# Contents

## Idea

A bi-brane is to a defect in a FQFT as a brane is to a boundary condition.

The term “bi-brane” was apparently introduced in

• Jürgen Fuchs, Christoph Schweigert, Konrad Waldorf, Bi-branes: Target Space Geometry for World Sheet topological Defects (arXiv)

The description below approaches the concept in a slightly more abstract context.

The notion of brane and bi-brane can be made very abstract, but to get the main idea it is useful to start with considering what is usually called the geometric case.

Recall for instance from

• Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, Richard J. Szabo, D-Branes, RR-Fields and Duality on Noncommutative Manifolds (arXiv, blog)

that a geometric brane on some space $X$ and a bundle gerbe $\mathcal{G}$, regarded as a special kind of 2-vector bundle, on $X$ is

• a map $\iota : Q \to X$

• a morphism $\sigma : 1 \to \iota^* \mathcal{G}$ from the trivial 2-vector bundle on $Q$ into the pullback of $\mathcal{G}$ to $Q$ – this morphism is called a gerbe module or twisted vector bundle.

If we write this more diagrammatically using the classifying (fiber-assigning) cocycle $g : X \to 2 Vect$ of $\mathcal{G}$, then this data of a brane is a transformation

$\array{ && Q \\ & \swarrow && \searrow^{\iota} \\ pt &&\stackrel{\sigma}{\Rightarrow}&& X \\ & \searrow && \swarrow_{g} \\ && 2 Vect } \,.$

Conceived in this form the notion has an obvious generalizations:

let $X$ and $Y$ be two possibly different spaces with two possibly different 2-vector bundles on them, classified by cocycles $g_1$ and $g_2$, then a bi-brane for this situation is

• a span $x \stackrel{\iota_1}{\leftarrow} Q \stackrel{\iota_2}{\rightarrow} Y$;

• and a transformation between the two pulled back bundles

$\array{ && Q \\ & {}^{\iota_1}\swarrow && \searrow^{\iota_2} \\ X &&\stackrel{\bibrane}{\Rightarrow}&& Y \\ & {}_{g_1}\searrow &&& \swarrow_{g_2} \\ && 2 Vect } \,.$

The description of branes in the above diagrammatic form was first given in

and described in more detail in

• Urs Schreiber, Konrad Waldorf, Connections on Nonabelian Gerbes and their Holonomy (arXiv blog).

The generalization to bi-branes is developed at

This is very closely related to the spans appearing in

The relation is discussed a bit at this blog entry.

At least some aspects of the concept have more or less implicitly been considered before, notably in the context of topological T-duality. A translation of the construction in topological T-duality to the above diagrammatic formulation was originally given here.

The interpretation of T-duality in terms of bi-branes is discussed in more detail in

• Gor Sarkissian, Christoph Schweigert, Some remarks on defects and T-duality (arXiv)

## Bi-branes motivated from 2d CFT

Recall that from a 2-dimensional CFT one induces a (generalized) target space geometry in generalization of how a spectral triple induces such a generalized geometry.

From category-algebraic considerations one obtains defect line?s in 2-d CFT, which are encoded by bimodules as [brane]s are encoded by modules. Bi-branes are the answer to the question: “What is the target space structure corresponding to defect lines in the 2d CFT”?

For certain 2-d CFTs based on current algebras the bi-branes corresponding to certain defect lines in these theories have been introduced and discussed in

• Jürgen Fuchs, Christoph Schweigert, Konrad Waldorf, Bi-branes: Target Space Geometry for World Sheet topological Defects (arXiv)

## In WZW theories

In the above article it is found that, just as symmetric conformal branes in WZW models, whose target space is a Lie group $G$, correspond to submanifolds of $G$ given by conjugacy classes in $G$, bi-branes in WZW model correspond to spans or correspondences

$\array{ && B \\ & \swarrow &&\searrow \\ G &&&& G }$

where $B$ is a biconjugacy class of $G$.

## References

Revised on January 8, 2014 15:32:14 by Urs Schreiber (82.113.106.24)