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\begin{document}

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\section*{moduli space of monopoles}

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

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

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

\hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory}

[[!include AQFT and operator algebra contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{ScatteringAmplitudesofMonopoles}{Scattering amplitudes of monopoles}\dotfill \pageref*{ScatteringAmplitudesofMonopoles} \linebreak
\noindent\hyperlink{charge_quantization_in_cohomotopy}{Charge quantization in Cohomotopy}\dotfill \pageref*{charge_quantization_in_cohomotopy} \linebreak
\noindent\hyperlink{relation_to_braid_groups}{Relation to braid groups}\dotfill \pageref*{relation_to_braid_groups} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{as_transversal_ddbrane_intersections}{As transversal D$p$/D$(p+2)$-brane intersections}\dotfill \pageref*{as_transversal_ddbrane_intersections} \linebreak
\noindent\hyperlink{relation_to_braids}{Relation to braids}\dotfill \pageref*{relation_to_braids} \linebreak


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

By the [[Nahm transform]], the [[moduli space]] of time-translation invariant [[self-dual Yang-Mills theory]] [[solitons]] on 4d [[Minkowski spacetime]] $\mathbb{R}^{3,1}$ is equivalently the space of solutions to the [[Bogomolny equations]] on 3d [[Euclidean space]], which in tur\# iN n may be thought of as [[magnetic monopoles]] in 3d [[Euclidean field theory|Euclidean]] [[Yang-Mills theory]] coupled to a charged [[scalar field]] (a ``[[Higgs field]]''). Therefore this moduli space is traditionally referred to simply as the \emph{moduli space of magnetic monopoles} (e.g. \hyperlink{AtiyahHitchin88}{Atiyah-Hitchin 88}) or just the \emph{moduli space of monopoles}.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

The moduli space

\begin{equation}
\mathcal{M}_k
  \;\coloneqq\;
  \cdots
\label{ModuliSpaceOfkMonopoles}\end{equation}
of $k$ monopoles is \ldots{} (\hyperlink{AtiyahHitchin88}{Atiyah-Hitchin 88, p. 15-16}).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{ScatteringAmplitudesofMonopoles}{}\subsubsection*{{Scattering amplitudes of monopoles}}\label{ScatteringAmplitudesofMonopoles}

Write

\begin{equation}
Maps_{cplx\,rtnl}^{\ast/}\big( \mathbb{C}P^1, \mathbb{C}P^1 \big)_k
  \;\subset\;
  Maps_{cplx\,rtnl}^{\ast/}\big( \mathbb{C}P^1, \mathbb{C}P^1 \big)
  \;\subset\;
  Maps^{\ast/}\big( S^2, S^2 \big)
\label{SpaceOfRationalFunctionsOfDegreek}\end{equation}
for the [[space of maps|space of]] [[pointed topological space|pointed]] [[rational functions]] from the [[Riemann sphere]] to itself, of [[degree of a continuous function|degree]] $k \in \mathbb{N}$, inside the full [[Cohomotopy]] [[cocycle space]]. The [[homotopy type]] of $R_k$ is discussed in \hyperlink{Segal79}{Segal 79}.

To each configuration $c \in \mathcal{M}_k$ of $k \in \mathbb{N}$ magnetic monopoles is associated a [[scattering amplitude]]

\begin{equation}
S(c) 
    \in    
  Maps_{cplx\,rtnl}^{\ast/}\big( \mathbb{C}P^1, \mathbb{C}P^1 \big)_k
\label{ScatteringAmplitudes}\end{equation}
(\hyperlink{AtiyahHitchin88}{Atiyah-Hitchin 88 (2.8)})

$\backslash$linebreak

\hypertarget{charge_quantization_in_cohomotopy}{}\subsubsection*{{Charge quantization in Cohomotopy}}\label{charge_quantization_in_cohomotopy}

\begin{prop}
\label{ModuliSpaceOfKMonopolesIsSpaceOfComplexRationalFunctions}\hypertarget{ModuliSpaceOfKMonopolesIsSpaceOfComplexRationalFunctions}{}
\textbf{([[moduli space of monopoles|moduli space of k monopoles]] is [[space of maps|space of]] [[degree of a continuous function|degree]] $k$ [[complex number|complex]]-[[rational functions]] from [[Riemann sphere]] to itself)}

