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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 nn-particle, a section of the bundle it is charged under is essentially what in physics you would call a quantum state of the nn-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 nn to n+1n+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)par={ab} \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:=Bim(Vect) VectMod T := \mathrm{Bim}(\mathrm{Vect}) \stackrel{\subset}{\to} {}_{\mathrm{Vect}}\mathrm{Mod}

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

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

which transgresses # to configuration space

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

as

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

then, clearly, sections

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

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

(7) Id C e(x) e(γ) e(y) tra(x) tra(γ) tra(y) \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 Bim\mathrm{Bim}. But this means that the naively expected section e(γ)e(\gamma) of the fiber over γ\gamma is accompanied by modules e(x)e(x) and e(y)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 *tra *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 UXU \to X of target space XX, we may consider the bundle gerbe with connection as a 2-functor from the 2-category of paths in the transition groupoid #

(8)tra:P 2(U )Σ(1dVect) \mathrm{tra} : P_2(U^\bullet) \to \Sigma(1d\mathrm{Vect})

to 1-dimensional vector spaces #.

Notice the canonical inclusion

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

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

(10)conf[par,P 2(U )] \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 )P_2(U^\bullet), this involves in particular morphisms that come from 2-cells

(11)(γ,i) (γ,j):=(x,i) (γ,i) (y,i) (x,j) (γ,j) (y,j), \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 γ\gamma may “jump” from being regarded as sitting in U iU_i to being regarded as sitting in U jU_j.

One then has to unwrap the definition of a morphism e:1 *tra *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 ee with morphisms (γ,i)(γ,j)(\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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