# Contents

## Idea

An abstractly defined $n$-dimensional quantum field theory is a consistent assignment of state-space and correlators to $n$-dimensional cobordisms with certain structure (topological structure, conformal structure, Riemannian structure, etc. see FQFT/AQFT). In an open-closed QFT the cobordisms are allowed to have boundaries.

In this abstract formulation of QFT a D-brane is a type of data assigned by the QFT to boundaries of cobordisms.

### In $2d$ rational CFT

A well understood class of examples is this one: among all 2-dimensional conformal field theory that case of full rational 2d CFT has been understood completely, using FFRS-formalism. It is then a theorem that full 2-rational CFTs are classified by

1. a modular tensor category $𝒞$ (to be thought of as being the category of representaitons of the vertex operator algebra of the 2d CFT);

2. a special symmetric Frobenius algebra object $A$ internal to $𝒞$.

In this formulation a type of brane of the theory is precisely an $A$-module in $𝒞$ (an $A$-bimodule is a bi-brane or defect line ):

the 2d cobordisms with boundary on which the theory defined by $A\in 𝒞$ carry as extra structure on their connected boundary pieces a label given by an equivalence class of an $A$-module in $𝒞$. The assignment of the CFT to such a cobordism with boundary is obtained by

• triangulating the cobordism,

• labeling all internal edges by $A$

• labelling all boundary pieces by the $A$-module

• all vertices where three internal edges meet by the multiplication operation

• and all points where an internal edge hits a moundary by the corresponding action morphism

• and finally evaluating the resulting string diagram in $𝒞$.

So in this abstract algebraic formulation of QFT on the worldvolume, a brane is just the datum assigned by the QFT to the boundary of a cobordism. But abstractly defined QFTs may arise from quantization of sigma models. This gives these boundary data a geometric interpretation in some space. This we discuss in the next section.

### In $2d$ TFT

Another case where the branes of a QFT are under good mathematical control is TCFT: the (infinity,1)-category-version of a 2d TQFT.

Particularly the A-model and the B-model are well understood.

There is also a mathematical structure called string topology with D-branes. At present this is more “string inspired” than actually derived from string theory, though.

### In terms of geometric data of the $\sigma$-model background

An abstractly defined QFT (as a consistent assignment of state spaces and propagators to cobordisms as in FQFT) may be obtained by quantization from geometric data :

Sich a sigma-model QFT is the quantization of an action functional on a space of maps $\Sigma \to X$ from a cobordims (“worldvolume”) $\Sigma$ to some target space $X$ that may carry further geoemtric data such as a Riemannian metric, or other background gauge fields.

One may therefore try to match the geometric data on $X$ that encodes the $\sigma$-model with the algebraic data of the FQFT that results after quantization. This gives a geometric interpretation to many of the otherwise purely abstract algebraic properties of the worldvolume QFT.

It turns out that if one checks which geometric data corresponds to the $A$-modules in the above discussion, one finds that these tend to come from structures that look at least roughly like submanifolds of the target space $X$. And typically these submanifolds themselves carry their own background gauge field data.

A well-understood case is the Wess-Zumino-Witten model: for this the target space $X$ is a simple Lie group $X=G$ and the background field is a circle 2-bundle with connection (a bundle gerbe) on $G$, representing the background field that is known as the Kalb-Ramond field.

In this case it turns out that branes for the sigma model on $X$ are given in the smplest case by conjugacy classes $D\subset G$ inside the group, and that these carry twisted vector bundle with the twist given by the Kalb-Ramond background bundle. These vector bundles are known in the string theory literature as Chan-Paton vector bundles . The geometric intuition is that a QFT with certain boundary condition comes form a quantization of spaces of maps $\Sigma \to G$ that are restricted to take the boundary of $\Sigma$ to these submanifolds.

More generally, one finds that the geometric data that corresponds to the branes in the algebraically defined 2d QFT is given by cocycles in the twisted differential K-theory of $G$. These may be quite far from having a direct interpretation as submanifolds of $G$.

The case of rational 2d CFT considered so far is only the best understood of a long sequence of other examples. Here the collection of all D-branes – identified with the colleciton of all internal modules over an internal frobenius algebra, forms an ordinary category.

