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.
For a broader perspective see at brane.
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 representations 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 boundary 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.
graphics grabbed from Ibanez-Uranga 12
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 :
Such a sigma-model QFT is the quantization of an action functional on a space of maps $\Sigma \to X$ from a cobordism (“worldvolume”) $\Sigma$ to some target space $X$ 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 $X$ 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 simplest 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 from 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 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 …
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 13, Chapter 18.5)
For D-branes in the WZW-model see WZW-model – D-branes.
from M-branes to F-branes: superstrings, D-branes and NS5-branes
(e.g. Johnson 97, Blumenhagen 10)
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).
See at K-theory classification of D-brane charge
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, traditonally 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 4) 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 $E_2$ pages, K-theory classes on spacetime $X$ 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)
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).
∞-Chern-Simons theory from binary and non-degenerate invariant polynomial
(adapted from Ševera 00)
The original article is
Joseph Polchinski, Dirichlet-Branes and Ramond-Ramond Charges, Phys. Rev. Lett. 75:4724-4727, 1995 (arXiv:hep-th/9510017)
from p.7:
although it appears that we have modified the type II theory by adding something new to it, we are now arguing that these objects are actually intrinsic to any nonperturbative formulation of the type II theory; presumably one should think of them as an alternate representation of the black $p$-branes
General review:
Joseph Polchinski, TASI Lectures on D-Branes (arXiv:hep-th/9611050)
Clifford Johnson, Études on D-Branes, in: Mike Duff et. al. (eds.) Nonperturbative aspects of strings, branes and supersymmetry, Proceedings, Trieste, Italy, March 23-April 3, 1998 (arXiv:hep-th/9812196, spire:481393)
Koji Hashimoto, D-Brane – Superstrings and New Perspective of Our World, Springer 2012 (doi:10.1007/978-3-642-23574-0, spire:1188897)
Pietro Fre, The Branes: Three Viewpoints, In: Gravity, a Geometrical Course Springer 2013 (spire:1242195, doi:10.1007/978-94-007-5443-0_7)
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
Review includes
Frederik Roose, Strings and D-branes on orbifolds: from boundary states to geometry, 2001 (pdf)
Nikolas Prezas, Aspects of branes and orbifolds in string theory, 2002 (pdf, web)
See also the references at orientifold.
The Green-Schwarz sigma model with DBI action for D-branes is discussed in
Martin Cederwall, Alexander von Gussich, Bengt Nilsson, Per Sundell, Anders Westerberg, The Dirichlet Super-p-Branes in Ten-Dimensional Type IIA and IIB Supergravity, Nucl.Phys. B490 (1997) 179-201 (arXiv:hep-th/9611159)
Mina Aganagic, Jaemo Park, Costin Popescu, John Schwarz, Dual D-Brane Actions, Nucl. Phys. B496 (1997) 215-230 (arXiv:hep-th/9702133)
Discussion of Green-Schwarz action functionals for super D-branes and the interpretation of the WZW cocycles for the D-branes as cocycles on “extended super-Minkowski spacetime” is due to
C. Chryssomalakos, José de Azcárraga, J. M. Izquierdo and C. Pérez Bueno, The geometry of branes and extended superspaces, Nuclear Physics B Volume 567, Issues 1–2, 14 February 2000, Pages 293–330 (arXiv:hep-th/9904137)
Makoto Sakaguchi, IIB-Branes and New Spacetime Superalgebras, JHEP 0004 (2000) 019 (arXiv:hep-th/9909143)
José de Azcárraga, J. M. Izquierdo, Superalgebra cohomology, the geometry of extended superspaces and superbranes (arXiv:hep-th/0105125)
See also division algebras and supersymmetry.
