group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The quarge quantization of D-brane charge/RR-fields in twisted equivariant K-theory differential topological K-theory.
(…)
Under AdS/CFT duality in solid state physics the K-theory classification of topological phases of matter corresponds to the K-theory classification of D-brane charge (Ryu-Takayanagi 10a, Ryu-Takayanagi 10v).
The idea that D-branes have Dirac charge quantization in topological 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)
Daniel Freed, Michael Hopkins, On Ramond-Ramond fields and K-theory, JHEP 0005 (2000) 044 (arXiv:hep-th/0002027)
See also at anti-D-brane.
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):
But there remain conceptual issues with the proposal that D-brane charge is in K-theory, as highlighted for in
Jan de Boer, Robbert Dijkgraaf, Kentaro Hori, Arjan Keurentjes, John Morgan, David Morrison, Savdeep Sethi, section 4.5 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, JHEP0511:004, 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.
Further review and discussion of D-brane charge in K-theory includes the following
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 (arXiv:hep-th/0304018)
Juan José Manjarín, Topics on D-brane charges with B-fields, Int.J.Geom.Meth.Mod.Phys. 1 (2004) (arXiv:hep-th/0405074)
A textbook account of D-brane charge in (twisted) topological K-theory is
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
Specifically for D-branes in WZW models see
More on this, with more explicit relation to noncommutative motives, is in
Snigdhayan Mahanta, Noncommutative correspondence categories, simplicial sets and pro $C^\ast$-algebras (arXiv:0906.5400)
Snigdhayan Mahanta, Higher nonunital Quillen $K'$-theory, KK-dualities and applications to topological $\mathbb{T}$-duality, Journal of Geometry and Physics, Volume 61, Issue 5 2011, p. 875-889. (pdf)
Discussion of D-brane matrix models taking these K-theoretic effects into account (K-matrix model) is in
The proposal that D-brane charge on orbifolds is measured in equivariant K-theory goes back to
but 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)
An elaborate proposal for the correct flavour of real equivariant K-theory needed for orientifolds is sketched in
Discussion of the alleged K-theory classification of D-brane charge in relation to the M-theory supergravity C-field is in
See also
For more on this perspective as 10d type II as a self-dual higher gauge theory in the boudnary of a kind of 11-d Chern-Simons theory is in
More complete discussion of the decomposition of the supergravity C-field as one passes from 11d to 10d is in
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