More generally, at least for 2-dimensional TQFTs analogous considerations yield not just categories but stable (∞,1)-categories of boundary condition objects. For instance for what is called the B-model 2-d TQFT the category of D-branes is the derived category of coherent sheaves on some Calabi-Yau space.

Starting with Kontsevich’s homological algebra reformulation of mirror symmetry the study of (derived) D-brane categories has become a field in its own right in pure mathematics.

… lots of further things to say …

## Examples

### In the WZW model

For D-branes in the WZW-model see WZW-model – D-branes.

## Characterizations

### In terms of Dirac structures

D-branes may be identified with Dirac structures on a Courant Lie 2-algebroid over spacetime related to the type II geometry (Asakawa-Sasa-Watamura). See at Dirac structure for more on this.

Table of branes appearing in supergravity/string theory

branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
$\left(D=2n\right)$type IIA$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D0-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$BFSS matrix model
D2-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D4-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$D=5 super Yang-Mills theory with Khovanov homology observables
$\left(D=2n+1\right)$type IIB$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D1-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$2d CFT with BH entropy
D3-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$N=4 D=4 super Yang-Mills theory
D5-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D7-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
NS-branetype I, II, heteroticcircle n-connection$\phantom{\rule{thinmathspace}{0ex}}$
string$\phantom{\rule{thinmathspace}{0ex}}$B2-field2d SCFT
NS5-brane$\phantom{\rule{thinmathspace}{0ex}}$B6-fieldlittle string theory
M-brane11D SuGra/M-theorycircle n-connection$\phantom{\rule{thinmathspace}{0ex}}$
M2-brane$\phantom{\rule{thinmathspace}{0ex}}$C3-fieldABJM theory, BLG model
M5-brane$\phantom{\rule{thinmathspace}{0ex}}$C6-field6d (2,0)-superconformal QFT
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane$\phantom{\rule{thinmathspace}{0ex}}$C6-field on G2-manifold

∞-Chern-Simons theory from binary and non-degenerate invariant polynomial

$n\in ℕ$symplectic Lie n-algebroidLie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $\left(n+1\right)$-d sigma-modelhigher symplectic geometry$\left(n+1\right)$d sigma-modeldg-Lagrangian submanifold/ real polarization leaf= brane(n+1)-module of quantum states in codimension $\left(n+1\right)$discussed in:
0symplectic manifoldsymplectic manifoldsymplectic geometryLagrangian submanifoldordinary space of states (in geometric quantization)geometric quantization
1Poisson Lie algebroidsymplectic groupoid2-plectic geometryPoisson sigma-modelcoisotropic submanifold (of underlying Poisson manifold)brane of Poisson sigma-model2-module = category of modules over strict deformation quantiized algebra of observablesextended geometric quantization of 2d Chern-Simons theory
2Courant Lie 2-algebroidsymplectic 2-groupoid3-plectic geometryCourant sigma-modelDirac structureD-brane in type II geometry
$n$symplectic Lie n-algebroidsymplectic n-groupoid(n+1)-plectic geometry$d=n+1$ AKSZ sigma-model

## References

### General

A classical text describing how the physics way to think of D-branes leads to seeing that they are objects in derived categories is

This can to a large extent be read as a dictionary from homological algebra terminology to that of D-brane physics.

More recent similar material with the emphasis on the K-theory aspects is

### KK-theoretic description

Discussion of D-branes in KK-theory is reviewed in

based on

### For rational CFT

For exhaustive details on D-branes in 2-dimensional rational CFT see the references given at

### For topological strings

A discussion of topological D-branes in the context of higher category theory is in

### Open string worldsheet Anomaly cancellation

The need for twisted spin^c structures as quantum anomaly-cancellaton condition on the worldvolume of D-branes was first discussed in

More details are in

A clean review is provided in

• Kim Laine, Geometric and topological aspects of Type IIB D-branes (arXiv:0912.0460)

For more see at Freed-Witten anomaly cancellation.

### Relation to Dirac structures

• Tsuguhiko Asakawa, Shuhei Sasa, Satoshi Watamura, D-branes in Generalized Geometry and Dirac-Born-Infeld Action (arXiv:1206.6964)

Revised on May 17, 2013 13:02:52 by Urs Schreiber (82.169.65.155)