\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{Kaluza-Klein monopole}

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

\hypertarget{gravity}{}\paragraph*{{Gravity}}\label{gravity}

[[!include gravity contents]]

\hypertarget{riemannian_geometry}{}\paragraph*{{Riemannian geometry}}\label{riemannian_geometry}

[[!include Riemannian geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{in_supergravity}{In supergravity}\dotfill \pageref*{in_supergravity} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{NearHorizonGeometry}{Near-horizon geometry}\dotfill \pageref*{NearHorizonGeometry} \linebreak
\noindent\hyperlink{RelationToTheD6Brane}{Relation to the D6-brane in type IIA string theory}\dotfill \pageref*{RelationToTheD6Brane} \linebreak
\noindent\hyperlink{relation_to_the_d7brane_in_type_iib_string_theoryftheory}{Relation to the D7-brane in type IIB string theory/F-theory}\dotfill \pageref*{relation_to_the_d7brane_in_type_iib_string_theoryftheory} \linebreak
\noindent\hyperlink{other_brane_charges_}{Other Brane charges (?)}\dotfill \pageref*{other_brane_charges_} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{in_5d_gravity}{In 5d gravity}\dotfill \pageref*{in_5d_gravity} \linebreak
\noindent\hyperlink{in_11d_supergravity}{In 11d supergravity}\dotfill \pageref*{in_11d_supergravity} \linebreak
\noindent\hyperlink{original_articles}{Original articles}\dotfill \pageref*{original_articles} \linebreak
\noindent\hyperlink{review}{Review}\dotfill \pageref*{review} \linebreak
\noindent\hyperlink{relation_to_black_holes}{Relation to black holes}\dotfill \pageref*{relation_to_black_holes} \linebreak


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

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

In the context of [[Kaluza-Klein theory]] where an [[electromagnetic field]] in [[Einstein-Maxwell theory]] in [[dimension]] $d$ is modeled by a configuration of pure Einstein [[gravity]] in dimension $d+1$, a \emph{Kaluza-Klein monopole} is a configuration of gravity in dimension $d+1$ which in dimension $d$ looks like a magnetic [[monopole]] (\hyperlink{Sorkin83}{Sorkin 83}, \hyperlink{GrossPerry83}{Gross-Perry 83}).

\hypertarget{in_supergravity}{}\subsubsection*{{In supergravity}}\label{in_supergravity}

This situation is of particular interest in the reduction of [[11-dimensional supergravity]] (or [[M-theory]], where one also speaks of the [[M-brane|MK6-brane]]) where the Kaluza-Klein magnetic monopole charge is interpreted as [[D6-brane]] charge under [[duality between M-theory and type IIA string theory]].

The \emph{Kaluza-Klein monopole} (\hyperlink{HanKoh85}{Han-Koh 85}) is one type of solution of the [[equations of motion]] of [[11-dimensional supergravity]]. It is given by the [[product]] $N_4\times \mathbb{R}^{11-5,1}$ of Euclidean [[Taub-NUT spacetime]] with [[Minkowski spacetime]]. Upon [[Kaluza-Klein compactification]] this looks like a [[monopole]], whence the name. (For discussion as an [[ADE-singularity]] see \hyperlink{IMSY98}{IMSY 98, section 9}, \hyperlink{Asano00}{Asano 00, section 3}.)

Upon [[KK-compactification]] on a 6-dimensional [[fiber]], with the 11d KK-monopole / [[D6-brane]] completely [[wrapped brane|wrapping]] the fiber, the KK-monopole in [[11d supergravity]] becomes the KK-monopole in [[5d supergravity]]. Further compactifying on a circle leads to a [[black hole]] in 4d, incarnated as a D0/D6 bound state (e.g. \hyperlink{Nelson93}{Nelson 93}).

[[!include KK-monopole geometries -- table]]

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

(For the moment the following is all about the KK-monopoles in [[11d supergravity]]/[[M-theory]].)

\hypertarget{NearHorizonGeometry}{}\subsubsection*{{Near-horizon geometry}}\label{NearHorizonGeometry}

We discuss the [[near horizon geometry]] of coincident MK6-branes.

