## November 28, 2006

### D-Branes from Tin Cans, II

#### Posted by Urs Schreiber

A brief note on how a 2-section of a transport 2-functor transgressed to the configuration space of the open 2-particle (string) encodes gerbe modules (Chan-Paton bundles) associated to the endpoints of the 2-particle.

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

(1)$\mathrm{par} = \{a \to b\}$

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

(2)$T := \mathrm{Bim}(\mathrm{Vect}) \stackrel{\subset}{\to} {}_{\mathrm{Vect}}\mathrm{Mod}$

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

(3)$\mathrm{tra} : \mathrm{tar} \to T$

which transgresses # to configuration space

(4)$\mathrm{conf} \subset [\mathrm{par},\mathrm{tar}]$

as

(5)$\mathrm{tra}_ * : \mathrm{conf} \to [\mathrm{par},T] \,,$

then, clearly, sections

(6)$e : 1_* \to \mathrm{tra}_*$

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

(7)$\array{ \mathbb{C} &\stackrel{\mathrm{Id}}{\to}& \C \\ e(x) \downarrow \;\; &\;\Downarrow e(\gamma)& \;\; \downarrow e(y) \\ \mathrm{tra}(x) &\stackrel{\mathrm{tra}(\gamma)}{\to}& \mathrm{tra}(y) }$

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 #

(8)$\mathrm{tra} : P_2(U^\bullet) \to \Sigma(1d\mathrm{Vect})$

to 1-dimensional vector spaces #.

Notice the canonical inclusion

(9)$\Sigma(1d\mathrm{Vect}) \stackrel{\subset}{\to} \mathrm{Bim}(\mathrm{Vect}) \,.$

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

(10)$\mathrm{conf} \subset [\mathrm{par},P_2(U^\bullet)]$

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

(11)$\array{ (\gamma,i) \\ \downarrow \\ (\gamma,j) } \;\;\;\; := \;\;\;\, \array{ (x,i) &\stackrel{(\gamma,i)}{\to}& (y,i) \\ \downarrow &\Downarrow& \downarrow \\ (x,j) &\stackrel{(\gamma,j)}{\to}& (y,j) } \,,$

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.

Posted at November 28, 2006 8:43 PM UTC

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