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\section*{type I string theory}

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

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

[[!include string theory - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

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

[[!include gravity contents]]

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

[[!include functorial quantum field theory - contents]]

\hypertarget{higher_spin_geometry}{}\paragraph*{{Higher spin geometry}}\label{higher_spin_geometry}

[[!include higher spin geometry - contents]]

\hypertarget{elliptic_cohomology}{}\paragraph*{{Elliptic cohomology}}\label{elliptic_cohomology}

[[!include elliptic cohomology -- 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{TadpoleCancellationAndSO32GUT}{Tadpole cancellation and $SO(32)$-GUT in Type I}\dotfill \pageref*{TadpoleCancellationAndSO32GUT} \linebreak
\noindent\hyperlink{tadpole_cancellation_and_gut_in_type_i_2}{Tadpole cancellation and $SO(16) \times SO(16)$-GUT in Type I'}\dotfill \pageref*{tadpole_cancellation_and_gut_in_type_i_2} \linebreak
\noindent\hyperlink{orbifolds_of_type_i}{Orbifolds of type I}\dotfill \pageref*{orbifolds_of_type_i} \linebreak
\noindent\hyperlink{dualities}{Dualities}\dotfill \pageref*{dualities} \linebreak
\noindent\hyperlink{stringstring_dualities}{String-string dualities}\dotfill \pageref*{stringstring_dualities} \linebreak
\noindent\hyperlink{horavawitten_theory}{Horava-Witten theory}\dotfill \pageref*{horavawitten_theory} \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{type_i}{Type I'}\dotfill \pageref*{type_i} \linebreak
\noindent\hyperlink{Phenomenology}{Phenomenology}\dotfill \pageref*{Phenomenology} \linebreak
\noindent\hyperlink{duality}{Duality}\dotfill \pageref*{duality} \linebreak
\noindent\hyperlink{geometric_engineering_of___scft}{Geometric engineering of $D=6$ $\mathcal{N}=(1,0)$ SCFT}\dotfill \pageref*{geometric_engineering_of___scft} \linebreak


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

What is called \emph{type I string theory} is [[type IIB string theory]] on [[orientifold]] [[spacetimes]], hence on [[O9-planes]].

Its [[T-duality|T-dual]], called \emph{type I' string theory}, is [[type IIA string theory]] on [[O8-planes]], which under the [[duality between M-theory and type IIA string theory]] is M-theory [[KK-compactification|KK-compactified]] on the [[orientifold]] $S^1 \times S^1 \sslash \mathbb{Z}_2$ (see also \emph{[[M-theory on S1/G}HW times H/G\_ADE]]\_):

\begin{displaymath}
\itexarray{
    M 
    \\
    {}^{ \mathllap{S^1 \times S^1/\mathbb{Z}_2 }}\big\downarrow
    \\
    I' &\underset{T}{\leftrightarrow}& I
  }
\end{displaymath}
\begin{quote}%
table from \hyperlink{BlumenhagenLustTheisen13}{BLT 13}


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

\hypertarget{TadpoleCancellationAndSO32GUT}{}\subsubsection*{{Tadpole cancellation and $SO(32)$-GUT in Type I}}\label{TadpoleCancellationAndSO32GUT}

For type I string theory on [[flat orbifold|flat]] ([[toroidal orbifold|toroidal]]) [[target spacetime]] [[orientifolds]] $\mathbb{R}^{9,1}$ (i.e. for [[type IIB string theory]] on flat toroidal [[O9-planes]]) [[RR-field tadpole cancellation]] requires 32 [[D-branes]] (see \href{orientifold+plane#ContingOfDBranesOnOrientifolds}{this Remark} for counting D-branes in [[orientifolds]]) to cancel the [[O-plane charge]] of -32 (\href{orientifold+plane#OPlaneChargeForFlatOrientifolds}{here}).

Under the [[duality between type I and heterotic string theory]] this translates to the [[semi-spin group|semi-spin]] [[gauge group]] [[SemiSpin(32)]] of [[heterotic string theory]].

Discussion of type-I [[string phenomenology]] and [[grand unified theory]] based on [[SO(32)]] type-I strings: (\hyperlink{MMRB86}{MMRB 86}, \hyperlink{IbanezMunozRigolin98}{Ibanez-Munoz-Rigolin 98}, \hyperlink{Yamatsu17}{Yamatsu 17}).

