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\section*{MO5}

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{charge_anomaly_cancellation_and_duality_to_o4planes}{Charge, anomaly cancellation and duality to O4-planes}\dotfill \pageref*{charge_anomaly_cancellation_and_duality_to_o4planes} \linebreak
\noindent\hyperlink{as_a_sector_of_heterotic_mtheory_on_adesingularities}{As a sector of heterotic M-theory on ADE-singularities}\dotfill \pageref*{as_a_sector_of_heterotic_mtheory_on_adesingularities} \linebreak
\noindent\hyperlink{as_geometric_engineering_of___scfts}{As geometric engineering of $D=6$ $\mathcal{N} = (2,0)$ SCFTs}\dotfill \pageref*{as_geometric_engineering_of___scfts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{as_a_sector_of_the__in_heterotic_mtheory_at_adesingularities}{As a sector of the $-\tfrac{1}{2}M5$ in heterotic M-theory at ADE-singularities}\dotfill \pageref*{as_a_sector_of_the__in_heterotic_mtheory_at_adesingularities} \linebreak
\noindent\hyperlink{relation_to_o4planes}{Relation to O4-planes}\dotfill \pageref*{relation_to_o4planes} \linebreak
\noindent\hyperlink{geometric_engineering_of_orthogonal___scfts}{Geometric engineering of orthogonal $D=6$, $\mathcal{N}=(2,0)$ SCFTs}\dotfill \pageref*{geometric_engineering_of_orthogonal___scfts} \linebreak


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

The \emph{MO5} (\hyperlink{DasguptaMukhil95}{Dasgupta-Mukhil 95, Sec. 2FFurther}, \hyperlink{Witten95}{Witten 95, Sec. 3.3}) is one of the types of $\tfrac{1}{2}$ [[BPS state|BPS]] [[orientifold]] [[fixed loci]] in [[M-theory]]/on [[D=11 N=1 supergravity|D=11 N=1]] [[super-spacetime]]. Locally on [[super Minkowski spacetime]] it is the [[super-geometry|super]]-[[orientifold]] $\mathbb{R}^{10,1\vert \mathbf{32}} \sslash \mathbb{Z}^{\Gamma^{56789}}_2$ given by the global [[orbifold quotient]] by the [[group of order 2]] which is generated by the [[Clifford algebra]]-element $\Gamma_5 \Gamma_6 \Gamma_7 \Gamma_8 \Gamma_9 \in Pin^+(10,1)$ in the positive [[pin group]] (for precise details see \hyperlink{HSS18}{HSS 18, Lemma 4.12}).

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

\hypertarget{charge_anomaly_cancellation_and_duality_to_o4planes}{}\subsubsection*{{Charge, anomaly cancellation and duality to O4-planes}}\label{charge_anomaly_cancellation_and_duality_to_o4planes}

Under [[duality between M-theory and type IIA string theory]], for [[KK-compactification]] along a [[circle]]-[[fiber]] parallel to the MO5, a plain MO5 becomes the [[O4-plane]], specifically the $O^- 4$-plane, while an MO5 with an [[M5-brane]] on top of it becomes the $O^+ 4$-plane (\hyperlink{Gimon98}{Gimon 9, Sec. III8},\hyperlink{HananyKol00}{Hanany-Kol 00, Sec. 3.1}).

Hence the spacetime [[dimensions]] of the MO5 is the same that of the [[black brane|black]] [[M5-brane]], but its charge as an [[O-plane]] is $- \tfrac{1}{2}$ that of a single [[M5-brane]] (\hyperlink{DasguptaMukhil95}{Dasgupta-Mukhil 95, Sec. 2}, \hyperlink{Witten95}{Witten 95, 3.3}, \hyperlink{Hori98}{Hori 98, 2.1}), so that am [[M-theory|M-theoretic]] lift of [[RR-field tadpole cancellation]] requires the presence of 16 [[M5-branes]] in an [[toroidal orbifold|toroidal]] MO5-type [[orientifold]] $\mathbb{T}^{\mathbf{5}_{sgn}} \!\sslash\! \mathbb{Z}_2$.

\hypertarget{as_a_sector_of_heterotic_mtheory_on_adesingularities}{}\subsubsection*{{As a sector of heterotic M-theory on ADE-singularities}}\label{as_a_sector_of_heterotic_mtheory_on_adesingularities}

By the classification of \href{M5-brane#MFF12}{MFF 12, Sec. 8.3}, the MO5-plane is \emph{not} in fact a $\tfrac{1}{2}$ [[BPS state|BPS]] solution of [[D=11 N=1 supergravity]]. But the [[brane intersection|intersection]] of an [[MK6]] with an [[MO9-plane]] is a $\tfrac{1}{4}$ [[BPS state|BPS]] solution to [[D=11 N=1 supergravity]].

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


\end{quote}
Thus, [[heterotic M-theory on ADE-orbifolds]] the MO5-plane becomes the [[fixed locus]] of the [[diagonal]] $\mathbb{Z}_2 \overset{ diag }{\hookrightarrow } \mathbb{Z}_2 \times G^{DE}$, hence the [[brane intersection|intersection]] of an [[MO9]] with an [[MK6]]${}_{DE}$ (\hyperlink{FLO99}{FLO 99, Sec. 4}), also called the $-\tfrac{1}{2}M5$, since under [[duality between M-theory and type IIA string theory]] this becomes the [[half NS5-brane]] of [[type I' string theory]].

