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 the observation that their charge expressed in RR-field flux densities resembles the image of a Chern character:
Michael Green, Jeffrey A. Harvey, Gregory Moore, I-Brane Inflow and Anomalous Couplings on D-Branes, Class. Quant. Grav. 14 (1997) 47-52 $[$arXiv:hep-th/9605033, doi:10.1088/0264-9381/14/1/008$]$
Ruben Minasian, Gregory Moore, K-theory and Ramond-Ramond charge, JHEP 9711:002 (1997) $[$arXiv:hep-th/9710230, doi:10.1088/1126-6708/1997/11/002$]$
Further early discussion:
Edward Witten, D-Branes And K-Theory, JHEP 9812:019 (1998) [arXiv:hep-th/9810188, doi:10.1088/1126-6708/1998/12/019]
Petr Hořava, Type IIA D-Branes, K-Theory, and Matrix Theory (1998). (hep-th/9812135).
Daniel Freed, Michael Hopkins, On Ramond-Ramond fields and K-theory, JHEP 0005 (2000) 044 $[$arXiv:hep-th/0002027, doi:10.1088/1126-6708/2000/05/044$]$
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
Understanding the solitonic (non-singular) D-branes and their T-duality in K-theory:
Towards a matrix model taking these K-theoretic effects into account (K-matrix model):
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. A 16 (2001) 693-706 $[$arXiv:hep-th/0002023, doi:10.1088/1126-6708/2000/03/007$]$
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) 1-9 $[$doi:10.1007/978-3-540-74956-1, pdf$]$
Fabio Ruffino, Topics on topology and superstring theory (arXiv:0910.4524)
Hisham Sati, Urs Schreiber: Flux Quantization, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2402.18473]
Amplification of torsion-charges implied by charge quantization in Ktheory
Volker Braun, K-Theory Torsion [arXiv:hep-th/0005103]
Ilka Brunner, Jacques Distler, Torsion D-Branes in Nongeometrical Phases, Adv. Theor. Math. Phys. 5 (2002) 265-309 [doi:10.4310/ATMP.2001.v5.n2.a3, arXiv:hep-th/0102018]
Ilka Brunner, Jacques Distler, Rahul Mahajan, Return of the Torsion D-Branes, Adv. Theor. Math. Phys. 5 (2002) 311-352 [doi:10.4310/ATMP.2001.v5.n2.a4, arXiv:hep-th/0106262]
Review of D-branes charge seen in KK-theory:
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 (2008) 643-706 $[$arXiv:hep-th/0607020, doi:10.1007/s00220-007-0396-y$]$
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, doi:10.1088/1742-6596/103/1/012004$]$
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, pdf]
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, doi:10.1088/1126-6708/2005/11/004]
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. […] it is fair to say that a satisfactory answer is still missing.
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. A 23 (2008) 933-974 [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)
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