The assignment \eqref{ScatteringAmplitudes} is a [[diffeomorphism]] identifying the moduli space \eqref{ModuliSpaceOfkMonopoles} of $k$ magnetic monopoles with the space \eqref{SpaceOfRationalFunctionsOfDegreek} of complex-[[rational functions]] from the [[Riemann sphere]] to itself, of [[degree of a continuous function|degree]] $k$ (hence the [[cocycle space]] of complex-rational 2-[[Cohomotopy]])

\begin{displaymath}
\mathcal{M}_k
  \;
  \underoverset{
    \simeq_{diff}
  }{ 
    S 
  }{
    \;\;\longrightarrow\;\;
  }
  \;
  Maps_{cplx\,rtnl}^{\ast/}
  \big( 
    \mathbb{C}P^1, \mathbb{C}P^1 
  \big)_k
\end{displaymath}
\end{prop}
(due to \hyperlink{Donaldson84}{Donaldson 84}, see also \hyperlink{AtiyahHitchin88}{Atiyah-Hitchin 88, Theorem 2.10}).

\begin{prop}
\label{SpaceOfComplexRationalMapsOnRiemannSpherekEquivalentToCohomotopyCocycleSpace}\hypertarget{SpaceOfComplexRationalMapsOnRiemannSpherekEquivalentToCohomotopyCocycleSpace}{}
\textbf{([[space of maps|space of]] [[degree of a continuous function|degree]] $k$ [[complex number|complex]]-[[rational functions]] from [[Riemann sphere]] to itself is [[n-equivalence|k-equivalent]] to [[Cohomotopy]] [[cocycle space]] in degree $k$)}

The inclusion of the complex rational self-maps maps of [[degree of a continuous function|degree]] $k$ into the full based [[space of maps]] of [[degree of a continuous function|degree]] $k$ (hence the $k$-component of the second [[iterated loop space]] of the [[2-sphere]], and hence the plain [[Cohomotopy]] [[cocycle space]]) induces an [[isomorphism]] of [[homotopy groups]] in degrees $\leq k$ (in particular a [[n-equivalence|k-equivalence]]):

\begin{displaymath}
Maps_{cplx\,rtnl}^{\ast/}
  \big( 
    \mathbb{C}P^1, \mathbb{C}P^1 
  \big)_k
  \;
  \underset{
    \simeq_{\leq k}
  }{
    \;\;\hookrightarrow\;\;
  }
  \;
  Maps^{\ast/}\big( S^2, S^2 \big)_k
\end{displaymath}
\end{prop}
(\hyperlink{Segal79}{Segal 79, Prop. 1.1})

Hence, Prop. \ref{ModuliSpaceOfKMonopolesIsSpaceOfComplexRationalFunctions} and Prop. \ref{SpaceOfComplexRationalMapsOnRiemannSpherekEquivalentToCohomotopyCocycleSpace} together say that the moduli space of $k$-monopoles is $k$-equivalent to the Cohomotopy cocycle space $\mathbf{\pi}^2\big( S^2 \big)_k$.

\begin{displaymath}
\mathcal{M}_k
  \;
  \underoverset{
    \simeq_{diff}
  }{ 
    S 
  }{
    \;\;\longrightarrow\;\;
  }
  \;
  Maps_{cplx\,rtnl}^{\ast/}
  \big( 
    \mathbb{C}P^1, \mathbb{C}P^1 
  \big)_k
  \;
  \underset{
    \simeq_{\leq k}
  }{
    \;\;\hookrightarrow\;\;
  }
  \;
  Maps^{\ast/}\big( S^2, S^2 \big)_k
\end{displaymath}
This is a [[non-abelian group|non-abelian]] analog of the [[Dirac charge quantization]] of the [[electromagnetic field]], with [[ordinary cohomology]] replaced by [[Cohomotopy]] [[generalized cohomology theory|cohomology theory]]:

