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\section*{heterotic M-theory on ADE-orbifolds}

\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{DualStringTheoryPerspectives}{Dual string theory perspectives}\dotfill \pageref*{DualStringTheoryPerspectives} \linebreak
\noindent\hyperlink{HetEWithADESingularities}{$HET_{E}$-Theory on ADE-singularities}\dotfill \pageref*{HetEWithADESingularities} \linebreak
\noindent\hyperlink{TypeIPrimeWithD6OnO8}{$I'$-Theory with D6-branes on O8-planes}\dotfill \pageref*{TypeIPrimeWithD6OnO8} \linebreak
\noindent\hyperlink{IPrimeTheoryOnADESingularitiesIntersectingO8Planes}{$I'$-theory on ADE-singularities intersecting O8-planes}\dotfill \pageref*{IPrimeTheoryOnADESingularitiesIntersectingO8Planes} \linebreak
\noindent\hyperlink{GeometricEngineeringOfDIs6NIs1SCFTs}{Geometric engineering of $D = 6$ $N=(1,0)$ SCFTs on M5-branes}\dotfill \pageref*{GeometricEngineeringOfDIs6NIs1SCFTs} \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_theory_with_adesingularities}{As $HET_E$-theory with ADE-singularities}\dotfill \pageref*{as_theory_with_adesingularities} \linebreak
\noindent\hyperlink{as_theory_with_d6branes}{As $I'$-theory with D6-branes}\dotfill \pageref*{as_theory_with_d6branes} \linebreak
\noindent\hyperlink{as_theory_with_adesingularities_2}{As $I'$-theory with ADE-singularities}\dotfill \pageref*{as_theory_with_adesingularities_2} \linebreak
\noindent\hyperlink{ftheory_perspective}{F-theory perspective}\dotfill \pageref*{ftheory_perspective} \linebreak
\noindent\hyperlink{ReferencesGeometricEngineeringOfDIs6NIs1SCFT}{Geometric engineering of $D=6, \mathcal{N}=(1,0)$ SCFT}\dotfill \pageref*{ReferencesGeometricEngineeringOfDIs6NIs1SCFT} \linebreak


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

The [[KK-compactification]] of [[M-theory]] on [[fibers]]

\begin{displaymath}
S^1 \!\sslash\! G_{HW} \times \mathbb{H}\!\sslash\! G_{ADE}
\end{displaymath}
which are, locally, the [[Cartesian product]] of

\begin{enumerate}%
\item the [[circle]] [[orientifold|orientifolded]] by $G_{HW} \simeq \mathbb{Z}_2$ as in [[Horava-Witten theory]];


\item the [[quaternions]] [[orbifold|orbifolded]] by a [[finite subgroup of SU(2)]] $G_{ADE}$.



\end{enumerate}
(\hyperlink{Sen97}{Sen 97, Sec 3}, \hyperlink{FLO99}{Faux-Lüst-Ovrut 99}, \hyperlink{KSTY99}{Kaplunovsky-Sonnenschein-Theisen-Yankielowicz 99}, \hyperlink{FLO00a}{Faux-Lüst-Ovrut 00a}, \hyperlink{FLO00b}{00b}, \hyperlink{FLO00c}{00c})

\begin{quote}%
graphics grabbed from \hyperlink{HSS18}{HSS18, Example 2.2.7}


\end{quote}
For $G_{ADE} = \mathbb{Z}_2$ this subsumes M-theory on [[K3]] times $S^1 \sslash G_{HW}$ (\hyperlink{SeibergWitten96}{Seiberg-Witten 96})

$\backslash$linebreak

\hypertarget{DualStringTheoryPerspectives}{}\subsection*{{Dual string theory perspectives}}\label{DualStringTheoryPerspectives}

Under [[duality in string theory]] (specifically: [[duality between M-theory and type IIA string theory]] and [[duality between M-theory and heterotic string theory]]) M-theory on $\mathbb{S}^{1}\sslash \mathbb{Z}_2^{HW} \times \mathbb{T}^{4}\sslash G^{ADE}$ appears through the following [[string theory]]-perspectives

\begin{enumerate}%
\item \hyperlink{HetEWithADESingularities}{$HET_{E}$-Theory on ADE-singularities}


\item \hyperlink{TypeIPrimeWithD6OnO8}{$I'$-Theory with D6-branes on O8-planes}


\item \hyperlink{IPrimeTheoryOnADESingularitiesIntersectingO8Planes}{$I'$-Theory on ADE-singularities intersecting O8-planes}