The [[metric tensor]] of $N$ coincident [[KK-monopoles]] in [[11-dimensional supergravity]] in the limit that $\ell_{th} \coloneqq N \ell_P \to 0$ is

\begin{equation}
g_{MK6} 
    \;=\;
  g_{\mathbb{R}^{6,1}}
  +
  (d y)^2
  + 
  y^2
  \big(
    (d \theta)^2
    + 
    (\sin \theta)^2 (d \varphi)^2
    + 
    (\cos \theta)^2 (d \phi)^2
  \big)
\label{SmallNMK6}\end{equation}
subject to the identification

\begin{equation}
(\varphi, \phi)
  \;\sim\;
  (\varphi, \phi) + (2\pi/N ,2\pi/N)
  \,.
\label{OrbifoldIdentificationForKKMonopole}\end{equation}
This is equation (47) in \hyperlink{IMSY98}{IMSY 98}, which applies subject to the condition

\begin{displaymath}
U/\left(\frac{N}{g^{2/3}_{YM}}\right) 
  \;=\; 
  U/\left(\frac{N}{(2\pi)^{4/3} \ell_P}\right) 
  \;\gg\; 
  1
\end{displaymath}
from a few lines above. Inserting this condition into the definition $y^2 \coloneqq 2 N \ell^3_P U$ right above (47) shows that

\begin{displaymath}
\begin{aligned}
  y^2 
  & =
  2 N \ell^3_P U
  \\
  & =
  2(2\pi)^{-4/3} N^2 \ell_P^2 
  \;
  \underset{
    \gg 1
  }{
  \underbrace{
     \left(U/\left(\frac{N}{ (2 \pi)^{4/3} \ell_P}\right)\right)
  }}
  \end{aligned}
\end{displaymath}
hence that the distance $y$ from the locus of the MK6-brane is large in units of

\begin{displaymath}
\ell_{th} \;=\; \sqrt{2} (2\pi)^{-2/3} N \ell_P
  \,.
\end{displaymath}
The identification \eqref{OrbifoldIdentificationForKKMonopole} means that this is the [[orbifold]] [[metric cone]] $\mathbb{R}^{6,1} \times \left( \mathbb{R}^4/(\mathbb{Z}_N)\right)$, hence an [[ADE classification| A-type]] [[ADE-singularity]]. To make this more explicit, introduce the complex coordinates

\begin{displaymath}
v \;\coloneqq\; y \, e^{i \varphi} \sin \theta
  \;\;\;
  w \;\coloneqq\; y \, e^{i \phi} \cos \theta
\end{displaymath}
on $\mathbb{R}^4 \simeq \mathbb{C}^2$, in terms of which \eqref{SmallNMK6} becomes

\begin{displaymath}
g_{MK6} 
  \;\coloneqq\;
  d v d \overline v
  + 
  d w d \overline w
\end{displaymath}
and which exhibit the identification \eqref{OrbifoldIdentificationForKKMonopole} as indeed that of the [[ADE classification|A-type]] $\mathbb{Z}_N$-action (\hyperlink{Asano00}{Asano 00, around (18)}).

\hypertarget{RelationToTheD6Brane}{}\subsubsection*{{Relation to the D6-brane in type IIA string theory}}\label{RelationToTheD6Brane}

Under the relation between [[M-theory]] and [[type IIA superstring theory]] an [[ADE orbifold]] of the 11d KK-monopole corresponds to [[D6-branes]] combined with [[O6-planes]] (\hyperlink{Townsend95}{Townsend 95, p. 6}, \hyperlink{AtiyahWitten01}{Atiyah-Witten 01, p. 17-18} see also e.g. \hyperlink{BerglundBrandhuber02}{Berglund-Brandhuber 02, around p. 15}).

