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\begin{document}

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\section*{electric-magnetic duality}

\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{duality_in_string_theory}{}\paragraph*{{Duality in string theory}}\label{duality_in_string_theory}

[[!include duality in string theory -- contents]]

\hypertarget{langlands_correspondence}{}\paragraph*{{Langlands correspondence}}\label{langlands_correspondence}

[[!include Langlands correspondence -- contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{description}{Description}\dotfill \pageref*{description} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Electric-magnetic duality is a lift of [[Hodge star operator|Hodge duality]] from [[de Rham cohomology]] to [[ordinary differential cohomology]].

\hypertarget{description}{}\subsection*{{Description}}\label{description}

Consider a [[circle n-bundle with connection]] $\nabla$ on a space $X$. Its [[higher parallel transport]] is the [[action functional]] for the [[sigma-model]] of $(n-1)$-dimensional objects ($(n-1)$-branes) propagating in $X$.

For $n = 1$ this is the coupling of the [[electromagnetic field]] to particles. For $n = 2$ this is the coupling of the [[Kalb-Ramond field]] to strings.

The [[curvature]] $F_\nabla \in \Omega^{n+1}(X)$ is a closed $(n+1)$-form. The condition that its image $\star F_\nabla$ under the [[Hodge star operator]] is itself closed

\begin{displaymath}
d_{dR} \star F_\nabla = 0
\end{displaymath}
is the [[Euler-Lagrange equation]] for the standard (abelian [[Yang-Mills theory]]-[[action functional]] on the space of circle n-bundle with connection.

If this is the case, it makes sense to ask if $\star F_\nabla$ itself is the curvature $(d-(n+1))$-form of a circle $(d-(n+1)-1)$-bundle with connection $\tilde \nabla$, where $d = dim X$ is the [[dimension]] of $X$.

If such $\tilde \nabla$ exists, its [[higher parallel transport]] is the gauge interaction [[action functional]] for $(d-n-3)$-dimensional objects propagating on $X$.

In the special case of ordinary [[electromagnetism]] with $n=1$ and $d = 4$ we have that electrically charged 0-dimensional particles couple to $\nabla$ and magnetically charged $(4-(1+1)-2) = 0$-dimensional particles couple to $\tilde \nabla$.

In analogy to this case one calls generally the $d-n-3$-dimensional objects coupling to $\tilde \nabla$ the \textbf{magnetic duals} of the $(n-1)$-dimensional objects coupling to $\nabla$.

\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

For $d= 4$ EM-duality is the special abelian case of [[S-duality]] for [[Yang-Mills theory]]. Witten and Kapustin argued that this is governed by the [[geometric Langlands correspondence]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item In [[N=2 D=4 super Yang-Mills theory]] electric-magnetic duality is studied as \emph{[[Seiberg-Witten theory]]}.


\item In [[heterotic string theory]] one considers 1-dimensional objects in $d=10$-dimensional spaces electrically charged (under the [[Kalb-Ramond field]]). Their magnetic duals are 5-dimensional objects (fivebranes), studied in [[dual heterotic string theory]].


\item [[!include electric-magnetic duality -- table]]


\item [[M2-brane]] and [[M5-brane]]: the 4-form $G_4$ which the M2-brane couples to is the Hodge-dual of the 7-form $G_7$ which the M5-brane couples to



\end{itemize}
\begin{displaymath}
G_7 = \star G_4
\end{displaymath}
as imposed by the equations of motion for 11d [[supergravity]] (see \href{https://ncatlab.org/nlab/show/torsion+constraints+in+supergravity#Examples11dSuGra}{here}). The associated [[C3-field]] and [[C6-field]] are electric-magnetic duals.

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

[[duality in physics]], [[duality in string theory]]

\begin{itemize}%
\item [[parent action functional]]


\item [[S-duality]]

\begin{itemize}%
\item \textbf{electro-magnetic duality}

\begin{itemize}%
\item [[Montonen-Olive duality]]
\item [[dual graviton]]
\item [[dual photon]]

\end{itemize}

\item [[Seiberg duality]]


\item [[geometric Langlands correspondence]]


\item [[quantum geometric Langlands correspondence]]



\end{itemize}


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

Detailed review is in

\begin{itemize}%
\item [[José Figueroa-O'Farrill]], \emph{Electromagnetic Duality for Children} (1998) ([[FigueroaElectromagneticDuality.pdf:file]])

\end{itemize}
It was originally noticed 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}
that where [[electric charge]] in [[Yang-Mills theory]] 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 the electric/magnetic duality conjecture formulation in

\begin{itemize}%
\item [[Claus Montonen]], [[David Olive]], \emph{Magnetic Monopoles As Gauge Particles?} Phys. Lett. B72 (1977) 117-120.