A corresponding discussion as ∞-Wess-Zumino-Witten theory and refinement of the brane scan to a “brane bouquet” of super L-∞ algebra extensions (hence in infinity-Lie theory via ∞-Wess-Zumino-Witten theory) is discussed in
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields, International Journal of Geometric Methods in Modern Physics Volume 12, Issue 02 (2015) 1550018 (arXiv:1308.5264)
Domenico Fiorenza, Hisham Sati, Urs Schreiber The WZW term of the M5-brane and differential cohomotopy, J. Math. Phys. 56, 102301 (2015) (arXiv:1506.07557)
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Rational sphere valued supercocycles in M-theory and type IIA string theory (arXiv:1606.03206)
Vincent Braunack-Mayer, Hisham Sati, Urs Schreiber, Gauge enhancement of Super M-Branes (arXiv:1806.01115)
On the conjectural D-brane charge quantization in topological K-theory:
The idea that D-branes have Dirac charge quantization in topological K-theory originates with:
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)
Daniel Freed, Michael Hopkins, On Ramond-Ramond fields and K-theory, JHEP 0005 (2000) 044 (arXiv:hep-th/0002027)
and with emphasis on the full picture of twisted differential K-theory in:
Here:
Expression of these D-brane K-theory classes via the Atiyah-Hirzebruch spectral sequence:
Juan Maldacena, Gregory Moore, Nathan Seiberg, D-Brane Instantons and K-Theory Charges, JHEP 0111:062,2001 (arXiv:hep-th/0108100)
Jarah Evslin, Hisham Sati, Can D-Branes Wrap Nonrepresentable Cycles?, JHEP0610:050,2006 (arXiv:hep-th/0607045)
Specifically for D-branes in WZW models see
Discussion of D-brane matrix models taking these K-theoretic effects into account (K-matrix model) is in
Discussion of charge quantization in twisted K-theory for the case of non-vanishing B-field:
Witten 98, Sec. 5.3 (for torsion twists)
Peter Bouwknegt, Varghese Mathai, D-branes, B-fields and twisted K-theory, Int. J. Mod. Phys. A16:693-706, 2001 (arXiv:hep-th/0002023)
An elaborate proposal for the correct flavour of equivariant KR-theory needed for orientifolds is sketched in:
Discussion of full-blown twisted differential K-theory and its relation to D-brane charge in type II string theory
Discussion of full-blown twisted differential orthogonal K-theory and its relation to D-brane charge in type I string theory (on orientifolds):
Kasper Olsen, Richard Szabo, Brane Descent Relations in K-theory, Nucl.Phys. B566 (2000) 562-598 (arXiv:hep-th/9904153)
Kasper Olsen, Richard Szabo, Constructing D-Branes from K-Theory, Adv. Theor. Math. Phys. 3 (1999) 889-1025 (arXiv:hep-th/9907140)
John Schwarz, TASI Lectures on Non-BPS D-Brane Systems (arXiv:hep-th/9908144)
Edward Witten, Overview Of K-Theory Applied To Strings, Int. J. Mod. Phys. A16:693-706, 2001 (arXiv:hep-th/0007175)
Greg Moore, K-Theory from a physical perspective, in: Ulrike Tillmann (ed.) Topology, Geometry and Quantum Field Theory, Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal, Cambridge University Press (2004) (arXiv:hep-th/0304018, doi:10.1017/CBO9780511526398.011)
Juan José Manjarín, Topics on D-brane charges with B-fields, Int. J. Geom. Meth. Mod. Phys. 1 (2004) (arXiv:hep-th/0405074)
Jarah Evslin, What Does(n’t) K-theory Classify?, Modave Summer School in Mathematical Physics (arXiv:hep-th/0610328, spire:730502)
Stefan Fredenhagen, Physical Background to the K-Theory Classification of D-Branes: Introduction and References (doi:10.1007/978-3-540-74956-1_1), chapter in: Dale Husemoeller, Michael Joachim, Branislav Jurčo, Martin Schottenloher, Basic Bundle Theory and K-Cohomology Invariants, Lecture Notes in Physics, Springer 2008 (doi:10.1007/978-3-540-74956-1, pdf)
Fabio Ruffino, Topics on topology and superstring theory (arXiv:0910.4524)
See also for instance
Discussion of D-branes in KK-theory is reviewed in
based on
Rui Reis, Richard Szabo, Geometric K-Homology of Flat D-Branes ,Commun.Math.Phys. 266 (2006) 71-122 (arXiv:hep-th/0507043)
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)
Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, Richard Szabo, D-branes, KK-theory and duality on noncommutative spaces, J. Phys. Conf. Ser. 103:012004, 2008 (arXiv:0709.2128)
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
But there remain conceptual issues with the proposal that D-brane charge is in K-theory, as highlighted in
Jan de Boer, Robbert Dijkgraaf, Kentaro Hori, Arjan Keurentjes, John Morgan, David Morrison, Savdeep Sethi, section 4.5.2 and 4.6.5 of Triples, Fluxes, and Strings, Adv. Theor. Math. Phys. 4 (2002) 995-1186 (arXiv:hep-th/0103170)
Jarah Evslin, section 8 of: What Does(n’t) K-theory Classify?, Second Modave Summer School in Mathematical Physics (arXiv:hep-th/0610328)
In particular, actual checks of the proposal that D-brane charge is given by K-theory, via concrete computation in boundary conformal field theory, have revealed some subtleties:
Stefan Fredenhagen, Thomas Quella, Generalised permutation branes, JHEP 0511:004, 2005 (arXiv:hep-th/0509153)
It might surprise that despite all the progress that has been made in understanding branes on group manifolds, there are usually not enough D-branes known to explain the whole charge group predicted by (twisted) K-theory.
