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\section*{Dirac charge quantization}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{chernweil_theory}{}\paragraph*{{$\infty$-Chern-Weil theory}}\label{chernweil_theory}

[[!include infinity-Chern-Weil theory - contents]]

\hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology}

[[!include differential cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{for_monopole_charges_in_electromagnetism}{For monopole charges in electromagnetism}\dotfill \pageref*{for_monopole_charges_in_electromagnetism} \linebreak
\noindent\hyperlink{for_monopole_charges_in_nonabelian_yangmills_theory}{For monopole charges in non-abelian Yang-Mills theory}\dotfill \pageref*{for_monopole_charges_in_nonabelian_yangmills_theory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{reference}{Reference}\dotfill \pageref*{reference} \linebreak
\noindent\hyperlink{for_the_electromagnetic_field}{For the electromagnetic field}\dotfill \pageref*{for_the_electromagnetic_field} \linebreak
\noindent\hyperlink{for_the_weak_nuclear_force_field}{For the weak nuclear force field}\dotfill \pageref*{for_the_weak_nuclear_force_field} \linebreak
\noindent\hyperlink{ReferencesForBFieldAndRRFields}{For the B-field and RR-field in string theory}\dotfill \pageref*{ReferencesForBFieldAndRRFields} \linebreak
\noindent\hyperlink{for_the_cfield_in_mtheory}{For the C-field in M-theory}\dotfill \pageref*{for_the_cfield_in_mtheory} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\hypertarget{for_monopole_charges_in_electromagnetism}{}\subsubsection*{{For monopole charges in electromagnetism}}\label{for_monopole_charges_in_electromagnetism}

If the field of [[electromagnetism]] serves as a [[background gauge field]] for electrically charged [[quantum mechanics|quantum]] [[particles]] it is subject to a \emph{quantization condition}: Outside the locus of any [[magnetic charge]] -- for instance a magnetic [[monopole]] [[topological defect]] -- the [[electromagnetic field]] must be a [[circle bundle with connection|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 euqivalently the [[infinite complex projective space]] $B U(1) \simeq \mathbb{C}P^\infty$:

 $\,$

This goes back to an insight due to \hyperlink{Dirac31}{Dirac 31}, See \hyperlink{Heras18}{Heras 18} for traditional elementary review. See \hyperlink{Frankel}{Frankel} and \hyperlink{MangiarottiSardanashvily00}{Mangiarotti-Sardanashvily 00} for exposition of the modern picture in terms of [[fiber bundles in physics]]. See \hyperlink{Freed00}{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 \hyperlink{ReferencesForBFieldAndRRFields}{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]] (\hyperlink{Freed00}{Freed 00, Section 2}).

$\backslash$linebreak

\hypertarget{for_monopole_charges_in_nonabelian_yangmills_theory}{}\subsubsection*{{For monopole charges in non-abelian Yang-Mills theory}}\label{for_monopole_charges_in_nonabelian_yangmills_theory}

A similar charge quantization condition govers [[monopoles]] in [[SU(2)]]-[[Yang-Mills theory]], see at \emph{[[moduli space of monopoles]]}. Here the [[Atiyah-Hitchin charge quantization]] (\hyperlink{AtiyahHitchin88}{Atiyah-Hitchin 88, Theorem 2.10}) says that the [[moduli space]] of [[monopoles]] is the [[holomorphic function|complex]]-[[rational function|rational]] 2-[[Cohomotopy]] of an asymptotic 2-sphere enclosing the monopoles:

 $\,$

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[monopole]]

\begin{itemize}%
\item [[Dirac monopole]], [[Yang monopole]]

\end{itemize}

\item [[moduli space of monopoles]]



\end{itemize}
$\backslash$linebreak

$\backslash$linebreak

For more, see eventuall at \emph{\href{electromagnetic+field#DiracArgument}{electromagnetic field -- charge quantization}} (but still needs to be written\ldots{})

\hypertarget{reference}{}\subsection*{{Reference}}\label{reference}

\hypertarget{for_the_electromagnetic_field}{}\subsubsection*{{For the electromagnetic field}}\label{for_the_electromagnetic_field}