By (\hyperlink{Townsend95}{Townsend 95, (1)}, \hyperlink{Sen97c}{Sen 97c (1)-(4)}) the 11d [[spacetime]] describing the KK-monopole lift of a plain single D6 brane is $\mathbb{R}^{6,1}\times \mathbb{R}^3\times S^1$ with [[metric tensor]] away from the origin of the $\mathbb{R}^3$-factor (which is the locus of the lifted D6/monopole) being

\begin{displaymath}
d s_{11}^2 =
  - d t^2 
  + d s_{\mathbb{R}^6}^2
  + (1+\mu/r) d s_{\mathbb{R}^3}^2
  + (1+ \mu/r)^{-1} (d x^{11} - A_i d x^i)^2
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $A = A_i d x^i$ is any 3-form on $\mathbb{R}^3$ satisfying $d_{\mathbb{R}^3} A = \star d (1-\mu/r)$;


\item $r$ denotes the distance in $\mathbb{R}^3$ from the origin.


\item $\mu$ is the [[charge]] of the monopole.



\end{itemize}
Away from $\{0\} \in \mathbb{R}^3$ this gives a [[circle bundle]] with [[first Chern class]] measured by the integral of $R_0 \coloneqq d A$ (the [[RR-field]] of the [[D0-brane]]) over any [[sphere]] surrounding the singular locus in $\mathbb{R}^3 - \{0\}$. By the [[electric-magnetic duality]] between [[D0-branes]] and [[D6-branes]] (due to the [[self-dual higher gauge theory|self-duality]] of the [[RR-field]]) this means, from the 10-dimensional perspective, that at $0 \in \mathbb{R}^3$ a [[D6-brane]] is located.

\begin{quote}%
graphics grabbed from \hyperlink{AcharyaGukov04}{Acharya-Gukov 04}


\end{quote}
Notice that the circle bundle away from its degeneration locus as a bundle over $S^2 \hookrightarrow \mathbb{R}^3 -\{0\}$ is necessarily of the form $S^3 \to S^2$, a multiple of the [[complex Hopf fibration]] (see also \hyperlink{AtiyahMaldacenaVafa00}{Atiyah-Maldacena-Vafa 00, p. 10}).

Discussion as an [[ADE singularity|A-type ADE singularity]] is in (\hyperlink{Sen97c}{Sen 97c, section 2}). Generalization to [[ADE singularity|D-type singularities]] and hence D6-branes in [[orientifolds]] is in (\hyperlink{Sen97c}{Sen 97c ,section 3}). Discussion as the [[fixed point]] of the [[circle group]]-[[action]] on the M-theory circle fibers is in (\hyperlink{Townsend95}{Townsend 95, p. 6}, \hyperlink{AtiyahWitten01}{Atiyah-Witten 01, pages 17-18}). Witten emphasizes that it is important that the location of the D6 is not just a [[cyclic group]] [[orbifold]] singularity but really a [[circle group]]-[[action]] [[fixed point]] [[conical singularity]]:

\begin{quote}%
 Chiral fermions arise when the locus of A$-$D$-$E singularities passes through isolated points at which X has an isolated conical singularity that is not just an orbifold singularity (\hyperlink{Witten01}{Witten 01, p. 2}).


\end{quote}
Discussion dealing carefully with the perspective where the locus of the [[D6-brane]] is not taken out is in (\hyperlink{GaillardSchmude09}{Gaillard-Schmude 09}).

For more on this see at \emph{\href{D6-brane#RelationToOtherBranes}{D6-brane -- Relation to other branes}} and at \emph{[[M-theory lift of gauge enhancement on D6-branes]]}.

\hypertarget{relation_to_the_d7brane_in_type_iib_string_theoryftheory}{}\subsubsection*{{Relation to the D7-brane in type IIB string theory/F-theory}}\label{relation_to_the_d7brane_in_type_iib_string_theoryftheory}

Under further [[T-duality]] to [[type IIB superstring theory]]/[[F-theory]] these D6-branes become the [[D7-branes]].

[[!include F-branes -- table]]

\hypertarget{other_brane_charges_}{}\subsubsection*{{Other Brane charges (?)}}\label{other_brane_charges_}

In (\hyperlink{Hull97}{Hull 97}) it was argued that the KK-monopole in [[11-dimensional supergravity]] is the object which carries the 6-form charge [[Poincaré duality|Poincaré dual]] to the time-component of the 5-form charge of the [[M5-brane]] as appearing in the [[M-theory super Lie algebra]] via

\begin{displaymath}
\wedge^5 (\mathbb{R}^{10,1})^\ast
  \simeq
  \wedge^5 (\mathbb{R}^{10})^\ast \oplus \wedge^6 \mathbb{R}^{10}
  \,.
\end{displaymath}
(The same kind of relation identifies the time-component of the [[M2-brane]] charge with the charge of the [[M9-brane]], see there.)