The closest available towards an actual check of the argument for K-theory via open superstring tachyon condensation (Witten 98, Section 3) seems to be
which, however, concludes (on p. 32) with:
It would also be interesting to see if these developments can shed light on the long-speculated relation between string field theory and the K-theoretic description of D-brane charge $[$75, 76, 77$]$. We leave these questions for future work.
See also
which still lists (on p. 112) among open problems of string field theory:
“Are there topological invariants of the open string star algebra representing D-brane charges?”
The proposal that D-brane charge on orbifolds is measured in equivariant K-theory (orbifold K-theory) goes back to
It was pointed out that only a subgroup of equivariant K-theory can be physically relevant in
Further discussion of equivariant K-theory for D-branes on orbifolds includes the following:
Hugo García-Compeán, D-branes in orbifold singularities and equivariant K-theory, Nucl.Phys. B557 (1999) 480-504 (arXiv:hep-th/9812226)
Matthias Gaberdiel, Bogdan Stefanski, Dirichlet Branes on Orbifolds, Nucl.Phys.B578:58-84, 2000 (arXiv:hep-th/9910109)
Igor Kriz, Leopoldo A. Pando Zayas, Norma Quiroz, Comments on D-branes on Orbifolds and K-theory, Int.J.Mod.Phys.A23:933-974, 2008 (arXiv:hep-th/0703122)
Richard Szabo, Alessandro Valentino, Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory, Commun.Math.Phys.294:647-702, 2010 (arXiv:0710.2773)
Discussion of real K-theory for D-branes on orientifolds includes the following:
The original observation that D-brane charge for orientifolds should be in KR-theory is due to
and was then re-amplified in
Sergei Gukov, K-Theory, Reality, and Orientifolds, Commun.Math.Phys. 210 (2000) 621-639 (arXiv:hep-th/9901042)
Oren Bergman, E. Gimon, Shigeki Sugimoto, Orientifolds, RR Torsion, and K-theory, JHEP 0105:047, 2001 (arXiv:hep-th/0103183)
With further developments in
Discussion of orbi-orienti-folds using equivariant KO-theory is in
N. Quiroz, Bogdan Stefanski, Dirichlet Branes on Orientifolds, Phys.Rev. D66 (2002) 026002 (arXiv:hep-th/0110041)
Volker Braun, Bogdan Stefanski, Orientifolds and K-theory (arXiv:hep-th/0206158)
H. Garcia-Compean, W. Herrera-Suarez, B. A. Itza-Ortiz, O. Loaiza-Brito, D-Branes in Orientifolds and Orbifolds and Kasparov KK-Theory, JHEP 0812:007, 2008 (arXiv:0809.4238)
Discussion of the alleged K-theory classification of D-brane charge in relation to the M-theory C-field is in
See also
More complete discussion of double dimensional reduction of the supergravity C-field in 11d to the expected B-field and RR-field flux forms in 10d:
Varghese Mathai, Hisham Sati, Some Relations between Twisted K-theory and $E_8$ Gauge Theory, JHEP0403:016,2004 (arXiv:hep-th/0312033)
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Rational sphere valued supercocycles in M-theory and type IIA string theory, Journal of Geometry and Physics, Volume 114, Pages 91-108 April 2017 (arXiv:1606.03206, doi:10.1016/j.geomphys.2016.11.024)
The entropy of D-branes scales with the square of their number:
Around equation (3) in
Around (3) in:
Around equation (5.5) in:
See also:
Mark van Raamsdonk, The Amazing Matrix in String Theory (2006):
Most of the dramatic progress in string theory over the past decade has in some way involved the degrees of freedom of D-branes, solitonic objects in string theory whose low energy physics is described by quantum field theories living on the branes. Essential to many of the applications of D-brane physics is the fact that the number of degrees of freedom of a collection of branes scales not as the number of branes, but as the square of the number of branes.
Around (1.1) 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
4, no. 1, pp. 127–154 (2000), (arXiv:hep-th/9909089)
A clean review is provided in
For more see at Freed-Witten anomaly cancellation.