\end{enumerate}
The following graphics shows how the three perspectives arise from [[KK-compactification]] on three different choices of [[circle]]-[[fibers]]. Indicated also are the [[M5-branes]] and their string theoretic images at [[NS5-branes]]/[[D4-branes]] with [[geometric engineering of QFTs|geometrically engineer]] [[D=6 N=(1,0) SCFTs]] (see \hyperlink{GeometricEngineeringOfDIs6NIs1SCFTs}{further below}).

\begin{quote}%
graphics grabbed from \hyperlink{SatiSchreiber19}{SS19}


\end{quote}
\hypertarget{HetEWithADESingularities}{}\subsubsection*{{$HET_{E}$-Theory on ADE-singularities}}\label{HetEWithADESingularities}

(\ldots{})

[[Horava-Witten theory]], hence [[heterotic string theory]], on [[ADE-singularities]] $\mathbb{H} \sslash G_{ADE}$

(\hyperlink{Witten99}{Witten 99},\ldots{})

(\ldots{})

\hypertarget{TypeIPrimeWithD6OnO8}{}\subsubsection*{{$I'$-Theory with D6-branes on O8-planes}}\label{TypeIPrimeWithD6OnO8}



\begin{quote}%
from \hyperlink{GKSTY02}{GKSTY 02}


\end{quote}
If in addition the [[black brane|black]] [[NS5-brane]] sits at an [[O8-plane]], hence at the [[orientifold]] [[fixed point]]-locus, then in the ordinary $\mathbb{Z}/2$-[[quotient]] it appears as a ``[[half-brane]]'' -- the [[half M5-brane]] -- with only one copy of [[D6-branes]] ending on it:



\begin{quote}%
graphics grabbed from \hyperlink{GKSTY02}{GKSTY 02}


\end{quote}
(In \hyperlink{HananyZaffaroni99}{Hanany-Zaffaroni 99} this is interpreted in terms of the [[`t Hooft-Polyakov monopole]].)

 The lift to [[M-theory]] of this situation is an [[M5-brane]] [[brane intersection|intersecting]] an [[M9-brane]] (see at \emph{[[M-theory on S1/G}HW times H/G\_ADE]]\_):



\begin{quote}%
from \hyperlink{GKSTY02}{GKSTY 02}


\end{quote}
Alternatively the [[O8-plane]] may [[brane intersection|intersect]] the [[black brane|black]] [[D6-branes]] away from the [[black brane|black]] [[NS5-brane]]:



\begin{quote}%
from \hyperlink{HKLY15}{HKLY 15}


\end{quote}
In general, some of the NS5 sit away from the [[O8-plane]], while some sit on top of it:



\begin{quote}%
from \hyperlink{HananyZaffaroni98}{Hanany-Zaffaroni 98}


\end{quote}
\hypertarget{IPrimeTheoryOnADESingularitiesIntersectingO8Planes}{}\subsubsection*{{$I'$-theory on ADE-singularities intersecting O8-planes}}\label{IPrimeTheoryOnADESingularitiesIntersectingO8Planes}

(\hyperlink{BergmanRodriguezGomez12}{Bergman \& Rodriguez-Gomez 12, Sec. 3})

(\ldots{})

\hypertarget{GeometricEngineeringOfDIs6NIs1SCFTs}{}\subsection*{{Geometric engineering of $D = 6$ $N=(1,0)$ SCFTs on M5-branes}}\label{GeometricEngineeringOfDIs6NIs1SCFTs}

The [[M5-branes]]-configurations as \hyperlink{DualStringTheoryPerspectives}{above} are supposed to [[geometric engineering of QFTs|geometrically engineer]] [[D=6 N=(1,0) SCFTs]].

See the references \hyperlink{ReferencesGeometricEngineeringOfDIs6NIs1SCFT}{below}, for example \hyperlink{DHTV14}{DHTV 14, Section 6}, \hyperlink{GaiottoTomasiello14}{Gaiotto-Tomasiello 14}, \hyperlink{HKLY15}{HKLY 15}.

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

[[!include KK-compactifications of M-theory -- table]]

$\backslash$linebreak

[[!include Dp-D(p+4)-brane bound states -- contents]]

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

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

The first discussion of this compactification is possibly

\begin{itemize}%
\item [[Ashoke Sen]], Section 3 of: \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}
in the context of the [[M-theory lift of gauge enhancement on D6-branes]].