The original argument for charge quantization of the [[electromagnetic field]] is due to

\begin{itemize}%
\item [[P.A.M. Dirac]], \emph{Quantized Singularities in the Electromagnetic Field}, Proceedings of the Royal Society, A133 (1931) pp 60--72 (\href{http://rspa.royalsocietypublishing.org/content/133/821/60.short}{doi:10.1098/rspa.1931.0130})

\end{itemize}
Review:

\begin{itemize}%
\item [[Theodore Frankel]], section 16.4e of \emph{[[The Geometry of Physics - An Introduction]]} (\href{https://doi.org/10.1017/CBO9781139061377}{doi:10.1017/CBO9781139061377})


\item \hyperlink{Freed00}{Freed 00, Section 2}


\item L. Mangiarotti, [[Gennadi Sardanashvily]], \emph{Connections in Classical and Quantum Field Theory}, World Scientific, 2000 (\href{https://doi.org/10.1142/2524}{doi:10.1142/2524})


\item Ricardo Heras, \emph{Dirac quantisation condition: a comprehensive review}, Contemp. Phys. 59, 331 (2018) (\href{https://arxiv.org/abs/1810.13403}{arXiv:1810.13403})



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Magnetic_monopole#Dirac%27s_quantization}{Dirac's quantization}}

\end{itemize}
\hypertarget{for_the_weak_nuclear_force_field}{}\subsubsection*{{For the weak nuclear force field}}\label{for_the_weak_nuclear_force_field}

Discussion of the [[moduli space of monopoles]] for [[SU(2)]]-[[Yang-Mills theory]] ([[weak nuclear force]]):

\begin{itemize}%
\item [[Michael Atiyah]], [[Nigel Hitchin]], \emph{The geometry and dynamics of magnetic monopoles} M. B. Porter Lectures. Princeton University Press, Princeton, NJ, 1988 (\href{https://www.jstor.org/stable/j.ctt7zv206}{jstor:j.ctt7zv206})


\item [[Graeme Segal]], \emph{The topology of spaces of rational functions}, Acta Math. Volume 143 (1979), 39-72 (\href{https://projecteuclid.org/euclid.acta/1485890033}{euclid:1485890033})



\end{itemize}
\hypertarget{ReferencesForBFieldAndRRFields}{}\subsubsection*{{For the B-field and RR-field in string theory}}\label{ReferencesForBFieldAndRRFields}

Discussion in the broader context of the [[higher gauge fields]] in [[string theory]] ([[B-field]], [[RR-field]]) charge-quantized in [[generalized cohomology theories]] ([[twisted K-theory|twisted]] [[topological K-theory]]):

\begin{itemize}%
\item [[Daniel Freed]], \emph{[[Dirac charge quantization and generalized differential cohomology]]}, Surveys in Differential Geometry, Int. Press, Somerville, MA, 2000, pp. 129--194 (\href{http://arxiv.org/abs/hep-th/0011220}{arXiv:hep-th/0011220})

\end{itemize}
For a comprehensive list of literature in this case see at \emph{\href{D-brane#ReferencesKTheoryDescription}{D-brane -- Charge quantization in K-theory}}.

The idea that [[D-brane charge]] should be quantized in [[topological K-theory]] originates with these articles:

\begin{itemize}%
\item [[Ruben Minasian]], [[Gregory Moore]], \emph{K-theory and Ramond-Ramond charge}, JHEP9711:002,1997 (\href{http://arxiv.org/abs/hep-th/9710230}{arXiv:hep-th/9710230})


\item [[Edward Witten]], \emph{D-Branes And K-Theory}, JHEP 9812:019,1998 (\href{http://arxiv.org/abs/hep-th/9810188}{arXiv:hep-th/9810188})


\item [[Daniel Freed]], [[Michael Hopkins]], \emph{On Ramond-Ramond fields and K-theory}, JHEP 0005 (2000) 044 (\href{http://arxiv.org/abs/hep-th/0002027}{arXiv:hep-th/0002027})



\end{itemize}
See also at \emph{[[anti-D-brane]]}.

