nLab
D-brane

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.

• the branes of the B-model (“B-branes”) form the the stable (infinity,1)-category of chain complexes of quasicoherent sheaves on the target space (often considered just in terms of its homotopy category of an (infinity,1)-category, the derived category of quasicoherent sheaves);

• the branes of the A-model form the Fukaya category of the target space.

• the category of D-branes of the A-model on a symplectic Landau-Ginzburg model, is a Fukaya-Seidel category;

• the category of D-branes of the B-model on a complex Landau-Ginzburg model is a category of matrix factorizations.

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

Various dimensions

In type IIA supergravity

• D0-brane, D2-brane, D4-brane, D6-brane, D8-brane.

In type IIB supergravity

• D1-brane, D3-brane, D5-brane, D7-brane

In the WZW model

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

Properties

Characterization 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.

D-brane charge

In analogy to how in electromagnetism magnetic charge is given by a class in ordinary cohomology, so D-brane charge is given in (twisted) K-theory, or, if preferred, in its image under the Chern character.

The Chan-Paton bundle carried by a D-brane defines a class in twisted K-theory on the D-brane worldvolume and the D-brane charge is the push-forward (Umkehr map) of this class to spacetime, using a K-orientation of the embedding of the D-brane (a spin^c structure).

More in detail this means the following (BMRS2).

Let $X$ be a manifold regarded as spacetime and $i:Q\hookrightarrow X$ a submanifold regarded as the worldvolume of a D-brane. For ${\nabla }_B:X\to B^{2}U(1{)}_{\mathrm{conn}}$ the circle 2-bundle with connection which models the background B-field, write ${\chi }_B:X\to B^{2}U(1)$ for the underlying circle 2-group-principal 2-bundle.

The corresponding Chan-Paton bundle (a twisted line bundle for the case of a single D-brane) is the trivialization $\xi$ in

$$\begin{array}{ccc}& & Q\\ & \swarrow & & {\searrow }^i\\ *& & {⇙}_{\xi }& & X\\ & \searrow & & {\swarrow }_{{\chi }_B}\\ & & B^{2}U(1)\end{array}\simeq \begin{array}{ccc}& & Q\\ & \swarrow & \downarrow & {\searrow }^i\\ *& {⇙}_{\xi }& {\downarrow }^{i^{*}{\chi }_B}& {⇙}_{\mathrm{id}}& X\\ & \searrow & \downarrow & {\swarrow }_{{\chi }_B}\\ & & B^{2}U(1)\end{array}.$$\array{ && Q \\ & \swarrow && \searrow^{\mathrlap{i}} \\ \ast && \swArrow_{\xi} && X \\ & \searrow && \swarrow_{\mathrlap{\chi_B}} \\ && \mathbf{B}^2 U(1) } \;\;\;\;\; \simeq \;\;\;\;\; \array{ && Q \\ & \swarrow &\downarrow& \searrow^{\mathrlap{i}} \\ \ast &\swArrow_{\xi}& \downarrow^{\mathrlap{i^\ast \chi_B}} &\swArrow_{id}& X \\ & \searrow &\downarrow& \swarrow_{\mathrlap{\chi_B}} \\ && \mathbf{B}^2 U(1) } \,.

Assuming that $i:Q\to X$ is K-oriented in that for instance $X$ has a spin-structure and $Q$ a spin^c-structure, then under the groupoid convolution algebra functor $C^{*}$ this is incarnated as a Hilbert bimodule which defines a class in twisted operator K-theory, realized as the following comoposite in KK-theory

$$ℂ\stackrel{\Gamma (\xi )}{\to }C(Q{)}_{i^{*}{\chi }_B}\stackrel{i_{!}}{\to }C(X{)}_{{\chi }_B},$$\mathbb{C} \stackrel{\Gamma(\xi)}{\to} C(Q)_{i^\ast \chi_B} \stackrel{i_!}{\to} C(X)_{\chi_B} \,,

where

• $C(Q)$ and $C(X)$ are the C*-algebras of functions (vanishing at infinity) on the D-brane and on spacetime, respectively;