The original articles focusing on this situation:

\begin{itemize}%
\item Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], \emph{Intersecting Orbifold Planes and Local Anomaly Cancellation in M-Theory}, Nucl. Phys. B554: 437-483, 1999 (\href{https://arxiv.org/abs/hep-th/9903028}{arXiv:hep-th/9903028})


\item Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], \emph{Local Anomaly Cancellation, M-Theory Orbifolds and Phase-Transitions}, Nucl. Phys. B589: 269-291, 2000 (\href{https://arxiv.org/abs/hep-th/0005251}{arXiv:hep-th/0005251})


\item Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], \emph{An M-Theory Perspective on Heterotic K3 Orbifold Compactifications}, Int. J. Mod. Phys. A18:3273-3314, 2003 (\href{https://arxiv.org/abs/hep-th/0010087}{arXiv:hep-th/0010087})


\item Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], \emph{Twisted Sectors and Chern-Simons Terms in M-Theory Orbifolds}, Int. J. Mod. Phys. A18: 2995-3014, 2003 (\href{https://arxiv.org/abs/hep-th/0011031}{arXiv:hep-th/0011031})


\item [[Vadim Kaplunovsky]], J. Sonnenschein, [[Stefan Theisen]], S. Yankielowicz, \emph{On the Duality between Perturbative Heterotic Orbifolds and M-Theory on $T^4/Z_N$}, Nuclear Physics B Volume 590, Issues 1–2, 4 December 2000, Pages 123-160 Nuclear Physics B (\href{https://arxiv.org/abs/hep-th/9912144}{arXiv:hep-th/9912144}, )


\item E. Gorbatov, [[Vadim Kaplunovsky]], J. Sonnenschein, [[Stefan Theisen]], S. Yankielowicz, \emph{On Heterotic Orbifolds, M Theory and Type I' Brane Engineering}, JHEP 0205:015, 2002 (\href{https://arxiv.org/abs/hep-th/0108135}{arXiv:hep-th/0108135})



\end{itemize}
Discussion of [[heterotic M-theory]] on smooth [[K3]] originates around

\begin{itemize}%
\item [[Nathan Seiberg]], [[Edward Witten]], \emph{Comments on String Dynamics in Six Dimensions}, Nucl. Phys. B471:121-134, 1996 (\href{https://arxiv.org/abs/hep-th/9603003}{arXiv:hep-th/9603003})


\item Zygmunt Lalak, Steven Thomas, \emph{Gaugino Condensation, Moduli Potentials and Supersymmetry Breaking in M-Theory Models}, Nuclear Physics B Volume 515, Issues 1–2, 30 March 1998, Pages 55-72 Nuclear Physics B (\href{https://arxiv.org/abs/hep-th/9707223}{hep-th/9707223}, )

(on [[gaugino condensation]])



\end{itemize}
See also

\begin{itemize}%
\item Jacob Cole Claussen, \emph{The deconstruction of orbifold fixed points in heterotic M-theory}, 2016 (\href{http://hdl.handle.net/2152/41748}{hdl:2152/41748})


\item [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], Example 2.2.7 of: \emph{[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]}, CMP (2019) (\href{https://arxiv.org/abs/1805.05987}{arXiv:1805.05987}, \href{http://link.springer.com/article/10.1007/s00220-019-03442-3}{doi:10.1007/s00220-019-03442-3})


\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 4 of: \emph{[[schreiber:Super-exceptional embedding construction of the M5-brane|Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5]]} (\href{https://arxiv.org/abs/1908.00042}{arXiv:1908.00042})


\item [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], Section 4.1 of: \emph{[[schreiber:Equivariant Cohomotopy implies orientifold tadpole cancellation]]} (\href{https://arxiv.org/abs/1909.12277}{arXiv:1909.12277})



\end{itemize}
\hypertarget{as_theory_with_adesingularities}{}\subsubsection*{{As $HET_E$-theory with ADE-singularities}}\label{as_theory_with_adesingularities}

As [[heterotic string theory]] on [[orbifold]] [[ADE-singularities]]:

\begin{itemize}%
\item [[Edward Witten]], \emph{Heterotic String Conformal Field Theory And A-D-E Singularities}, JHEP 0002:025, 2000 (\href{https://arxiv.org/abs/hep-th/9909229}{arXiv:hep-th/9909229})

\end{itemize}
\hypertarget{as_theory_with_d6branes}{}\subsubsection*{{As $I'$-theory with D6-branes}}\label{as_theory_with_d6branes}

As [[type I' string theory]] with [[D6-branes]]:

\begin{itemize}%
\item [[Ilka Brunner]], [[Andreas Karch]], \emph{Branes at Orbifolds versus Hanany Witten in Six Dimensions}, JHEP 9803:003, 1998 (\href{https://arxiv.org/abs/hep-th/9712143}{arXiv:hep-th/9712143})


\item [[Amihay Hanany]], [[Alberto Zaffaroni]], \emph{Branes and Six Dimensional Supersymmetric Theories}, Nucl.Phys. B529 (1998) 180-206 (\href{https://arxiv.org/abs/hep-th/9712145}{arXiv:hep-th/9712145})


\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})


