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October 18, 2006

D-Branes 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:

  1. 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).

  2. 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 2-bundle with connection (\sim a line bundle gerbe # with connection and curving #), determined by a 2-vector transport # with values in Bim(Vect )\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

(1)hol(D)=exp(i DB)Tr (D) \mathrm{hol}(D) = \exp(i \int_D B)\mathrm{Tr}_\nabla(\partial D)

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 Bim(C)\mathrm{Bim}(C), for some monoidal category CC, 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 1e_1 coming in, propagating a little, and a state e 2e_2 coming out

such that for n=1n=1 it reproduces the familiar e 1|e itH|e 2\langle e_1 | e^{i t H}| e_2\rangle and such that categorifying it to n=2n=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

(2)hol(D)=exp(i DB)Tr (D) \mathrm{hol}(D) = \exp(i \int_D B)\mathrm{Tr}_\nabla(\partial D)

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.

Posted at October 18, 2006 2:17 PM UTC

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