• $C(X{)}_{{\chi }_B}$ is the groupoid convolution algebra of sections of ${\chi }_B$ regarded as a centrally extended groupoid over a Cech groupoid resolution of $X$ which supports a Cech cocycle for ${\chi }_B$, and similarly for $C(Q{)}_{i^{*}\chi B}$ and the pullback/restriction $i^{*}{\chi }_B$ of the background B-field to the brane;

• $i!$ is the push-forward (Umkehr map) dual to $i^{*}:C(X{)}_{{\chi }_B}\to C(Q{)}_{i^{*}{\chi }_B}$, realizes as a KK-theory class

  $$D_Q=i!=\in \mathrm{KK}(C(Q),C(X{)}_{{\chi }_B}).$$D_Q = i! = \in KK(C(Q), C(X)_{\chi_B}) \,.

The corresponding D-brane charge in KK-theory is the resulting composite (relative index)

$$i_{!}(\xi )=D_Q(\xi )\in \mathrm{KK}(ℂ,C(X{)}_{{\chi }_B})\simeq K^{[{\chi }_b]}(X)$$i_!(\xi) = D_Q(\xi) \in KK(\mathbb{C}, C(X)_{\chi_B}) \simeq K^{[\chi_b]}(X)

in twisted K-theory. Traditionally only the image of this under the Chern character

$$\mathrm{ch}:\mathrm{KK}\to \mathrm{HL}$$ch \colon KK \to HL

in real cohomology/cyclic cohomology is considered, $\mathrm{ch}(D_Q(\xi ))$. Moreover, traiditonally one thinks of first applying $\mathrm{ch}$ to $\xi$ and then pushing forward in $\mathrm{HL}$. By the C*-algebraic Grothendieck-Riemann-Roch theorem this gives the isomorphic expression

$$\mathrm{ch}(D_Q(\xi )){\otimes }_{C(X{)}_{{\chi }_B}}\mathrm{Todd}\in \mathrm{HL},$$ch(D_Q(\xi)) \otimes_{C(X)_{\chi_B}} Todd \in HL \,,

where on the right we have the relative Todd class. This is the form the D-brane charge was originally found in the physics literature and in which it is still often given.

(In (BMRS2, section 8) this is discussed for the untwisted case.)

Related concepts

• Chan-Paton bundle, twisted bundle, twisted K-theory, Chan-Paton gauge field

• Freed-Witten anomaly cancellation

• Dirac-Born-Infeld action

• black brane, black hole in string theory

• K-homology, KK-theory

Table of branes appearing in supergravity/string theory (for classification see at brane scan).

brane	in supergravity	charged under gauge field	has worldvolume theory
black brane	supergravity	higher gauge field	SCFT
D-brane	type II	RR-field	super Yang-Mills theory
$(D=2n)$	type IIA	$$	$$
D0-brane	$$	$$	BFSS matrix model
D2-brane	$$	$$	$$
D4-brane	$$	$$	D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane	$$	$$
D8-brane	$$	$$
$(D=2n+1)$	type IIB	$$	$$
D1-brane	$$	$$	2d CFT with BH entropy
D3-brane	$$	$$	N=4 D=4 super Yang-Mills theory
D5-brane	$$	$$	$$
D7-brane	$$	$$	$$
D9-brane	$$	$$	$$
NS-brane	type I, II, heterotic	circle n-connection	$$
string	$$	B2-field	2d SCFT
NS5-brane	$$	B6-field	little string theory
M-brane	11D SuGra/M-theory	circle n-connection	$$
M2-brane	$$	C3-field	ABJM theory, BLG model
M5-brane	$$	C6-field	6d (2,0)-superconformal QFT
topological M2-brane	topological M-theory	C3-field on G2-manifold
topological M5-brane	$$	C6-field on G2-manifold