\end{itemize}
According to (\hyperlink{KapustinWitten06}{Kapustin-Witten 06, pages 3-4}) the observation that the Montonen-Olive dual charge group coincides with the [[Langlands dual group]] is due to

\begin{itemize}%
\item [[Michael Atiyah]], private communication to [[Edward Witten]], 1977

\end{itemize}
See also the references at \emph{[[S-duality]]}.

The insight that the Montonen-Olive duality works more naturally in [[super Yang-Mills theory]] is due to

\begin{itemize}%
\item [[David Olive]], [[Edward Witten]], \emph{Supersymmetry Algebras That Include Topological Charges}, Phys. Lett. B78 (1978) 97-101.

\end{itemize}
and that it works particularly for [[N=4 D=4 super Yang-Mills theory]] is due to

\begin{itemize}%
\item H. Osborn, \emph{Topological Charges For $N = 4$ Supersymmetric Gauge Theories And Monopoles Of Spin 1}, Phys. Lett. B83 (1979) 321-326.

\end{itemize}
The observation that the $\mathbb{Z}_2$ electric/magnetic duality extends to an $SL(2,\mathbb{Z})$-action in this case is due to

\begin{itemize}%
\item [[John Cardy]], E. Rabinovici, \emph{Phase Structure Of Zp Models In The Presence Of A Theta Parameter}, Nucl. Phys. B205 (1982) 1-16;


\item [[John Cardy]], \emph{Duality And The Theta Parameter In Abelian Lattice Models}, Nucl. Phys. B205 (1982) 17-26.


\item A. Shapere and [[Frank Wilczek]], \emph{Selfdual Models With Theta Terms}, Nucl. Phys. B320 (1989) 669-695.



\end{itemize}
and specifically the embedding of this into [[string theory]] [[S-duality]] originates in

\begin{itemize}%
\item [[Ashoke Sen]], \emph{Dyon - Monopole Bound States, Self-Dual Harmonic Forms on the Multi-Monopole Moduli Space, and $SL(2,\mathbb{Z})$ Invariance in String Theory} (\href{http://arxiv.org/abs/hep-th/9402032}{arXiv:hep-th/9402032})

\end{itemize}
The understanding of this $SL(2,\mathbb{Z})$-symmetry as a remnant conformal transformation on a 6-dimensional [[principal 2-bundle]]-theory -- the [[6d (2,0)-superconformal QFT]] -- compactified on a torus is described in

\begin{itemize}%
\item [[Edward Witten]], pages 4-5 of \emph{Some Comments On String Dynamics}, Proceedings of \emph{String95} (\href{http://arxiv.org/abs/hepth/9507121}{arXiv:hepth/9507121})


\item [[Edward Witten]], \emph{On S-Duality in Abelian Gauge Theory} (\href{http://arxiv.org/abs/hep-th/9505186}{arXiv:hep-th/9505186})


\item [[Cumrun Vafa]], \emph{Geometric Origin of Montonen-Olive Duality}, Adv.Theor.Math.Phys.1:158-166, 1998 (\href{https://arxiv.org/abs/hep-th/9707131}{arXiv:hep-th/9707131})


\item [[Edward Witten]], \emph{[[Conformal field theory in four and six dimensions]]}, in [[Ulrike Tillmann]], \emph{Topology, Geometry and Quantum Field Theory: Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal}, London Mathematical Society Lecture Note Series (2004) (\href{http://arxiv.org/abs/0712.0157}{arXiv:0712.0157})



\end{itemize}
The relation of S-duality to [[geometric Langlands duality]] was understood in

\begin{itemize}%
\item [[Anton Kapustin]], [[Edward Witten]], \emph{Electric-Magnetic Duality And The Geometric Langlands Program} , Communications in number theory and physics, Volume 1, Number 1, 1--236 (2007) (\href{http://arxiv.org/abs/hep-th/0604151}{arXiv:hep-th/0604151})

\end{itemize}
Exposition of this is in

\begin{itemize}%
\item [[Edward Frenkel]], \emph{What Do Fermat's Last Theorem and Electro-magnetic Duality Have in Common?} KITP talk 2011 (\href{http://online.kitp.ucsb.edu/online/bblunch/frenkel/}{web})

\end{itemize}
A perspective unifying EM-duality of [[higher gauge theories]] with [[non-abelian T-duality]]:

\begin{itemize}%
\item [[Ján Pulmann]], [[Pavol Ševera]], [[Fridrich Valach]], \emph{A non-abelian duality for (higher) gauge theories} (\href{https://arxiv.org/abs/1909.06151}{arXiv:1909.06151})

\end{itemize}
[[!redirects electric-magnetic dualities]]

[[!redirects electro-magnetic duality]] [[!redirects electro-magnetic dualities]]

[[!redirects electromagnetic duality]] [[!redirects electromagnetic dualites]]



\end{document}