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

\begin{itemize}%
\item [[M-brane]]


\item [[Taub-NUT spacetime]]



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

\hypertarget{in_5d_gravity}{}\subsubsection*{{In 5d gravity}}\label{in_5d_gravity}

Original articles include

\begin{itemize}%
\item [[Rafael Sorkin]], \emph{Kaluza-Klein monopole} Phys. Rev. Lett. 51 (1983) 87 (\href{http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.51.87}{publisher})


\item [[David Gross]] and M. Perry, Nucl. Phys. B226 (1983) 29.


\item P. J. Ruback, \emph{The motion of Kaluza-Klein monopoles}, Comm. Math. Phys. Volume 107, Number 1 (1986), 93-102 (\href{https://projecteuclid.org/euclid.cmp/1104115933}{euclid:1104115933})


\item A. L. Cavalcanti de Oliveira, E. R. Bezerra de Mello, \emph{Kaluza-Klein Magnetic Monopole in Five-Dimensional Global Monopole Sapcetime}, Class.Quant.Grav. 21 (2004) 1685-1694 (\href{http://arxiv.org/abs/hep-th/0309189}{arXiv:hep-th/0309189})



\end{itemize}
Review includes

\begin{itemize}%
\item Emre Sakarya, \emph{Kaluza-Klein monopole}, 2007 (\href{https://inspirehep.net/record/875942/files/875942.pdf}{pdf})

\end{itemize}
Discussion of [[topological T-duality]] for KK-monopoles is in

\begin{itemize}%
\item Ashwin S. Pande, \emph{Topological T-duality and Kaluza-Klein Monopoles}, Adv. Theor. Math. Phys., 12, (2007), pp 185-215 (\href{https://arxiv.org/abs/math-ph/0612034}{arXiv:math-ph/0612034})

\end{itemize}
\hypertarget{in_11d_supergravity}{}\subsubsection*{{In 11d supergravity}}\label{in_11d_supergravity}

\hypertarget{original_articles}{}\paragraph*{{Original articles}}\label{original_articles}

\begin{itemize}%
\item Seung Kee Han, I.G. Koh, \emph{$N = 4$ Remaining Supersymmetry in Kaluza-Klein Monopole Background in D=11 Supergravity Theory}, Phys.Rev. D31 (1985) 2503, in [[Michael Duff]] (ed.), \emph{[[The World in Eleven Dimensions]]} 57-60 (\href{http://inspirehep.net/record/204521/?ln=en}{spire})


\item [[Paul Townsend]], \emph{The eleven-dimensional supermembrane revisited}, Phys.Lett.B350:184-187,1995 (\href{http://arxiv.org/abs/hep-th/9501068}{arXiv:hep-th/9501068})


\item [[Chris Hull]], \emph{Gravitational Duality, Branes and Charges}, Nucl.Phys. B509 (1998) 216-251 (\href{http://arxiv.org/abs/hep-th/9705162}{arXiv:hep-th/9705162})


\item [[Ashoke Sen]], \emph{Kaluza-Klein Dyons in String Theory}, Phys.Rev.Lett.79:1619-1621,1997 (\href{http://arxiv.org/abs/hep-th/9705212}{arXiv:hep-th/9705212})


\item [[Ashoke Sen]], \emph{Dynamics of Multiple Kaluza-Klein Monopoles in M- and String Theory}, Adv.Theor.Math.Phys.1:115-126, 1998 (\href{https://arxiv.org/abs/hep-th/9707042}{arXiv:hep-th/9707042})


\item [[Ashoke Sen]], \emph{A Note on Enhanced Gauge Symmetries in M- and String Theory}, JHEP 9709:001,1997 (\href{http://arxiv.org/abs/hep-th/9707123}{arXiv:hep-th/9707123})