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

$n\in \mathbb{N}$	symplectic Lie n-algebroid	Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $(n+1)$-d sigma-model	higher symplectic geometry	$(n+1)$d sigma-model	dg-Lagrangian submanifold/ real polarization leaf	= brane	(n+1)-module of quantum states in codimension $(n+1)$	discussed in:
0	symplectic manifold	symplectic manifold	symplectic geometry		Lagrangian submanifold	–	ordinary space of states (in geometric quantization)	geometric quantization
1	Poisson Lie algebroid	symplectic groupoid	2-plectic geometry	Poisson sigma-model	coisotropic submanifold (of underlying Poisson manifold)	brane of Poisson sigma-model	2-module = category of modules over strict deformation quantiized algebra of observables	extended geometric quantization of 2d Chern-Simons theory
2	Courant Lie 2-algebroid	symplectic 2-groupoid	3-plectic geometry	Courant sigma-model	Dirac structure	D-brane in type II geometry
$n$	symplectic Lie n-algebroid	symplectic n-groupoid	(n+1)-plectic geometry	$d=n+1$ AKSZ sigma-model

(adapted from Ševera 00)

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

• Paul Aspinwall, D-Branes on Calabi-Yau Manifolds (arXiv:hep-th/0403166)

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

• Richard Szabo, D-Branes and Bivariant K-Theory

KK-theoretic description and D-brane charge

The idea that the physics of D-branes is described by K-theory originates in

• Ruben Minasian, Gregory Moore, K-theory and Ramond-Ramond charge , JHEP9711:002,1997 (arXiv:hep-th/9710230)

• Edward Witten, D-Branes And K-Theory, JHEP 9812:019,1998 (arXiv:hep-th/9810188)

Reviews include

• Greg Moore, K-Theory from a physical perspective (arXiv:hep-th/0304018)

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

• Richard Szabo, D-branes and bivariant K-theory (arXiv:0809.3029)

based on

• Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, Richard Szabo, D-Branes, RR-Fields and Duality on Noncommutative Manifolds, Commun. Math. Phys. 277:643-706,2008 (arXiv:hep-th/0607020)

• Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, Richard Szabo, Noncommutative correspondences, duality and D-branes in bivariant K-theory, Adv. Theor. Math. Phys.13:497-552,2009 (arXiv:0708.2648)

• Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, Richard Szabo, D-branes, KK-theory and duality on noncommutative spaces, J. Phys. Conf. Ser. 103:012004,2008 (arXiv:0709.2128)

In particular (BMRS2) discusses the definition and construction of D-brane charge as a generalized index in KK-theory. The discussion there focuses on the untwisted case. Comments on the generalization of this to topologicall non-trivial B-field and hence twisted K-theory is in

• Richard Szabo, D-Branes, Tachyons and K-Homology, Mod. Phys. Lett. A17 (2002) 2297-2316 (arXiv:hep-th/0209210)

Specifically for D-branes in WZW models see

• Peter Bouwknegt, A note on equality of algebraic and geometric D-brane charges in WZW models (pdf)

More on this, with more explicit relation to noncommutative motives, is in

• Snigdhayan Mahanta?, Noncommutative correspondence categories, simplicial sets and pro $C^{*}$-algebras (arXiv:0906.5400)

• Snigdahayan Mahanta, Higher nonunital Quillen $K\prime$-theory, KK-dualities and applications to topological $𝕋$-duality, Journal of Geometry and Physics, Volume 61, Issue 5 2011, p. 875-889. (pdf)

For rational CFT

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

• FFRS-formalism.

For topological strings

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

• Anton Kapustin, Topological Field Theory, Higher Categories, and Their Applications (arXiv:1004.2307)

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

• Daniel Freed, Edward Witten, Anomalies in String Theory with D-Branes (arXiv:hep-th/9907189)

More details are in

• Anton Kapustin, D-branes in a topologically nontrivial B-field , Adv. Theor. Math. Phys. 4, no. 1, pp. 127–154 (2000), (arXiv:hep-th/9909089)

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)