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\begin{document}

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\section*{magnetic charge}

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

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

[[!include physicscontents]]

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

[[!include differential cohomology - contents]]

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

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

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{MagneticChargeAnomaly}{Magnetic charge anomaly}\dotfill \pageref*{MagneticChargeAnomaly} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Maxwell's equations for the [[electromagnetic field]] $F \in \Omega^2(X)$ on a $d$-dimensional [[spacetime]] $X$ in the presence of [[electric current]] $j_{el} \in \Omega^{d-1}(X)$ read

\begin{displaymath}
d F = 0
\end{displaymath}
\begin{displaymath}
d \star F = j_{el}
  \,.
\end{displaymath}
Formally this suggests an immediate generalization where a [[conserved current]] called the \textbf{magnetic current} $j_{mag} \in \Omega^{3}(X)$ is introduced and the first of the equations above is replaced by

\begin{displaymath}
d F = j_{mag}
  \,.
\end{displaymath}
The corresponding [[charge]] is the [[magnetic charge]].

One thing to notice about this is that

\begin{itemize}%
\item net magnetic charge has not been observed in nature.

\end{itemize}
The other thing is a more conceptual problem: the [[electromagnetic field|Dirac quantization condition]] says that the $2$-form $F$ is not entirely arbitrary, but constrained to be the characteristic curvature $2$-form of a degree-$2$ cocycle in [[ordinary differential cohomology]], for instance the curvature $2$-form of a $U(1)$-[[principal bundle]] [[connection on a bundle|with connection]]. But this necessarily implies that $d F = 0$.

Indeed, to circumvent dealing with this problem Dirac, in his original argument, has considered removing from $X$ the support of any magnetic charge density.

It was [[Dan Freed]] in (\hyperlink{Freed}{Freed}) who discussed that the global description of the [[electromagnetic field]] does make sense even in the presence of electric current if one generalizes the model of a degree-$2$ [[ordinary differential cohomology|differential cocycle]] to that of a [[twisted cohomology|twisted]] cocycle.

The magnetic current $3$-form $j_{mag}$ is then realized as the curvature characteristic $3$-form of a degree-$3$ cocycle in [[ordinary differential cohomology]] $\hat j_{mag}$, the [[electromagnetic field]] $F$ is the curvature characteristic $2$-form of a degree-$2$ [[twisted cohomology|twisted]] differential cocycle $\hat F$ and the equation

\begin{displaymath}
d F = j_{mag}
\end{displaymath}
expresses the twisting of $\hat F$ by $\hat j_{mag}$ at the level of curvature forms.

This means that Dirac almost found [[bundle gerbe]]s already in 1931, had he not discussed only the neighbourhood of magnetic monopoles.

\hypertarget{MagneticChargeAnomaly}{}\subsection*{{Magnetic charge anomaly}}\label{MagneticChargeAnomaly}

Short of an experimental detection of magnetic monopoles the above considerations are of little practical relevance for ordinary [[electromagnetism]]. In their (straightforward) generalization to higher abelian [[gauge theory]] they do, however, serve to provide a more complete conceptual picture that provides the conceptual bases for effects such as the [[Green–Schwarz mechanism]].

Namely from the very fact that in the presence of magnetic charge the gauge field $\hat F$ is a twisted cocycle, it follows that the standard [[path integral|action functional]] for electrically charged quantum particles -- the one whose consideration led Dirac to his Dirac quantization condition -- in general has a [[quantum anomaly]].

To see this, consider the following.

