### Branes, Bi-Branes, 2-Vectors, 2-Linear Maps

#### Posted by Urs Schreiber

The open charged 2-particle looks like $a \to b$ and its quantization, $q(\mathrm{tra})$, assigns to it a morphism $f$ of its 2-vector space $V \stackrel{f}{\to} V$ of 2-states $\psi : I \to q(\mathrm{tra})$ each of which is a generalized element $\psi : (a \to b) \;\; \mapsto \;\; \array{ I &\stackrel{\mathrm{Id}}{\to}& I \\ \psi(a) \downarrow \;\, &\;\;\Downarrow^{\psi(a\to b)}& \;\, \downarrow \psi(b) \\ V &\stackrel{f}{\to}& V } \,.$

When the 2-particle is charged under a line 2-bundle (a line bundle gerbe) the 2-vectors $\psi(a)$ and $\psi(b)$ are Chan-Paton bundles on D-branes, also known as modules for that gerbe.

The space of states is acted on 2-linearly by pull-push through spans $\array{ && \mathrm{hist} \\ & \swarrow &&& \searrow \\ \mathrm{conf} &&&&& \mathrm{conf} }\,,$ which may encode operation like time evolution or gauge transformations like T-duality.

In a chosen 2-basis for $V$, which is an algebra, 2-states appear as modules and 2-linear maps appear as bimodules.

The former fact harmonizes with the term “gerbe module” used for D-branes. In that sense, these bimodules could be addressed as **bi-branes**.

This is the language now chosen in

Fuchs, Schweigert, Waldorf
*Bi-branes: Target Space Geometry for World Sheet topological Defects*

Bi-branes: Target Space Geometry for World Sheet topological Defects.

Like an ordinary brane – at least in its geometric incarnation as a subspace with Chan-Paton bundle on it – is a submanifold of target space over wich the Kalb-Ramond field strength (the curvature of the gerbe) trivializes, a bi-brane is defined to be a submanifold of *two different* target spaces, over which the difference of two KR-fields trivializes.

While only very briefly toughed upon in the above paper, this is the familiar central structure of interest in topological T-duality, in which case the bi-brane bundle is the Poincaré-line bundle.

In fact, the condition on the KR fields now given for bi-branes is known in topological T-duality, as for instance discussed on p. 5 of

Peter Bouwknegt, Keith Hannabuss, Varghese Mathai
*T-duality for principal torus bundles and dimensionally reduced Gysin sequences*

hep-th/0412268.

New constructions along these lines, with Courant algeboroids and their morphisms, encoding gerbes and their morphisms, are in preparation by Cavalcant & Gualtieri: T-duality with NS-flux and generalized complex structures.

## Re: Branes, Bi-Branes, 2-Vectors, 2-Linear Maps

This is interesting and I feel somehow related to some work I’m doing on orthogonal hermite polynomials in 2-variables and pointwise fourier integral transforms and diagonalization via projective integral transforms via the Gram-Schmidt process. All my work is done via hypergeometric functions, generating functions, and quite a bit of analytic number theory. The nice thing is that every single step of the way I am only working with integer sequences and renormalized power series expansions. So little ole me has been able to do this stuff concretey without all this abstract nonsense. So, why is all this abstract stuff so popular when number theory is so much more concrete and applicable?