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

\begin{itemize}%
\item Daniel Grady, [[Hisham Sati]], \emph{Ramond-Ramond fields and twisted differential K-theory} (\href{https://arxiv.org/abs/1903.08843}{arXiv:1903.08843})

\end{itemize}
Discussion of full-blown [[twisted K-theory|twisted]] [[differential K-theory|differential]] [[KO-theory|orthogonal]] [[topological K-theory|K-theory]] and its relation to [[D-brane charge]] in [[type I string theory]] (on [[orientifolds]]):

\begin{itemize}%
\item Daniel Grady, [[Hisham Sati]], \emph{Twisted differential KO-theory} (\href{https://arxiv.org/abs/1905.09085}{arXiv:1905.09085})

\end{itemize}
 But there remain conceptual issues with the proposal that D-brane charge is in K-theory, as highlighted for in

\begin{itemize}%
\item [[Jan de Boer]], [[Robbert Dijkgraaf]], [[Kentaro Hori]], [[Arjan Keurentjes]], [[John Morgan]], [[David Morrison]], [[Savdeep Sethi]], section 4.5 and 4.6.5 of \emph{Triples, Fluxes, and Strings}, Adv.Theor.Math.Phys. 4 (2002) 995-1186 (\href{https://arxiv.org/abs/hep-th/0103170}{arXiv:hep-th/0103170})


\item [[Jarah Evslin]], section 8 of \emph{What Does(n't) K-theory Classify?}, Second Modave Summer School in Mathematical Physics (\href{https://arxiv.org/abs/hep-th/0610328}{arXiv:hep-th/0610328})



\end{itemize}
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:

\begin{itemize}%
\item [[Stefan Fredenhagen]], [[Thomas Quella]], \emph{Generalised permutation branes}, JHEP0511:004, 2005 (\href{https://arxiv.org/abs/hep-th/0509153}{arXiv:hep-th/0509153})

\begin{quote}%
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.


\end{quote}


\end{itemize}
Further review and discussion of D-brane charge in K-theory includes the following

\begin{itemize}%
\item Kasper Olsen, [[Richard Szabo]], \emph{Brane Descent Relations in K-theory}, Nucl.Phys. B566 (2000) 562-598 (\href{https://arxiv.org/abs/hep-th/9904153}{arXiv:hep-th/9904153})


\item Kasper Olsen, [[Richard Szabo]], \emph{Constructing D-Branes from K-Theory}, Adv.Theor.Math.Phys. 3 (1999) 889-1025 (\href{https://arxiv.org/abs/hep-th/9907140}{arXiv:hep-th/9907140})


\item [[John Schwarz]], \emph{TASI Lectures on Non-BPS D-Brane Systems} (\href{https://arxiv.org/abs/hep-th/9908144}{arXiv:hep-th/9908144})


\item [[Edward Witten]], \emph{Overview Of K-Theory Applied To Strings}, Int.J.Mod.Phys.A16:693-706,2001 (\href{https://arxiv.org/abs/hep-th/0007175}{arXiv:hep-th/0007175})


\item [[Greg Moore]], \emph{K-Theory from a physical perspective} (\href{http://arxiv.org/abs/hep-th/0304018}{arXiv:hep-th/0304018})


\item [[Juan José Manjarín]], \emph{Topics on D-brane charges with B-fields}, Int.J.Geom.Meth.Mod.Phys. 1 (2004) (\href{http://arxiv.org/abs/hep-th/0405074}{arXiv:hep-th/0405074})



\end{itemize}
A textbook account of D-brane charge in ([[twisted K-theory|twisted]]) [[topological K-theory]] is

\begin{itemize}%
\item [[Dale Husemoeller]], [[Michael Joachim]], [[Branislav Jurco]], [[Martin Schottenloher]], \emph{[[Basic Bundle Theory and K-Cohomology Invariants]]}, Lecture Notes in Physics, Springer 2008 (\href{http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf}{pdf})

\end{itemize}
See also for instance

\begin{itemize}%
\item [[Ilka Brunner]], [[Jacques Distler]], \emph{Torsion D-Branes in Nongeometrical Phases} (\href{https://arxiv.org/abs/hep-th/0102018}{arXiv:hep-th/0102018})