\item Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Masato Taki, Futoshi Yagi, \emph{A new 5d description of 6d D-type minimal conformal matter}, JHEP 1508:097, 2015 (\href{https://arxiv.org/abs/1505.04439}{arXiv:1505.04439})


\item Ibrahima Bah, Achilleas Passias, [[Alessandro Tomasiello]], \emph{$AdS_5$ compactifications with punctures in massive IIA supergravity}, JHEP11 (2017)050 (\href{https://arxiv.org/abs/1704.07389}{arXiv:1704.07389})



\end{itemize}
\hypertarget{as_theory_with_adesingularities_2}{}\subsubsection*{{As $I'$-theory with ADE-singularities}}\label{as_theory_with_adesingularities_2}

As [[type I' string theory]] at [[orbifold]] [[ADE-singularities]]:

\begin{itemize}%
\item [[Oren Bergman]], Diego Rodriguez-Gomez, Section 3 of: \emph{5d quivers and their $AdS_6$ duals}, JHEP07 (2012) 171 (\href{https://arxiv.org/abs/1206.3503}{arxiv:1206.3503})


\item Chiung Hwang, Joonho Kim, Seok Kim, Jaemo Park, Section 3.4.2 of: \emph{General instanton counting and 5d SCFT}, JHEP07 (2015) 063 (\href{https://arxiv.org/abs/1406.6793}{arxiv:1406.6793})



\end{itemize}
\hypertarget{ftheory_perspective}{}\subsubsection*{{F-theory perspective}}\label{ftheory_perspective}

The [[F-theory]] perspective:

\begin{itemize}%
\item Monika Marquart, [[Daniel Waldram]], \emph{F-theory duals of M-theory on $S^1/\mathbb{Z}_2 \times T^4 / \mathbb{Z}_N$} (\href{https://arxiv.org/abs/hep-th/0204228}{arXiv:hep-th/0204228})


\item Christoph Lüdeling, Fabian Ruehle, \emph{F-theory duals of singular heterotic K3 models}, Phys. Rev. D 91, 026010 (2015) (\href{https://arxiv.org/abs/1405.2928}{arXiv:1405.2928})



\end{itemize}
\hypertarget{ReferencesGeometricEngineeringOfDIs6NIs1SCFT}{}\subsubsection*{{Geometric engineering of $D=6, \mathcal{N}=(1,0)$ SCFT}}\label{ReferencesGeometricEngineeringOfDIs6NIs1SCFT}

On [[D=6 N=(1,0) SCFTs]] via [[geometric engineering of QFT|geometric engineering]] on [[M5-branes]]/[[NS5-branes]] at D-, E-type [[ADE-singularities]], notably from [[M-theory on S1/G\_HW times H/G\_ADE]], hence from [[orbifolds]] of [[type I' string theory]] (see at \href{NS5-brane#NSHalfBranes}{half NS5-brane}):

\begin{itemize}%
\item Michele Del Zotto, [[Jonathan Heckman]], [[Alessandro Tomasiello]], [[Cumrun Vafa]], Section 6 of: \emph{6d Conformal Matter}, JHEP02(2015)054 (\href{https://arxiv.org/abs/1407.6359}{arXiv:1407.6359})


\item [[Davide Gaiotto]], [[Alessandro Tomasiello]], \emph{Holography for $(1,0)$ theories in six dimensions}, JHEP12(2014)003 (\href{https://arxiv.org/abs/1404.0711}{arXiv:1404.0711})


\item Kantaro Ohmori, Hiroyuki Shimizu, \emph{$S^1/T^2$ Compactifications of 6d $\mathcal{N} = (1,0)$ Theories and Brane Webs}, J. High Energ. Phys. (2016) 2016: 24 (\href{https://arxiv.org/abs/1509.03195}{arXiv:1509.03195})


\item Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Futoshi Yagi, \emph{6d SCFTs, 5d Dualities and Tao Web Diagrams}, JHEP05 (2019)203 (\href{https://arxiv.org/abs/1509.03300}{arXiv:1509.03300})


\item Ibrahima Bah, Achilleas Passias, [[Alessandro Tomasiello]], \emph{$AdS_5$ compactifications with punctures in massive IIA supergravity}, JHEP11 (2017)050 (\href{https://arxiv.org/abs/1704.07389}{arXiv:1704.07389})


\item Santiago Cabrera, [[Amihay Hanany]], Marcus Sperling, \emph{Magnetic Quivers, Higgs Branches, and 6d $\mathcal{N}=(1,0)$ Theories}, J. High Energ. Phys. (2019) 2019: 71 (\href{https://arxiv.org/abs/1904.12293}{arXiv:1904.12293})



\end{itemize}
[[!redirects heterotic M-theory on ADE-singularities]]

[[!redirects M-theory on S1/G\_HW times H/G\_ADE]]



\end{document}