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If the field of electromagnetism serves as a background gauge field for electrically charged quantum particles it is subject to a quantization condition: Outside the locus of any magnetic charge – for instance a magnetic monopole topological defect – the electromagnetic field must be a connection on a principal U(1) bundle whose first Chern class is the discrete measure for the units of magnetic charge. Equivalently this means that the electromagnetic field is a cocycle in ordinary differential cohomology of degree 2.
In the underlying topological sector (“monopole”/“instantons”-sector) this is integral cohomology in degree-2, whose classifying space is equivalently the infinite complex projective space $B U(1) \simeq \mathbb{C}P^\infty$:
$\,$
This goes back to an insight due to Dirac 31. See Heras 18 for traditional elementary review, but see Alvarez 85, Frankel and Mangiarotti-Sardanashvily 00 for exposition of the modern picture in terms of fiber bundles in physics. See Freed 00, Section 2 for review in terms of differential cohomology with outlook to generalization to higher gauge fields in string theory (more on which in the references below).
A closely related phenomenon is the magnetic flux quantization in type II superconductors, see there.
On the locus of the magnetic charge itself the situation is more complex. There the magnetic current is given by a cocycle in ordinary differential cohomology of degree 3 (with compact support) and now the electromagnetic field is a connection on a twisted bundle (Freed 00, Section 2).
A similar charge quantization condition govers monopoles in SU(2)-Yang-Mills theory, see at moduli space of monopoles. Here the Atiyah-Hitchin charge quantization (Atiyah-Hitchin 88, Theorem 2.10) says that the moduli space of monopoles is the complex-rational 2-Cohomotopy of an asymptotic 2-sphere enclosing the monopoles:
$\,$
See at D-brane charge quantization in K-theory
See at supergravity C-field – shifted flux quantization condition
See also the references at fiber bundles in physics.
The original argument for charge quantization of the electromagnetic field is due to
Review:
Orlando Alvarez, Section 2 of: Topological quantization and cohomology, Comm. Math. Phys. Volume 100, Number 2 (1985), 279-309 (euclid:cmp/1103943448)
Theodore Frankel, section 16.4e of The Geometry of Physics - An Introduction, Cambridge University Press 2011 (doi:10.1017/CBO9781139061377)
L. Mangiarotti, Gennadi Sardanashvily, Connections in Classical and Quantum Field Theory, World Scientific, 2000 (doi:10.1142/2524)
Yakov Shnir, Section 2.1 of: Magnetic Monopoles, Springer 2005 (ISBN:978-3-540-29082-7)
Daniel S. Freed, Gregory W. Moore, Graeme Segal, p. 7 of: The Uncertainty of Fluxes, Commun. Math. Phys. 271:247-274, 2007 (arXiv:hep-th/0605198, doi:10.1007/s00220-006-0181-3)
Ricardo Heras, Dirac quantisation condition: a comprehensive review, Contemp. Phys. 59, 331 (2018) (arXiv:1810.13403)
See also
Discussion of the moduli space of monopoles for SU(2)-Yang-Mills theory (weak nuclear force):
Michael Atiyah, Nigel Hitchin, The geometry and dynamics of magnetic monopoles M. B. Porter Lectures. Princeton University Press, Princeton, NJ, 1988 (jstor:j.ctt7zv206)
Graeme Segal, The topology of spaces of rational functions, Acta Math. Volume 143 (1979), 39-72 (euclid:1485890033)
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)
Discussion of shifted C-field flux quantization of the C-field in D=11 supergravity/M-theory:
E. Diaconescu, Dan Freed, Greg Moore, The $M$-theory 3-form and $E_8$-gauge theory, chapter in Haynes Miller, Douglas Ravenel (eds.) Elliptic Cohomology Geometry, Applications, and Higher Chromatic Analogues, Cambridge University Press 2007 (arXiv:hep-th/0312069, doi:10.1017/CBO9780511721489)
Dan Freed, Greg Moore, Setting the quantum integrand of M-theory, Communications in Mathematical Physics, Volume 263, Number 1, 89-132, (arXiv:hep-th/0409135, doi:10.1007/s00220-005-1482-7)
Greg Moore, Anomalies, Gauss laws, and Page charges in M-theory (arXiv:hep-th/0409158)
Domenico Fiorenza, Hisham Sati, Urs Schreiber, The moduli 3-stack of the C-field, Communications in Mathematical Physics, Volume 333, Issue 1 (2015), Page 117-151, (arXiv:1202.2455, DOI 10.1007/s00220-014-2228-1)
Domenico Fiorenza, Hisham Sati, Urs Schreiber, M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory Advances in Theoretical and Mathematical Physics, Volume 18, Number 2 (2014) p. 229?321 (arXiv:1201.5277, doi:10.4310/ATMP.2014.v18.n2.a1)
Discussion in twisted Cohomotopy (“Hypothesis H”):
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Cohomotopy implies M-theory anomaly cancellation (arXiv:1904.10207)
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Cohomotopy implies M5 WZ term level quantization (arXiv:1906.07417)
and in equivariant Cohomotopy:
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