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.
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
a modular tensor category $\mathcal{C}$ (to be thought of as being the category of representaitons of the vertex operator algebra of the 2d CFT);
a special symmetric Frobenius algebra object $A$ internal to $\mathcal{C}$.
In this formulation a type of brane of the theory is precisely an $A$-module in $\mathcal{C}$ (an $A$-bimodule is a bi-brane or defect line ):
the 2d cobordisms with boundary on which the theory defined by $A \in \mathcal{C}$ carry as extra structure on their connected boundary pieces a label given by an equivalence class of an $A$-module in $\mathcal{C}$. 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 $\mathcal{C}$.
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.
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.
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
For D-branes in the WZW-model see WZW-model – D-branes.
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.
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 $i \colon Q \hookrightarrow X$ a submanifold regarded as the worldvolume of a D-brane. For $\nabla_B \colon X \to \mathbf{B}^2 U(1)_{conn}$ the circle 2-bundle with connection which models the background B-field, write $\chi_B \colon X \to \mathbf{B}^2 U(1)$ 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 $\xi$ in
Assuming that $i \colon Q \to X$ 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 $C^\ast$ this is incarnated as a Hilbert bimodule which defines a class in twisted operator K-theory, realized as the following comoposite in KK-theory
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 $C(Q)_{i^\ast \chi B}$ and the pullback/restriction $i^\ast \chi_B$ of the background B-field to the brane;
$i!$ is the push-forward (Umkehr map) dual to $i^\ast \colon C(X)_{\chi_B} \to C(Q)_{i^\ast \chi_B}$, realizes as a KK-theory class
The corresponding D-brane charge in KK-theory is the resulting composite (relative index)
in twisted K-theory. Traditionally only the image of this under the Chern character
in real cohomology/cyclic cohomology is considered, $ch(D_Q(\xi))$. Moreover, traiditonally one thinks of first applying $ch$ to $\xi$ and then pushing forward in $HL$. 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.
Table of branes appearing in supergravity/string theory (for classification see at brane scan).
∞-Chern-Simons theory from binary and non-degenerate invariant polynomial
(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
Discussion from the point of view of Green-Schwarz action functional-∞-Wess-Zumino-Witten theory is in
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
Discussion of D-branes in KK-theory is reviewed in
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)
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
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
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
For more see at Freed-Witten anomaly cancellation.