\end{itemize}
Discussion of D-branes in [[KK-theory]] is reviewed in

\begin{itemize}%
\item [[Richard Szabo]], \emph{D-branes and bivariant K-theory}, Noncommutative Geometry and Physics 3 1 (2013): 131. (\href{http://arxiv.org/abs/0809.3029}{arXiv:0809.3029})

\end{itemize}
based on

\begin{itemize}%
\item Rui Reis, [[Richard Szabo]], \emph{Geometric K-Homology of Flat D-Branes} ,Commun.Math.Phys. 266 (2006) 71-122 (\href{https://arxiv.org/abs/hep-th/0507043}{arXiv:hep-th/0507043})


\item [[Jacek Brodzki]], [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], \emph{D-Branes, RR-Fields and Duality on Noncommutative Manifolds}, Commun. Math. Phys. 277:643-706,2008 (\href{http://arxiv.org/abs/hep-th/0607020}{arXiv:hep-th/0607020})


\item [[Jacek Brodzki]], [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], \emph{Noncommutative correspondences, duality and D-branes in bivariant K-theory}, Adv. Theor. Math. Phys.13:497-552,2009 (\href{http://arxiv.org/abs/0708.2648}{arXiv:0708.2648})


\item [[Jacek Brodzki]], [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], \emph{D-branes, KK-theory and duality on noncommutative spaces}, J. Phys. Conf. Ser. 103:012004,2008 (\href{http://arxiv.org/abs/0709.2128}{arXiv:0709.2128})



\end{itemize}
In particular (\hyperlink{BMRS2}{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

\begin{itemize}%
\item [[Richard Szabo]], \emph{D-Branes, Tachyons and K-Homology}, Mod. Phys. Lett. A17 (2002) 2297-2316 (\href{http://arxiv.org/abs/hep-th/0209210}{arXiv:hep-th/0209210})

\end{itemize}
Specifically for D-branes in [[WZW models]] see

\begin{itemize}%
\item [[Peter Bouwknegt]], \emph{A note on equality of algebraic and geometric D-brane charges in WZW models} (\href{http://people.physics.anu.edu.au/~drt105/papers/BR0312259.pdf}{pdf})

\end{itemize}
More on this, with more explicit relation to [[noncommutative motives]], is in

\begin{itemize}%
\item [[Snigdhayan Mahanta]], \emph{Noncommutative correspondence categories, simplicial sets and pro $C^\ast$-algebras} (\href{http://arxiv.org/abs/0906.5400}{arXiv:0906.5400})


\item [[Snigdhayan Mahanta]], \emph{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. (\href{http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KQ.pdf}{pdf})



\end{itemize}
Discussion of D-brane [[matrix models]] taking these K-theoretic effects into account ([[K-matrix model]]) is in

\begin{itemize}%
\item T. Asakawa, S. Sugimoto, S. Terashima, \emph{D-branes, Matrix Theory and K-homology}, JHEP 0203 (2002) 034 (\href{https://arxiv.org/abs/hep-th/0108085}{arXiv:hep-th/0108085})

\end{itemize}
 The proposal that D-brane charge on [[orbifolds]] is measured in [[equivariant K-theory]] goes back to

\begin{itemize}%
\item \hyperlink{Witten98}{Witten 98, section 5.1}

\end{itemize}
but it was pointed out that only a subgroup of equivariant K-theory can be physically relevant in

\begin{itemize}%
\item [[Jan de Boer]], [[Robbert Dijkgraaf]], [[Kentaro Hori]], [[Arjan Keurentjes]], [[John Morgan]], [[David Morrison]], [[Savdeep Sethi]], around (137) of \emph{Triples, Fluxes, and Strings}, Adv.Theor.Math.Phys. 4 (2002) 995-1186 (\href{https://arxiv.org/abs/hep-th/0103170}{arXiv:hep-th/0103170})

\end{itemize}
Further discussion of [[equivariant K-theory]] for D-branes on [[orbifolds]] includes the following:

\begin{itemize}%
\item Hugo García-Compeán, \emph{D-branes in orbifold singularities and equivariant K-theory}, Nucl.Phys. B557 (1999) 480-504 (\href{https://arxiv.org/abs/hep-th/9812226}{arXiv:hep-th/9812226})