In the simplest case we consider a single electrically charged particle with worldline $\gamma : [0,1] \to X$ propagating inside a single patch $U_i  \to X$ of some [[cover]] over which the [[electromagnetic field]] is represented by a [[Deligne cohomology|Deligne cocycle]] with $1$-form $A_i \in \Omega^1(U_i)$. Then then action functional for the coupling of the particle to the background [[electromagnetic field]] is

\begin{displaymath}
(\hat F, \gamma) \mapsto
  \exp( -2 \pi i \int_\gamma A_i) \in U(1)
  \,.
\end{displaymath}
This may be rewritten by introducing the corresponding distributional $(d-1)$-form current $(j_{el})_i \in \Omega^{d-1}(U_i)$ as

\begin{displaymath}
(\hat F, \gamma) \mapsto
  \exp( -2 \pi i \int_{U_i} (j_{el})_i \wedge A_i) \in U(1)
  \,.
\end{displaymath}
This integrand may be recognized as the local connection $d$-form of the [[cup product]] of the Deligne $(d-1)$-cocycle $\hat {j_{el}}$ and the Deligne $2$-cocycle $\hat F$. This way

\begin{displaymath}
(\hat F, \gamma) \mapsto
  \exp( -2 \pi i \int_X \hat{j_{el}} \cdot \hat F) \in U(1)
  \,,
\end{displaymath}
where now the integral including the prefactor of $- 2 \pi i$ denotes push-forward in cohomology along $X \to {*}$, and we canonically identify degree-$0$ [[Deligne cohomology]] of the point with $U(1)$.

So far this is the formulation in [[differential cohomology]] of the ordinary action functional for the coupling of electric charges $\hat {j_{el}}$ to the [[electromagnetic field]] $\hat F$.

But in this formulation one can discuss what happens to this when a non-vanishing magnetic current $\hat {j_{mag}}$ is present. In that case $\hat F$ is no longer a plain $2$-cocycle, but a twisted $2$-cocycle. Accordingly, the above push-forward to the point produces not a $U(1)$-valued \emph{function} of $(\hat F, \hat {j_{el}})$, but a ``twisted function'', i.e. a section of a [[vector bundle|line bundle]].

Since $\hat F$ is twisted by $\hat {j_{mag}}$, this line bundle is the $2$-cocycle

\begin{displaymath}
\exp( -2 \pi i \int_X \hat{j_{el}} \cdot \hat {j_{mag}})  \,.
\end{displaymath}
(\ldots{}discussion of how this is a $2$-cocycle on configuration space goes here\ldots{})

Comparing with the discussion at [[quantum anomaly]], it follows that the non-triviality of this $2$-cocycle is the \emph{higher gauge theory anomaly} of the ordinary action functional describing the coupling of electric charges to the [[electromagnetic field]] in the presense of magnetic charges.

This phenomenon has traditionally been known somewhat implicitly in the context of the [[Green–Schwarz mechanism]].

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[conserved current]], [[charge]]


\item [[electric charge]], \textbf{magnetic charge}


\item [[color charge]]


\item [[flux]]


\item [[dyon]]


\item [[magnetic field]]


\item [[magnetic moment]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Magnetic charge for general [[compact Lie groups]] as [[gauge groups]] was first discussed in

\begin{itemize}%
\item [[François Englert]], P. Windey, \emph{Quantization Condition For `t Hooft Monopoles In Compact Simple Lie Groups}, Phys. Rev. D14 (1976) 2728-2731.

\end{itemize}
In

\begin{itemize}%
\item P. Goddard, J. Nuyts, and [[David Olive]], \emph{Gauge Theories And Magnetic Charge}, Nucl. Phys. B125 (1977) 1-28.

\end{itemize}
it was first noticed that when [[electric charge]] takes values in the [[weight lattice]] of the [[gauge group]], then [[magnetic charge]] takes values in the lattice of what is now called the [[Langlands dual group]]. (This led to Montonen-Olive's [[S-duality]] [[conjecture]]).

The interpretation of magnetic charge in terms of degree-3 cocycles in [[ordinary differential cohomology]] is in the introduction-section of

\begin{itemize}%
\item [[Dan Freed]], \emph{[[Dirac charge quantization and generalized differential cohomology]]} (\href{http://arxiv.org/abs/hep-th/0011220}{arXiv})

\end{itemize}
[[!redirects magnetic current]] [[!redirects magnetic charges]]



\end{document}