An abstractly defined nn-dimensional quantum field theory is a consistent assignment of state-space and correlators to nn-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.

For a broader perspective see at brane.

In 2d2d 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\mathcal{C} (to be thought of as being the category of representations of the vertex operator algebra of the 2d CFT);

  2. a special symmetric Frobenius algebra object AA internal to 𝒞\mathcal{C}.

In this formulation a type of brane of the theory is precisely an AA-module in 𝒞\mathcal{C} (an AA-bimodule is a bi-brane or defect line ):

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

  • triangulating the cobordism,

  • labeling all internal edges by AA

  • labelling all boundary pieces by the AA-module

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

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

  • and finally evaluating the resulting string diagram in 𝒞\mathcal{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.

graphics grabbed from Ibanez-Uranga 12

In 2d2d 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 σ\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 :

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

One may therefore try to match the geometric data on XX that encodes the σ\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 AA-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 XX. 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 XX is a simple Lie group X=GX = G and the background field is a circle 2-bundle with connection (a bundle gerbe) on GG, 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 XX are given in the simplest case by conjugacy classes DGD \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 from a quantization of spaces of maps ΣG\Sigma \to G that are restricted to take the boundary of Σ\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 GG. These may be quite far from having a direct interpretation as submanifolds of GG.

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 collection 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 …

As black branes

In perturbative string theory, hence for small string coupling constant the D-branes are incarnated as boundary conditions for the string’s worldsheet 2d CFT, exhibiting submanifolds in spacetime. As the string coupling constant increases and becomes non-perturbative, this picture of perturbative string theory breaks down, but at low energy (large scales) now supergravity becomes a good description, and now the D-branes are incarnated as black branes.

graphics grabbed from Ibanez-Uranga 12

This transition is also the key to understanding black holes in string theory.

1/2 BPS black branes in supergravity: D-branes, F1-brane, NS5-brane, M2-brane, M5-brane

(table taken from Blumenhagen-Lüst-Theisen “Basic concepts of string theory”)


Various dimensions

In type IIA supergravity

In type IIB supergravity

In the WZW model

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


As F-branes originating from M-branes

from M-branes to F-branes: superstrings, D-branes and NS5-branes

M-theory on S A 1×S B 1S^1_A \times S^1_B-elliptic fibrationKK-compactification on S A 1S^1_Atype IIA string theoryT-dual KK-compactification on S B 1S^1_Btype IIB string theoryF-theory on elliptically fibered-K3 fibrationduality between F-theory and heterotic string theoryheterotic string theory on elliptic fibration
M2-brane wrapping S A 1S_A^1double dimensional reduction \mapstotype IIA superstring\mapstotype IIB superstring\mapstoheterotic superstring
M2-brane wrapping S B 1S_B^1\mapstoD2-brane\mapstoD1-brane
M2-brane wrapping pp times around S A 1S_A^1 and qq times around S B 1S_B^1\mapstopp strings and qq D2-branes\mapsto(p,q)-string
M5-brane wrapping S A 1S_A^1double dimensional reduction \mapstoD4-brane\mapstoD5-brane
M5-brane wrapping S B 1S_B^1\mapstoNS5-brane\mapstoNS5-brane\mapstoNS5-brane
M5-brane wrapping pp times around S A 1S_A^1 and qq times around S B 1S_B^1\mapstopp D4-brane and qq NS5-branes\mapsto(p,q)5-brane
M5-brane wrapping S A 1×S B 1S_A^1 \times S_B^1\mapsto\mapstoD3-brane
KK-monopole/A-type ADE singularity (degeneration locus of S A 1S^1_A-circle fibration, Sen limit of S A 1×S B 1S^1_A \times S^1_B elliptic fibration)\mapstoD6-brane\mapstoD7-branesA-type nodal curve cycle degenertion locus of elliptic fibration ADE 2Cycle (Sen 97, section 2)SU-gauge enhancement
KK-monopole orientifold/D-type ADE singularity\mapstoD6-brane with O6-planes\mapstoD7-branes with O7-planesD-type nodal curve cycle degenertion locus of elliptic fibration ADE 2Cycle (Sen 97, section 3)SO-gauge enhancement
exceptional ADE-singularity\mapsto\mapstoexceptional ADE-singularity of elliptic fibration\mapstoE6-, E7-, E8-gauge enhancement

(e.g. Johnson 97, Blumenhagen 10)

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 XX be a manifold regarded as spacetime and i:QXi \colon Q \hookrightarrow X a submanifold regarded as the worldvolume of a D-brane. For B:XB 2U(1) conn\nabla_B \colon X \to \mathbf{B}^2 U(1)_{conn} the circle 2-bundle with connection which models the background B-field, write χ B:XB 2U(1)\chi_B \colon X \to \mathbf{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 ξ\xi in

