces existing theory, where available
(many people have thought about the categorified point before in different language  you should reproduce whatever is good about their results).

It is not a coincidence that parallel transport in a bundle as well as propagation in QFT both are functors taking values in vector spaces. Use the former as a guide to the latter and explain the relation.
Here I am asking myself in particular this:
Given a line 2bundle with connection ($\sim$ a line bundle gerbe # with connection and curving #), determined by a 2vector transport # with values in $\mathrm{Bim}(\mathrm{Vect}_\mathbb{C})$ #, what is the arrowtheory of
 the space of sections
 the transport over a disk with boundary insertions
 the boundary conditions that can be put on the disk
 the holonomy over a disk with given boundary conditions and given boundary insertions,
such that it leads to the familiar formula
for the abelian gerbe holonomy over a disk with given gerbe module on the boundary?
Analogously, given a 2dimensional QFT, determined by a 2vector transport with values in $\mathrm{Bim}(C)$, for some monoidal category $C$, what is the arrow theory of
 the twopoint disk amplitude for given boundary conditions.
In other words: What is a Dbrane?
More precisely, following the above hints, I want to formulate the concept of
a state $e_1$ coming in, propagating a little, and a state $e_2$ coming out
such that for $n=1$ it reproduces the familiar $\langle e_1  e^{i t H} e_2\rangle$ and such that categorifying it to $n=2$ makes Dbranes appear automatically.
In
Transport of Sections
(pdf, 6 pages)
I define an arrowtheory for the disk with given boundary conditions and two boundary insertions.
In
Disk Holonomy of a Line 2Bundle
(pdf, 4 pages)
I check that applied to the surface transport of a line 2bundle it reproduces the familiar formula for disk holonomy in the presence of a gerbe module on the boundary.
Even without taking the time to look at any of the details, you might enjoy looking at the diagram version of
on the last page.
While not too relevant for the general structure, I do freely use a couple of technical details on 2transport with values in rightinduced bimodules (described here) and in particular the description of gerbe curving in terms of 2functor to bimodules (described here).
The main point of it all culminates in definition 5.
There should be more to say about the nature of this definition. But not right now.