\item [[Matthias Gaberdiel]], [[Bogdan Stefanski]], \emph{Dirichlet Branes on Orbifolds}, Nucl.Phys.B578:58-84, 2000 (\href{https://arxiv.org/abs/hep-th/9910109}{arXiv:hep-th/9910109})


\item [[Igor Kriz]], Leopoldo A. Pando Zayas, Norma Quiroz, \emph{Comments on D-branes on Orbifolds and K-theory}, Int.J.Mod.Phys.A23:933-974, 2008 (\href{https://arxiv.org/abs/hep-th/0703122}{arXiv:hep-th/0703122})


\item [[Richard Szabo]], [[Alessandro Valentino]], \emph{Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory}, Commun.Math.Phys.294:647-702, 2010 (\href{https://arxiv.org/abs/0710.2773}{arXiv:0710.2773})



\end{itemize}
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

\begin{itemize}%
\item \hyperlink{Witten98}{Witten 98, section 5}

\end{itemize}
and was then re-amplified in

\begin{itemize}%
\item [[Sergei Gukov]], \emph{K-Theory, Reality, and Orientifolds}, Commun.Math.Phys. 210 (2000) 621-639 (\href{https://arxiv.org/abs/hep-th/9901042}{arXiv:hep-th/9901042})


\item [[Oren Bergman]], E. Gimon, [[Shigeki Sugimoto]], \emph{Orientifolds, RR Torsion, and K-theory}, JHEP 0105:047, 2001 (\href{https://arxiv.org/abs/hep-th/0103183}{arXiv:hep-th/0103183})



\end{itemize}
With further developments in

\begin{itemize}%
\item [[Varghese Mathai]], [[Michael Murray]], [[Daniel Stevenson]], \emph{Type I D-branes in an H-flux and twisted KO-theory}, JHEP 0311 (2003) 053 (\href{https://arxiv.org/abs/hep-th/0310164}{arXiv:hep-th/0310164})

\end{itemize}
Discussion of orbi-orienti-folds using [[equivariant K-theory|equivariant]] [[KO-theory]] is in

\begin{itemize}%
\item N. Quiroz, [[Bogdan Stefanski]], \emph{Dirichlet Branes on Orientifolds}, Phys.Rev. D66 (2002) 026002 (\href{https://arxiv.org/abs/hep-th/0110041}{arXiv:hep-th/0110041})


\item [[Volker Braun]], [[Bogdan Stefanski]], \emph{Orientifolds and K-theory} (\href{https://arxiv.org/abs/hep-th/0206158}{arXiv:hep-th/0206158})


\item H. Garcia-Compean, W. Herrera-Suarez, B. A. Itza-Ortiz, O. Loaiza-Brito, \emph{D-Branes in Orientifolds and Orbifolds and Kasparov KK-Theory}, JHEP 0812:007, 2008 (\href{https://arxiv.org/abs/0809.4238}{arXiv:0809.4238})



\end{itemize}
An elaborate proposal for the correct flavour of real equivariant K-theory needed for [[orientifolds]] is sketched in

\begin{itemize}%
\item [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], \emph{Orientifold Pr\'e{}cis} in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]} Proceedings of Symposia in Pure Mathematics, AMS (2011) (\href{http://arxiv.org/abs/0906.0795}{arXiv:0906.0795}, \href{http://www.ma.utexas.edu/users/dafr/bilbao.pdf}{slides})

\end{itemize}
Discussion of the alleged K-theory classification of D-brane charge in relation to the [[M-theory]] [[supergravity C-field]] is in

\begin{itemize}%
\item D. Diaconescu, [[Gregory Moore]], [[Edward Witten]], \emph{$E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory}, Adv.Theor.Math.Phys.6:1031-1134,2003 (\href{http://arxiv.org/abs/hep-th/0005090}{arXiv:hep-th/0005090}), summarised in \emph{A Derivation of K-Theory from M-Theory} (\href{http://arxiv.org/abs/hep-th/0005091}{arXiv:hep-th/0005091})

