nLab
brane

Context

Quantum field theory

String theory

Phyics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

The term brane in formal high energy physics, and in particular in string theory, refers to entities that one thinks of as physical objects that generalize the notion point particles to higher dimensional objects.

The term derives from the word membrane that was originally used to describe 2-dimensional “particles”. When the need was felt to speak also about 3-, 4- and higher dimensional such “particles” the usage “3-brane”, “4-brane” etc. was introduced. Ordinary particles would be 0-branes in this counting, the strings in string theory would be 1-branes and membranes themselves 2-branes.

There are two fundamentally different concrete realizations of this somewhat vague notion.

  1. D-branes and other boundary conditions

  2. fundamental or σ\sigma-model branes or NS-branes

Boundary conditions or D-branes

In terms of the algebraic data of the QFT on the worldvolume

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. See at boundary field theory for more on this.

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

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 representaitons 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 moundary 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.

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.

(…)

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 :

Sich a sigma-model QFT is the quantization of an action functional on a space of maps ΣX\Sigma \to X from a cobordims (“worldvolume”) Σ\Sigma to some target space XX 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 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 smplest 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 form 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 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 …

Fundamental or σ\sigma-model branes

In string theory one speaks apart from the D-branes also about fundamental branes . These are the objects Σ\Sigma in the nn-dimensional sigma model themselves.

  • For n=0n=0 this describes the ordinary quantum mechanics of a point particles on XX. And such point particles are the fundamental particles for instance of the standard model of particle physics.

  • For n=1n=1 this describes the quantum propagation of a string, and accordingly one speaks of the fundamental string or F1-brane (fundamental 1-brane).

  • For n=2n=2 this describes the quantum propagation of a membrane.

  • There are good indications that there is a way to describe heterotic string theory not in terms of fundamental 1-branes but in terms of the sigma-model of a fundamental 5-brane – the magnetic dual of the 1-brane in 10-dimensions.

  • etc.

The brane scan.

The Green-Schwarz type super pp-brane sigma-models (see at table of branes for further links and see at The brane bouquet for the full classification):

=D\stackrel{D}{=}p=p =123456789
11M2M5
10D0F1, D1D2D3D4NS5, D5D6D7D8D9
9*\ast
8*\ast
7M2 top{}_{top}
6F1 little{}_{little}, S1 sd{}_{sd}S3
5*\ast
4*\ast*\ast
3*\ast

(The first colums follow the exceptional spinors table.)

The corresponding exceptional super L-∞ algebra cocycles (schematically, without prefactors):

=D\stackrel{D}{=}p=p =123456789
11Ψ 2E 2\Psi^2 E^2 on sIso(10,1)Ψ 2E 5+Ψ 2E 2C 3\Psi^2 E^5 + \Psi^2 E^2 C_3 on m2brane
10Ψ 2E 1\Psi^2 E^1 on sIso(9,1)B 2 2+B 2Ψ 2+Ψ 2E 2B_2^2 + B_2 \Psi^2 + \Psi^2 E^2 on StringIIA\cdots on StringIIBB 2 3+B 2 2Ψ 2+B 2Ψ 2E 2+Ψ 2E 4B_2^3 + B_2^2 \Psi^2 + B_2 \Psi^2 E^2 + \Psi^2 E^4 on StringIIAΨ 2E 5\Psi^2 E^5 on sIso(9,1)B 2 4++Ψ 2E 6B_2^4 + \cdots + \Psi^2 E^6 on StringIIA\cdots on StringIIBB 2 5++Ψ 2E 8B_2^5 + \cdots + \Psi^2 E^8 in StringIIA\cdots on StringIIB
9Ψ 2E 4\Psi^2 E^4 on sIso(8,1)
8Ψ 2E 3\Psi^2 E^3 on sIso(7,1)
7Ψ 2E 2\Psi^2 E^2 on sIso(6,1)
6Ψ 2E 1\Psi^2 E^1 on sIso(5,1)Ψ 2E 3\Psi^2 E^3 on sIso(5,1)
5Ψ 2E 2\Psi^2 E^2 on sIso(4,1)
4Ψ 2E 1\Psi^2 E^1 on sIso(3,1)Ψ 2E 2\Psi^2 E^2 on sIso(3,1)
3Ψ 2E 1\Psi^2 E^1 on sIso(2,1)

Black branes

See black brane .

Properties

Worldvolume theories

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
D2-brane\,\,\,
D4-brane\,\,D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane\,\,
D8-brane\,\,
(D=2n+1)(D = 2n+1)type IIB\,\,
D(-1)-brane\,\,\,
D1-brane\,\,2d CFT with BH entropy
D3-brane\,\,N=4 D=4 super Yang-Mills theory
D5-brane\,\,\,
D7-brane\,\,\,
D9-brane\,\,\,
(p,q)-string\,\,\,
(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\,
A-brane\,
B-brane\,
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
M-wave
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

The super-brane scan

If the worldvolume QFT of the fundamental branes (for instance the worlsheet 2dCFT of the string) is required to be a supersymmetric QFT?, specifically if the Green-Schwarz action functional is used only particular combinations of the dimenion dimΣ=p+1dim \Sigma = p + 1 of the worldvolume and D=dimXD = dim X of spacetime are possible.

The corresponding table has been called the brane scan

The brane scan

singularityfield theory with singularities
boundary condition/braneboundary field theory
domain wall/bi-braneQFT with defects

∞-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)

References

General

  • Greg Moore, What is… a brane?, Notices of the AMS vol 52, no. 2 (pdf)

  • Joan Simon, Brane Effective Actions, Kappa-Symmetry and Applications (arXiv:1110.2422)

Boundary conditions / D-branes

(…)

See D-brane.

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

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

Fundamental branes

The “brane scan” table showing the consistent dimension pairs for the Green-Schwarz action functional was depicted in

going back to

  • A. Achúcarro, J. M. Evans, Paul Townsend and D. L. Wiltshire, Super pp-branes Physics Letters B Volume 198, Issue 4, 3 (1987)

Further developments are in

More along these lines is in

See also division algebras and supersymmetry.

Branes ending on branes

Revised on July 2, 2014 11:10:22 by Urs Schreiber (82.136.246.44)