Q i * ξ X χ B B 2U(1) Q i * ξ i *χ B id X χ B B 2U(1). \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:QXi \colon Q \to X is K-oriented in that for instance XX has a spin-structure and QQ a spin^c-structure, then under the groupoid convolution algebra functor C *C^\ast this is incarnated as a Hilbert bimodule which defines a class in twisted operator K-theory, realized as the following comoposite in KK-theory

Γ(ξ)C(Q) i *χ Bi !C(X) χ B, \mathbb{C} \stackrel{\Gamma(\xi)}{\to} C(Q)_{i^\ast \chi_B} \stackrel{i_!}{\to} C(X)_{\chi_B} \,,


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

i !(ξ)=D Q(ξ)KK(,C(X) χ B)K [χ 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

ch:KKHL ch \colon KK \to HL

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

ch(D Q(ξ)) C(X) χ BToddHL, 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.)

For more general discussion see at Freed-Witten anomaly – Details as well as at Poincaré duality algebra – Properties – K-Orientation and Umkehr maps.

Via the Atiyah-Hirzebruch spectral sequence

The Atiyah-Hirzeburch spectral sequence expresses, starting from its E 2E_2 pages, K-theory classes on spacetime XX as kernels of certain differential acting on ordinary cohomology in all even degrees (for type IIA strings) or all odd degrees (for type IIB strings)

E 2 p,q=H p(X,KU q(*))KU (X). E_2^{p,q} = H^p(X, KU^q(\ast)) \Rightarrow KU^\bullet(X) \,.

Discussion of D-brane charge this way is in (Maldacena-Moore-Seiberg 01, Evslin-Sati 06).

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

branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
(D=2n)(D = 2n)type IIA\,\,
D0-brane\,\,BFSS matrix model
D4-brane\,\,D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane\,\,D=7 super Yang-Mills theory
(D=2n+1)(D = 2n+1)type IIB\,\,
D1-brane\,\,2d CFT with BH entropy
D3-brane\,\,N=4 D=4 super Yang-Mills theory
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection\,
string\,B2-field2d SCFT
NS5-brane\,B6-fieldlittle string theory
D-brane for topological string\,
M-brane11D SuGra/M-theorycircle n-connection\,
M2-brane\,C3-fieldABJM theory, BLG model
M5-brane\,C6-field6d (2,0)-superconformal QFT
M9-brane/O9-planeheterotic string theory
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane\,C6-field on G2-manifold
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d

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

nn \in \mathbb{N}symplectic Lie n-algebroidLie integrated smooth ∞-groupoid = moduli ∞-stack of fields of (n+1)(n+1)-d sigma-modelhigher symplectic geometry(n+1)(n+1)d sigma-modeldg-Lagrangian submanifold/ real polarization leaf= brane(n+1)-module of quantum states in codimension (n+1)(n+1)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
nnsymplectic Lie n-algebroidsymplectic n-groupoid(n+1)-plectic geometryd=n+1d = n+1 AKSZ sigma-model

(adapted from Ševera 00)



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

Discussion with an eye towards string phenomenology is in

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

Comments on the role of D-branes in mathematical physics and mathematics is in

As higher super-GS-WZW type σ\sigma-models

Discussion from the point of view of Green-Schwarz action functional-∞-Wess-Zumino-Witten theory is in

K-theoretic description and D-brane charge

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

See also at anti-D-brane.

Review of the physical lpicture includes

A textbook account of D-brane charge in (twisted) topological K-theory is

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

based on

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

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 *C^\ast-algebras (arXiv:0906.5400)

  • Snigdhayan Mahanta, Higher nonunital Quillen KK'-theory, KK-dualities and applications to topological 𝕋\mathbb{T}-duality, Journal of Geometry and Physics, Volume 61, Issue 5 2011, p. 875-889. (pdf)

Discussion of D-brane matrix models taking these K-theoretic effects into account (K-matrix model) is in

  • T. Asakawa, S. Sugimoto, S. Terashima, D-branes, Matrix Theory and K-homology, JHEP 0203 (2002) 034 (arXiv:hep-th/0108085)

Via the Atiyah-Hirzebruch spectral sequence

Expression of these D-brane K-theory classes via the Atiyah-Hirzebruch spectral sequence is discussed in

Detailed review of this is in

For rational CFT

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

Branes within branes

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)

Last revised on April 13, 2018 at 04:34:07. See the history of this page for a list of all contributions to it.