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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. See 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).
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 K-theory classification of D-brane charge
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)
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)
Discussion in the broader context of the higher gauge fields in string theory (B-field, RR-field) charge-quantized in generalized cohomology theories (twisted topological K-theory):
For a comprehensive list of literature in this case see at D-brane – Charge quantization in K-theory.
The idea that D-brane charge should be quantized in topological K-theory originates with these articles:
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
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:
Last revised on October 19, 2020 at 14:24:24. See the history of this page for a list of all contributions to it.