 $\,$

$\backslash$linebreak

\hypertarget{relation_to_braid_groups}{}\subsubsection*{{Relation to braid groups}}\label{relation_to_braid_groups}

\begin{prop}
\label{ModuliSpaceOfkMonopolesStablyWeakHomotopyEquivbalentToClassifyingSpaceOfBraids}\hypertarget{ModuliSpaceOfkMonopolesStablyWeakHomotopyEquivbalentToClassifyingSpaceOfBraids}{}
\textbf{([[moduli space of monopoles]] is [[stable weak homotopy equivalence|stably weak homotopy equivalent]] to [[classifying space]] of [[braid group]])}

For $k \in \mathbb{N}$ there is a [[stable weak homotopy equivalence]] between the [[moduli space of k monopoles]] \eqref{ModuliSpaceOfkInstantons} and the [[classifying space]] of the [[braid group]] $Braids_{2k}$ on $2 k$ strands:

\begin{displaymath}
\Sigma^\infty \mathcal{M}_k
  \;\simeq\;
  \Sigma^\infty Braids_{2k}
\end{displaymath}
\end{prop}
(\hyperlink{CohenCohenMannMilgram91}{Cohen-Cohen-Mann-Milgram 91})

$\backslash$linebreak

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

\begin{itemize}%
\item [[Nahm transform]], [[Bogomolny equation]]

\end{itemize}
[[!include moduli spaces -- contents]]

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item [[Simon Donaldson]], \emph{Nahm's Equations and the Classification of Monopoles}, Comm. Math. Phys., Volume 96, Number 3 (1984), 387-407, (\href{https://projecteuclid.org/euclid.cmp/1103941858}{euclid:cmp.1103941858})


\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})


\item [[Michael Atiyah]], [[Nigel Hitchin]], J. T. Stuart and M. Tabor, \emph{Low-Energy Scattering of Non-Abelian Magnetic Monopoles}, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 315, No. 1533, New Developments in the Theory and Application of Solitons (Aug. 13, 1985), pp. 459-469 (\href{https://www.jstor.org/stable/37546}{jstor:37546})


\item [[Gary Gibbons]], [[Nicholas Manton]], \emph{Classical and Quantum Dynamics of BPS Monopoles}, Nucl. Phys. B274 (1986) 183-224 (\href{http://inspirehep.net/record/18322}{spire:18322}, )


\item [[Ralph Cohen]], \emph{Stability phenomena in the topology of moduli spaces} (\href{https://arxiv.org/abs/0908.1938}{arXiv:0908.1938})



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Monopole_moduli_space}{Monopole moduli space}}

\end{itemize}
\hypertarget{as_transversal_ddbrane_intersections}{}\subsubsection*{{As transversal D$p$/D$(p+2)$-brane intersections}}\label{as_transversal_ddbrane_intersections}

In [[string theory]] [[Yang-Mills monopoles]] are [[geometric engineering of QFT|geometrically engineeted]] as transversally [[intersecting brane|intersecting]] [[Dp-D(p+2)-brane bound states]]:

For transversal [[D1-D3-brane bound states]]:

\begin{itemize}%
\item Duiliu-Emanuel Diaconescu, \emph{D-branes, Monopoles and Nahm Equations}, Nucl. Phys. B503 (1997) 220-238 (\href{https://arxiv.org/abs/hep-th/9608163}{arxiv:hep-th/9608163})


\item [[Amihay Hanany]], [[Edward Witten]], \emph{Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics}, Nucl. Phys. B492:152-190, 1997 (\href{https://arxiv.org/abs/hep-th/9611230}{arxiv:hep-th/9611230})


\item Jessica K. Barrett, Peter Bowcock, \emph{Using D-Strings to Describe Monopole Scattering} (\href{https://arxiv.org/abs/hep-th/0402163}{arxiv:hep-th/0402163})


\item Jessica K. Barrett, Peter Bowcock, \emph{Using D-Strings to Describe Monopole Scattering - Numerical Calculations} (\href{https://arxiv.org/abs/hep-th/0512211}{arxiv:hep-th/0512211})



\end{itemize}
For transversal [[D2-D4-brane bound states]] (with an eye towards [[AdS/QCD]]):

\begin{itemize}%
\item Alexander Gorsky, Valentin Zakharov, Ariel Zhitnitsky, \emph{On Classification of QCD defects via holography}, Phys. Rev. D79:106003, 2009 (\href{https://arxiv.org/abs/0902.1842}{arxiv:0902.1842})

