FQFT and cohomology
Types of quantum field thories
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.
An abstractly defined -dimensional quantum field theory is a consistent assignment of state-space and correlators to -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 brane is a type of data assigned by the QFT to boundaries of cobordisms.
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
the 2d cobordisms with boundary on which the theory defined by carry as extra structure on their connected boundary pieces a label given by an equivalence class of an -module in . The assignment of the CFT to such a cobordism with boundary is obtained by
triangulating the cobordism,
labeling all internal edges by
labelling all boundary pieces by the -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 .
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.
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.
Sich a sigma-model QFT is the quantization of an action functional on a space of maps from a cobordims (“worldvolume”) to some target space 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 that encodes the -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 -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 . 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 is a simple Lie group and the background field is a circle 2-bundle with connection (a bundle gerbe) on , 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 are given in the smplest case by conjugacy classes 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 that are restricted to take the boundary of 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 . These may be quite far from having a direct interpretation as submanifolds of .
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.
… lots of further things to say …
For this describes the ordinary quantum mechanics of a point particles on . And such point particles are the fundamental particles for instance of the standard model of particle physics.
For this describes the quantum propagation of a string, and accordingly one speaks of the fundamental string or F1-brane (fundamental 1-brane).
For 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.
The brane scan.
|10||D0||F1, D1||D2||D3||D4||NS5, D5||D6||D7||D8||D9|
|11||on sIso(10,1)||on m2brane|
|10||on sIso(9,1)||on StringIIA||on StringIIB||on StringIIA||on sIso(9,1)||on StringIIA||on StringIIB||in StringIIA||on StringIIB|
|6||on sIso(5,1)||on sIso(5,1)|
|4||on sIso(3,1)||on sIso(3,1)|
See black brane .
|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|
|D0-brane||BFSS matrix model|
|D4-brane||D=5 super Yang-Mills theory with Khovanov homology observables|
|D1-brane||2d CFT with BH entropy|
|D3-brane||N=4 D=4 super Yang-Mills theory|
|NS-brane||type I, II, heterotic||circle n-connection|
|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|
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 of the worldvolume and of spacetime are possible.
The corresponding table has been called the brane scan
|singularity||field theory with singularities|
|boundary condition/brane||boundary field theory|
|domain wall/bi-brane||QFT with defects|
|symplectic Lie n-algebroid||Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of -d sigma-model||higher symplectic geometry||d sigma-model||dg-Lagrangian submanifold/ real polarization leaf||= brane||(n+1)-module of quantum states in codimension||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|
|symplectic Lie n-algebroid||symplectic n-groupoid||(n+1)-plectic geometry||AKSZ sigma-model|
(adapted from Ševera 00)
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
The “brane scan” table showing the consistent dimension pairs for the Green-Schwarz action functional was depicted in
going back to
Further developments are in
More along these lines is in
See also division algebras and supersymmetry.