nLab bi-brane

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

• Urs Schreiber, Quantum 2-states: sections of 2-vector bundles, talk at Fields Institute workshop Higher categories and their applications (2007)

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

• Nonabelian cocycles and their quantum symmetries.

This is very closely related to the spans appearing in

• geometric function theory.

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 $G$.

Related concepts

• motivic quantization

singularity	field theory with singularities
boundary condition/brane	boundary field theory
domain wall/bi-brane	QFT with defects

References

• Joost Nuiten, Cohomological quantization of local prequantum boundary field theory