\item Nissan Itzhaki, [[Juan Maldacena]], Jacob Sonnenschein, Shimon Yankielowicz, \emph{Supergravity and The Large $N$ Limit of Theories With Sixteen Supercharges}, Phys. Rev. D 58, 046004 1998 (\href{https://arxiv.org/abs/hep-th/9802042}{arXiv:hep-th/9802042})


\item Masako Asano, \emph{Compactification and Identification of Branes in the Kaluza-Klein monopole backgrounds} (\href{https://arxiv.org/abs/hep-th/0003241}{arXiv:hep-th/0003241})


\item [[Michael Atiyah]], [[Juan Maldacena]], [[Cumrun Vafa]], \emph{An M-theory Flop as a Large N Duality}, J.Math.Phys.42:3209-3220,2001 (\href{http://arxiv.org/abs/hep-th/0011256}{arXiv:hep-th/0011256})


\item [[Michael Atiyah]], [[Edward Witten]] \emph{$M$-Theory dynamics on a manifold of $G_2$-holonomy}, Adv. Theor. Math. Phys. 6 (2001) (\href{http://arxiv.org/abs/hep-th/0107177}{arXiv:hep-th/0107177})


\item [[Edward Witten]], \emph{Anomaly Cancellation On Manifolds Of $G_2$ Holonomy} (\href{http://arxiv.org/abs/hep-th/0108165}{arXiv:hep-th/0108165})


\item Per Berglund, Andreas Brandhuber, \emph{Matter From $G_2$ Manifolds}, Nucl.Phys. B641 (2002) 351-375 (\href{http://arxiv.org/abs/hep-th/0205184}{arXivLhep-th/0205184})


\item Jerome Gaillard, [[Johannes Schmude]], \emph{The lift of type IIA supergravity with D6 sources: M-theory with torsion}, JHEP 1002:032,2010 (\href{http://arxiv.org/abs/0908.0305}{arXiv:0908.0305})



\end{itemize}
Discussion in terms of [[G-structures]]:

\begin{itemize}%
\item [[Ulf Danielsson]], Giuseppe Dibitetto, Adolfo Guarino, \emph{KK-monopoles and G-structures in M-theory/type IIA reductions}, JHEP 1502 (2015) 096 (\href{https://arxiv.org/abs/1411.0575}{arXiv:1411.0575})

\end{itemize}
\hypertarget{review}{}\paragraph*{{Review}}\label{review}

\begin{itemize}%
\item [[Clifford Johnson]], section 10.5 of \emph{D-brane primer} (\href{http://arxiv.org/abs/hep-th/0007170}{arXiv:hep-th/0007170})


\item [[Katrin Becker]], [[Melanie Becker]], [[John Schwarz]], p. 333 of \emph{String Theory and M-Theory: A Modern Introduction}, 2007


\item [[Luis Ibáñez]], [[Angel Uranga]], section 6.3.3 of \emph{[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]}, Cambridge University Press 2012


\item [[Bobby Acharya]], [[Sergei Gukov]], p. 45 of \emph{M theory and Singularities of Exceptional Holonomy Manifolds}, Phys.Rept.392:121-189,2004 (\href{http://arxiv.org/abs/hep-th/0409191}{arXiv:hep-th/0409191})



\end{itemize}
\hypertarget{relation_to_black_holes}{}\subsubsection*{{Relation to black holes}}\label{relation_to_black_holes}

Relation to [[black holes in string theory]]

\begin{itemize}%
\item William Nelson, \emph{Kaluza-Klein Black Holes in String Theory}, Phys.Rev.D49:5302-5306,1994 (\href{http://arxiv.org/abs/hep-th/9312058}{arXiv:hep-th/9312058})

\end{itemize}
[[!redirects Kaluza-Klein monopoles]]

[[!redirects KK monopole]] [[!redirects KK monopoles]]

[[!redirects KK-monopole]] [[!redirects KK-monopoles]]

[[!redirects MK6]] [[!redirects MK6-brane]] [[!redirects MK6-branes]]



\end{document}