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\begin{document}

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\section*{M-theory lift of gauge enhancement on D6-branes}

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

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{construction}{Construction}\dotfill \pageref*{construction} \linebreak
\noindent\hyperlink{singularity_and_gauge_enhancement}{$A_{N-1}$-singularity and $SU(N)$-gauge enhancement}\dotfill \pageref*{singularity_and_gauge_enhancement} \linebreak
\noindent\hyperlink{singularity_and_gauge_enhancement_2}{$D_{N}$-singularity and $SO(2N)$-gauge enhancement}\dotfill \pageref*{singularity_and_gauge_enhancement_2} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{refrences}{Refrences}\dotfill \pageref*{refrences} \linebreak


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

It is well understood that in [[type IIA string theory]] there appears

\begin{itemize}%
\item $SU(N)$ [[enhanced gauge symmetry]] on $N$-coincident [[D-branes]];


\item $SO(2N)$ enhanced gauge symmetry on $N$-coincident [[D-branes]] at an [[orientifold plane]].



\end{itemize}
The physical picture of this effect is that the gauge bosons of these [[gauge fields]] are the modes of the [[open strings]] stretching between these D-branes which become massless as the branes coincide.

Now, as the situation is lifted to [[M-theory]], the [[D0-branes]], [[D2-branes]] and [[D4-branes]] lift to [[M2-branes]] and [[M5-branes]], and the gauge enhancement is thought to be similarly reflected on these [[M-branes]] (as exhibited for the [[M2-branes]] by the [[BLG-model]] and [[ABJM-model]]).

But for the [[D6-brane]] the situation is different: the [[D6-brane]] lifts not to an [[M-brane]], but to a configuration of the [[field (physics)|field]] of [[11-dimensional supergravity]]: the 11d [[Kaluza-Klein monopole]].

 Here we discuss the picture of how [[gauge enhancement]] on [[D6-branes]] in [[type IIA string theory]] is reflected on [[Kaluza-Klein monopoles]] at [[ADE-singularities]]. The explicit realization reviewed below is due to (\hyperlink{Sen97}{Sen 97}).



\begin{quote}%
graphics grabbed from \href{http://ncatlab.org/schreiber/show/Equivariant+homotopy+and+super+M-branes}{HSS18}


\end{quote}
\hypertarget{construction}{}\subsection*{{Construction}}\label{construction}

\hypertarget{singularity_and_gauge_enhancement}{}\subsubsection*{{$A_{N-1}$-singularity and $SU(N)$-gauge enhancement}}\label{singularity_and_gauge_enhancement}

The 11d multi-centered [[Kaluza-Klein monopole]] [[spacetime]] has [[metric tensor]] of the form

\begin{displaymath}
ds^2 
     \coloneqq 
    - d t^2 
    + \underoverset{n = 5}{10}{\sum} d y^n d y^n
    + ds^2_{TN}
  \,,
\end{displaymath}
where

\begin{displaymath}
ds^2_{TN} 
    \coloneqq
  V^{-1}( d x^4 + \vec \omega \cdot d \vec r )^2 + V d \vec r^2
\end{displaymath}
is the metric tensor of the multi-centered [[Taub-NUT space]], with $x^4$ the canonical coordinate on a [[circle]] and with

\begin{displaymath}
V \coloneqq 1 + \underoverset{I = 1}{N}{\sum} \frac{4m}{{\vert \vec r - \vec r_I\vert}}
\end{displaymath}
and

\begin{displaymath}
\vec \nabla \times \vec \omega = - \vec \nabla V
\end{displaymath}
for $\{\vec r_I\}_{I =1}^N$ the set of positions of the KK-monopoles of mass $m$.