\end{itemize}
For transversal [[D6-D8-brane bound states]] (with an eye towards [[AdS/QCD]]):

\begin{itemize}%
\item Deog Ki Hong, Ki-Myeong Lee, Cheonsoo Park, Ho-Ung Yee, Section V of: \emph{Holographic Monopole Catalysis of Baryon Decay}, JHEP 0808:018, 2008 (\href{https://arxiv.org/abs/0804.1326}{https:arXiv:0804.1326})

\end{itemize}
With emphasis on [[half NS5-branes]] in [[type I' string theory]]:

\begin{itemize}%
\item [[Amihay Hanany]], [[Alberto Zaffaroni]], \emph{Monopoles in String Theory}, JHEP 9912 (1999) 014 (\href{https://arxiv.org/abs/hep-th/9911113}{arxiv:hep-th/9911113})

\end{itemize}
The moduli space of monopoles appears also in the [[KK-compactification]] of the [[M5-brane]] on a complex surface ([[AGT-correspondence]]):

\begin{itemize}%
\item Benjamin Assel, Sakura Schafer-Nameki, Jin-Mann Wong, \emph{M5-branes on $S^2 \times M_4$: Nahm's Equations and 4d Topological Sigma-models}, J. High Energ. Phys. (2016) 2016: 120 (\href{https://arxiv.org/abs/1604.03606}{arxiv:1604.03606})

\end{itemize}
\hypertarget{relation_to_braids}{}\subsubsection*{{Relation to braids}}\label{relation_to_braids}

Relation to [[braid groups]]:

\begin{itemize}%
\item [[Fred Cohen]], [[Ralph Cohen]], B. M. Mann, R. J. Milgram, \emph{The topology of rational functions and divisors of surfaces}, Acta Math (1991) 166: 163 (\href{https://doi.org/10.1007/BF02398886}{doi:10.1007/BF02398886})


\item [[Ralph Cohen]], John D. S. Jones \emph{Monopoles, braid groups, and the Dirac operator}, Comm. Math. Phys. Volume 158, Number 2 (1993), 241-266 (\href{https://projecteuclid.org/euclid.cmp/1104254240}{euclid:cmp/1104254240})



\end{itemize}
Relation of [[Dp-D(p+2)-brane bound states]] (\href{Dp-Dp+2-brane+bound+states#ReferencesRelationToMonopoles}{hence} [[Yang-Mills monopoles]]) to [[Vassiliev braid invariants]] via [[chord diagrams]] computing [[radii]] of [[fuzzy spheres]]:

\begin{itemize}%
\item [[Sanyaje Ramgoolam]], [[Bill Spence]], S. Thomas, Section 3.2 of: \emph{Resolving brane collapse with $1/N$ corrections in non-Abelian DBI}, Nucl. Phys. B703 (2004) 236-276 (\href{https://arxiv.org/abs/hep-th/0405256}{arxiv:hep-th/0405256})


\item [[Simon McNamara]], [[Constantinos Papageorgakis]], [[Sanyaje Ramgoolam]], [[Bill Spence]], Appendix A of: \emph{Finite $N$ effects on the collapse of fuzzy spheres}, JHEP 0605:060, 2006 (\href{https://arxiv.org/abs/hep-th/0512145}{arxiv:hep-th/0512145})


\item [[Simon McNamara]], Section 4 of: \emph{Twistor Inspired Methods in Perturbative FieldTheory and Fuzzy Funnels}, 2006 (\href{http://inspirehep.net/record/1351861}{spire:1351861}, \href{https://strings.ph.qmul.ac.uk/sites/default/files/Mcnamaraphd.pdf}{pdf}, [[McNamara06.pdf:file]])


\item [[Constantinos Papageorgakis]], p. 161-162 of: \emph{On matrix D-brane dynamics and fuzzy spheres}, 2006 ([[Papageorgakis06.pdf:file]])



\end{itemize}
[[!redirects moduli spaces of monopoles]]

[[!redirects moduli space of k monopoles]] [[!redirects moduli spaces of k monopoles]]

[[!redirects Atiyah-Hitchin charge quantization]]

[[!redirects Yang-Mills monopole]] [[!redirects Yang-Mills monopoles]]



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