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 D-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 boundary 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.
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.
Such a sigma-model QFT is the quantization of an action functional on a space of maps from a cobordism (“worldvolume”) to some target space 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 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 simplest 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 from 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 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.
… lots of further things to say …
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 be a manifold regarded as spacetime and a submanifold regarded as the worldvolume of a D-brane. For the circle 2-bundle with connection which models the background B-field, write for the underlying circle 2-group-principal 2-bundle.
Assuming that is K-oriented in that for instance has a spin-structure and a spin^c-structure, then under the groupoid convolution algebra functor this is incarnated as a Hilbert bimodule which defines a class in twisted operator K-theory, realized as the following comoposite in KK-theory
is the groupoid convolution algebra of sections of regarded as a centrally extended groupoid over a Cech groupoid resolution of which supports a Cech cocycle for , and similarly for and the pullback/restriction of the background B-field to the brane;
The corresponding D-brane charge in KK-theory is the resulting composite (relative index)
in real cohomology/cyclic cohomology is considered, . Moreover, traiditonally one thinks of first applying to and then pushing forward in . By the C*-algebraic Grothendieck-Riemann-Roch theorem this gives the isomorphic expression
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.
The Atiyah-Hirzeburch spectral sequence expresses, starting from its pahes, K-theory classes on spacetime 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)
|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|
|(D25-brane)||(bosonic string theory)|
|NS-brane||type I, II, heterotic||circle n-connection|
|NS5-brane||B6-field||little string theory|
|D-brane for topological string|
|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|
|M9-brane/O9-plane||heterotic string theory|
|topological M2-brane||topological M-theory||C3-field on G2-manifold|
|topological M5-brane||C6-field on G2-manifold|
|solitons on M5-brane||6d (2,0)-superconformal QFT|
|self-dual string||self-dual B-field|
|3-brane in 6d|
|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)
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 idea that the physics of D-branes is described by K-theory originates in
Discussion of D-branes in KK-theory is reviewed in
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
More on this, with more explicit relation to noncommutative motives, is in
Expression of these D-brane K-theory classes via the Atiyah-Hirzebruch spectral sequence is discussed in
Detailed review of this is in
For exhaustive details on D-branes in 2-dimensional rational CFT see the references given at
A discussion of topological D-branes in the context of higher category theory is in
More details are in
A clean review is provided in
For more see at Freed-Witten anomaly cancellation.