Notice that the radius $16 \pi m V^{-1/2}$ of the $x^4$-circle vanishes precisely at the positions $\vec r_I$. Nevertheless, as long as all the $\vec r_I$ are distinct, then the above is a smooth spacetime also at the positions of the $\vec r_I$ -- if the periodicity of $x^4$ is taken to be $16 \pi m$. But with this periodicity fixed, then as the $N$ monopole positions $\vec r_I$ coincide, then the resulting metric has a [[conical singularity]] at that point, of [[ADE-singularity]] type $A_{N-1}$.

To see where the gauge enhancement arises from in this singular case, observe that in the non-singular configuration there are $N-1$ linearly indepent 2-[[cycles]] $S_{i j}$ in the multi-center KK-monopole spacetime, represented by the the [[2-spheres]] that are swept out by the circle fiber as it moves from $\vec r_i$ to $\vec r_j$ (remembering that the circle fiber radius vanishes precisely at the positions $\vec r_I$).

For the canonical choice of straight path between $\vec r_i$ and $\vec r_j$ (and arbitrary fixed position in the remaining 7 dimensions) then the [[surface]] [[area]] of these [[2-spheres]] is

\begin{displaymath}
vol(S_{i j}) = 16 \pi m  {\vert \vec r_i - \vec r_j\vert}
  \,.
\end{displaymath}
For any other choice of path the surface area will be larger. Hence an [[M2-brane]] with tension $T_{M2}$ [[wrapped brane|swrapping]] the 2-[[cycle]] $S_{i j}$ has minimal tension energy when in the configuration of these spheres, namely

\begin{displaymath}
m_{i j} = 16 \pi m T_{M2} {\vert \vec r_i - \vec r_j\vert}
  \,.
\end{displaymath}
The type IIA limit is given by $m \to 0$. In this limit the [[M2-branes]] wrapping the above cycles become the type IIA [[superstring]] by [[double dimensional reduction]], the [[KK-monopoles]] become the [[D6-branes]], and it is evident from the geometry that the membrane warpping $S_{i j}$ becomes an [[open string]] strentching between the $i$th and the $j$th D6-brane.

In the limit $m \to 0$ the D6-branes coincide,the strings stretching between them become massless (in accord with the above formula for the wrapped M2-brane mass), and become the gauge bosons of an $SU(N)$ [[Chan-Paton gauge field]].

(\href{Sen97}{Sen 97, section 2})

\hypertarget{singularity_and_gauge_enhancement_2}{}\subsubsection*{{$D_{N}$-singularity and $SO(2N)$-gauge enhancement}}\label{singularity_and_gauge_enhancement_2}

Now consider the above setupmodified by replacing the [[Taub-NUT space]] with coordinates $(\vec r, x^4)$ by its $\mathbb{Z}/2$-[[orbifold]] given by the $\mathbb{Z}/2$-[[action]] with nontrivial operation given by

\begin{displaymath}
(\vec r, x^4) \mapsto (- \vec r, - x^4)
\end{displaymath}
and in the Taub-NUT metric replace $V$ by

\begin{displaymath}
V 
    \coloneqq 
  1 
    - 
  \frac{16m}{r}
    +
  \underoverset{i = 1}{N}{\sum}
  \left(
    \frac{4m}{\vert \vec r - \vec r_i\vert}
    +
    \frac{4m}{\vert \vec r + \vec r_i\vert}
  \right)
  \,.
\end{displaymath}
The type IIA image of the origin of this configuration is an [[orientifold plane]].

Now as the $\vec r_i$ all approach 0 we get an 11d spacetime with a $D_N$-type [[ADE-singularity]], whose type IIA image is $N$ D6-branes coincident on an [[orientifold plane]].

(\href{Sen97}{Sen 97, section 3})

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

\begin{itemize}%
\item [[heterotic M-theory on ADE-orbifolds]]


\item [[membrane triple junction]]



\end{itemize}
[[!include F-branes -- table]]

\hypertarget{refrences}{}\subsection*{{Refrences}}\label{refrences}

\begin{itemize}%
\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})

\end{itemize}


\end{document}