\hypertarget{as_geometric_engineering_of___scfts}{}\subsubsection*{{As geometric engineering of $D=6$ $\mathcal{N} = (2,0)$ SCFTs}}\label{as_geometric_engineering_of___scfts}

Coincident [[M5-branes]] on an [[MO5-plane]] are supposed to [[geometric engineering of QFTs|geometrically engineer]] [[D=6 N=(2,0) SCFTs]] with [[classification of simple Lie groups|D-series]] [[gauge groups]] (\hyperlink{AKY98}{AKY 98, Sec. IIB}).

But if these are really $-\tfrac{1}{2} M5 = MK6 \cap MO9$-branes of [[heterotic M-theory on ADE-orbifolds]], then they will [[geometric engineering of QFTs|geometrically engineer]] [[D=6 N=(1,0) SCFTs]], instead, see \href{heterotic+M-theory+on+ADE-orbifolds#GeometricEngineeringOfDIs6NIs1SCFTs}{there}.

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

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

The original articles are

\begin{itemize}%
\item Keshav Dasgupta, [[Sunil Mukhi]], \emph{Orbifolds of M-theory}, Nucl. Phys. B465 (1996) 399-412 (\href{https://arxiv.org/abs/hep-th/9512196}{arXiv:hep-th/9512196})


\item [[Edward Witten]], \emph{Five-branes And M-Theory On An Orbifold}, Nucl. Phys. B463:383-397, 1996 (\href{https://arxiv.org/abs/hep-th/9512219}{arXiv:hep-th/9512219})


\item [[Kentaro Hori]], \emph{Consistency Conditions for Fivebrane in M Theory on $\mathbb{R}^5/\mathbb{Z}_2$ Orbifold}, Nucl.Phys.B539:35-78, 1999 (\href{https://arxiv.org/abs/hep-th/9805141}{arXiv:hep-th/9805141})



\end{itemize}
Classification view of [[rational equivariant homotopy theory|rational]] [[equivariant Cohomotopy]] of [[super-spacetime]]:

\begin{itemize}%
\item [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], Example 2.2.2 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})

\end{itemize}
\hypertarget{as_a_sector_of_the__in_heterotic_mtheory_at_adesingularities}{}\subsubsection*{{As a sector of the $-\tfrac{1}{2}M5$ in heterotic M-theory at ADE-singularities}}\label{as_a_sector_of_the__in_heterotic_mtheory_at_adesingularities}

Discussion in the more general context of [[heterotic M-theory on ADE-orbifolds]] is in

\begin{itemize}%
\item Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], Section 4 of \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})

\end{itemize}
and in view of [[super-embedding formalism|super-embeddings]] of [[M5-branes]] in

\begin{itemize}%
\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})

\end{itemize}
\hypertarget{relation_to_o4planes}{}\subsubsection*{{Relation to O4-planes}}\label{relation_to_o4planes}

Relation of the MO5 to the [[O4-plane]] under [[duality between M-theory and type IIA string theory]]:

\begin{itemize}%
\item [[Kentaro Hori]], Section 3 of: \emph{Consistency Conditions for Fivebrane in M Theory on $\mathbb{R}^5/\mathbb{Z}_2$ Orbifold}, Nucl. Phys. B539:35-78, 1999 (\href{https://arxiv.org/abs/hep-th/9805141}{arXiv:hep-th/9805141})


\item [[Eric Gimon]], \emph{On the M-theory Interpretation of Orientifold Planes} (\href{https://arxiv.org/abs/hep-th/9806226}{arXiv:hep-th/9806226}, \href{http://inspirehep.net/record/472499}{spire:472499})


\item [[Amihay Hanany]], [[Barak Kol]], Section 3.1 in: \emph{On Orientifolds, Discrete Torsion, Branes and M Theory}, JHEP 0006 (2000) 013 (\href{https://arxiv.org/abs/hep-th/0003025}{arXiv:hep-th/0003025})


\item Yoonseok Hwang, Joonho Kim, Seok Kim, \emph{M5-branes, orientifolds, and S-duality}, J. High Energ. Phys. (2016) 2016: 148 (\href{https://arxiv.org/abs/1607.08557}{arXiv:1607.08557})



\end{itemize}
\hypertarget{geometric_engineering_of_orthogonal___scfts}{}\subsubsection*{{Geometric engineering of orthogonal $D=6$, $\mathcal{N}=(2,0)$ SCFTs}}\label{geometric_engineering_of_orthogonal___scfts}

As [[geometric engineering of QFTs|geometric engineering]] of [[D=6 N=(2,0) SCFTs]] with [[classification of simple Lie groups|D-series]] [[gauge groups]]:

\begin{itemize}%
\item Changhyun Ahn, Hoil Kim, Hyun Seok Yang, \emph{$SO(2N)$ $(0,2)$ SCFT and M Theory on $AdS_7 \times \mathbb{R}P^4$}, Phys.Rev. D59 (1999) 106002 (\href{https://arxiv.org/abs/hep-th/9808182}{arXiv:hep-th/9808182})

\end{itemize}
[[!redirects MO5-plane]]



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