\end{itemize}
See also

\begin{itemize}%
\item Inaki Garcia-Etxebarria, [[Angel Uranga]], \emph{From F/M-theory to K-theory and back}, JHEP 0602:008,2006 (\href{https://arxiv.org/abs/hep-th/0510073}{arXiv:hep-th/0510073})

\end{itemize}
For more on this perspective as 10d type II as a [[self-dual higher gauge theory]] in the boudnary of a kind of [[higher dimensional Chern-Simons theory|11-d Chern-Simons theory]] is in

\begin{itemize}%
\item Dmitriy Belov, [[Greg Moore]], \emph{Type II Actions from 11-Dimensional Chern-Simons Theories} (\href{http://arxiv.org/abs/hep-th/0611020}{arXiv:hep-th/0611020})

\end{itemize}
More complete discussion of the decomposition of the [[supergravity C-field]] as one passes from 11d to 10d is in

\begin{itemize}%
\item [[Varghese Mathai]], [[Hisham Sati]], \emph{Some Relations between Twisted K-theory and E8 Gauge Theory}, JHEP0403:016,2004 (\href{http://arxiv.org/abs/hep-th/0312033}{arXiv:hep-th/0312033})

\end{itemize}
\hypertarget{for_the_cfield_in_mtheory}{}\subsubsection*{{For the C-field in M-theory}}\label{for_the_cfield_in_mtheory}

Discussion of charge quantization of the [[C-field]] in [[D=11 supergravity]]/[[M-theory]]:

\begin{itemize}%
\item E. Diaconescu, [[Dan Freed]], [[Greg Moore]], \emph{The $M$-theory 3-form and $E_8$-gauge theory}, chapter in [[Haynes Miller]], [[Douglas Ravenel]] (eds.) \emph{Elliptic Cohomology Geometry, Applications, and Higher Chromatic Analogues}, Cambridge University Press 2007 (\href{http://arxiv.org/abs/hep-th/0312069}{arXiv:hep-th/0312069}, \href{https:
//doi.org/10.1017/CBO9780511721489}{doi:10.1017/CBO9780511721489})


\item [[Dan Freed]], [[Greg Moore]], \emph{Setting the [[quantum integrand]] of M-theory}, Communications in Mathematical Physics, Volume 263, Number 1, 89-132, (\href{http://arxiv.org/abs/hep-th/0409135}{arXiv:hep-th/0409135}, \href{https://doi.org/10.1007/s00220-005-1482-7}{doi:10.1007/s00220-005-1482-7})


\item [[Greg Moore]], \emph{Anomalies, Gauss laws, and Page charges in M-theory} (\href{http://arxiv.org/abs/hep-th/0409158}{arXiv:hep-th/0409158})


\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:The moduli 3-stack of the C-field]]}, Communications in Mathematical Physics, Volume 333, Issue 1 (2015), Page 117-151, (\href{http://arxiv.org/abs/1202.2455}{arXiv:1202.2455}, \href{http://link.springer.com/article/10.1007%2Fs00220-014-2228-1}{DOI 10.1007/s00220-014-2228-1})


\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:7d Chern-Simons theory and the 5-brane|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 (\href{http://arxiv.org/abs/1201.5277}{arXiv:1201.5277}, \href{https://dx.doi.org/10.4310/ATMP.2014.v18.n2.a1}{doi:10.4310/ATMP.2014.v18.n2.a1})



\end{itemize}
Discussion in [[twisted Cohomotopy]] (``[[schreiber:Hypothesis H]]''):

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation]]} (\href{https://arxiv.org/abs/1904.10207}{arXiv:1904.10207})


\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization]]} (\href{https://arxiv.org/abs/1906.07417}{arXiv:1906.07417})



\end{itemize}
and in [[equivariant Cohomotopy]]:

\begin{itemize}%
\item [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], \emph{[[schreiber:Equivariant Cohomotopy implies orientifold tadpole cancellation]]} (\href{https://arxiv.org/abs/1909.12277}{arXiv:1909.12277})

\end{itemize}
[[!redirects Dirac's charge quantization]]

[[!redirects charge quantization]] [[!redirects charge quantizations]]



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