The n-Category Café

Here I am asking myself in particular this:

Given a line 2-bundle with connection ($\sim$ a line bundle gerbe # with connection and curving #), determined by a 2-vector transport # with values in $\mathrm{Bim}(\mathrm{Vect}_\mathbb{C})$ #, what is the arrow-theory 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 2-dimensional QFT, determined by a 2-vector transport with values in $\mathrm{Bim}(C)$, for some monoidal category $C$, what is the arrow theory of

• the two-point disk amplitude for given boundary conditions.

In other words: What is a D-brane?

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 D-branes appear automatically.

In

Transport of Sections
(pdf, 6 pages)

I define an arrow-theory for the disk with given boundary conditions and two boundary insertions.

In

Disk Holonomy of a Line 2-Bundle
(pdf, 4 pages)

I check that applied to the surface transport of a line 2-bundle 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 2-transport with values in right-induced bimodules (described here) and in particular the description of gerbe curving in terms of 2-functor 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.