nLab D-brane charge quantization in topological K-theory -- references

D-brane charge quantization in topological K-theory

On the conjectural D-brane charge quantization in topological K-theory:

Origin and basics

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:

Daniel Freed, Dirac charge quantization and generalized differential cohomology, Surveys in Differential Geometry, Int. Press, Somerville, MA, 2000, pp. 129–194 (arXiv:hep-th/0011220, doi:10.4310/SDG.2002.v7.n1.a6, spire:537392)

Here:

Minasian-Moore 97 observe that differential form-expressions derived for D-brane charge look like images of K-theory classes under the Chern character;

Witten 98, Section 3 adds the observation that the tachyon condensation – which is expected (this is Sen's conjecture from Sen 98) for open strings between D-brane/anti-D-branes – plausibly implements on their Chan-Paton vector bundles the defining equivalence relation (here) of topological K-theory.

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

Peter Bouwknegt, A note on equality of algebraic and geometric D-brane charges in WZW models (pdf)

Discussion of D-brane matrix models taking these K-theoretic effects into account (K-matrix model) is in

T. Asakawa, S. Sugimoto, S. Terashima, D-branes, Matrix Theory and K-homology, JHEP 0203 (2002) 034 (arXiv:hep-th/0108085)

Twisted, equivariant and differential refinement

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:

Jacques Distler, Dan Freed, Greg Moore, Orientifold Précis in: Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory Proceedings of Symposia in Pure Mathematics, AMS (2011) (arXiv:0906.0795, slides)

Discussion of full-blown twisted differential K-theory and its relation to D-brane charge in type II string theory

Daniel Grady, Hisham Sati, Ramond-Ramond fields and twisted differential K-theory (arXiv:1903.08843)

Discussion of full-blown twisted differential orthogonal K-theory and its relation to D-brane charge in type I string theory (on orientifolds):

Daniel Grady, Hisham Sati, Twisted differential KO-theory (arXiv:1905.09085)

Reviews

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)

Conceptual problems

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

Theodore Erler, Analytic Solution for Tachyon Condensation in Berkovits’ Open Superstring Field Theory, JHEP 1311 (2013) 007 (arXiv:1308.4400)

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.

For orbifolds in equivariant K-theory

The proposal that D-brane charge on orbifolds is measured in equivariant K-theory (orbifold K-theory) goes back to

Witten 98, section 5.1

It was pointed out that only a subgroup of equivariant K-theory can be physically relevant in

Jan de Boer, Robbert Dijkgraaf, Kentaro Hori, Arjan Keurentjes, John Morgan, David Morrison, Savdeep Sethi, around (137) of: Triples, Fluxes, and Strings, Adv.Theor.Math.Phys. 4 (2002) 995-1186 (arXiv:hep-th/0103170)

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

Witten 98, section 5

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

Varghese Mathai, Michael Murray, Daniel Stevenson, Type I D-branes in an H-flux and twisted KO-theory, JHEP 0311 (2003) 053 (arXiv:hep-th/0310164)

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

Duiliu-Emanuel Diaconescu, Gregory Moore, Edward Witten, $E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory, Adv.Theor.Math.Phys.6:1031-1134,2003 (arXiv:hep-th/0005090), summarised in A Derivation of K-Theory from M-Theory (arXiv:hep-th/0005091)