\hypertarget{tadpole_cancellation_and_gut_in_type_i_2}{}\subsubsection*{{Tadpole cancellation and $SO(16) \times SO(16)$-GUT in Type I'}}\label{tadpole_cancellation_and_gut_in_type_i_2}

For type I' string theory on [[flat orbifold|flat]] ([[toroidal orbifold|toroidal]]) [[target spacetime]] [[orientifolds]] $X^{8,1} \times \mathbb{S}^1/\mathbb{Z}_2$ (i.e. for [[type IIA string theory]] on two flat toroidal [[O8-planes]]) [[RR-field tadpole cancellation]] requires 16 [[D-branes]] (see \href{orientifold+plane#ContingOfDBranesOnOrientifolds}{this Remark} for counting D-branes in [[orientifolds]]) on each of the two [[O8-planes]] to cancel the total [[O-plane charge]] of $-32 = 2 \cdot (-16)$ (\href{orientifold+plane#OPlaneChargeForFlatOrientifolds}{here}).

Discussion of [[Spin(16)]]-[[GUT]] phenomenology:

(\ldots{})

\hypertarget{orbifolds_of_type_i}{}\subsubsection*{{Orbifolds of type I}}\label{orbifolds_of_type_i}

Type I' on [[toroidal orbifold|toroidal]] [[orientifolds]] with [[ADE-singularities]] (e.g. \hyperlink{BergmanRodriguezGomez12}{Bergman\&Rodriguez-Gomez 12, Sec. 3})

[[duality in string theory|dual]] to [[heterotic M-theory on ADE-orbifolds]].

(\ldots{})

\hypertarget{dualities}{}\subsubsection*{{Dualities}}\label{dualities}

\hypertarget{stringstring_dualities}{}\paragraph*{{String-string dualities}}\label{stringstring_dualities}

See at \emph{[[duality between type I and heterotic string theory]]}

\hypertarget{horavawitten_theory}{}\paragraph*{{Horava-Witten theory}}\label{horavawitten_theory}

One considers the [[KK-compactification]] of [[M-theory]] on a [[cyclic group of order 2|Z/2]]-[[orbifold]] of a [[torus]], hence of the [[Cartesian product]] of two [[circles]]

\begin{displaymath}
\itexarray{
    & S^1_A &\times& S^1_B
    \\
    \text{radius}: &  R_{11} && R_{10}
  }
\end{displaymath}
such that the reduction on the first factor $S^1_A$ corresponds to the [[duality between M-theory and type IIA string theory]], hence so that subsequent [[T-duality]] along the second factor yields [[type IIB string theory]] (in its [[F-theory]]-incarnation). Now the diffeomorphism which exchanges the two circle factors and hence should be a symmetry of M-theory is interpreted as [[S-duality]] in [[type II string theory]]:

\begin{displaymath}
IIB \overset{S}{\leftrightarrow} IIB
\end{displaymath}
\begin{quote}%
graphics taken from \hyperlink{HoravaWitten95}{Horava-Witten 95, p. 15}


\end{quote}
If one considers this situation additionally with a $\mathbb{Z}/2\mathbb{Z}$-[[orbifold]] quotient of the first circle factor, one obtains the [[duality between M-theory and heterotic string theory]] ([[Horava-Witten theory]]). If instead one performs it on the second circle factor, one obtains [[type I string theory]].

Here in both cases the [[involution]] [[action]] is by [[reflection]] of the circle at a line through its center. Hence if we identify $S^1 \simeq \mathbb{R} / \mathbb{Z}$ then the action is by multiplication by /1 on the [[real line]].

In summary:

M-theory on

\begin{itemize}%
\item $(S^1_A \sslash \mathbb{Z}_2 ) \times S^1_B$ yields [[heterotic string theory]]


\item $S^1_A \times \left( S^1_B \sslash \mathbb{Z}_2 \right)$ yields [[type I string theory]]



\end{itemize}
Hence the [[S-duality]] that swaps the two circle factors corresponds to \emph{[[duality between type I and heterotic string theory]]}.

\begin{displaymath}
\itexarray{
     HE 
       &\overset{KK/\mathbb{Z}^A_2}{\leftrightarrow}& 
     M 
       &\overset{KK/\mathbb{Z}^B_2}{\leftrightarrow}& 
     I'
     \\
     \mathllap{T}\updownarrow
     &&  && 
     \updownarrow \mathrlap{T}  
     \\
     HO 
     && \underset{\phantom{A}S\phantom{A}}{\leftrightarrow} && 
     I
  }
\end{displaymath}
\begin{quote}%
graphics taken from \hyperlink{HoravaWitten95}{Horava-Witten 95, p. 16}


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

[[!include string theory and cohomology theory -- table]]

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

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

\begin{itemize}%
\item [[Luis Ibáñez]], [[Angel Uranga]], section 4.4.3 of: \emph{[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]}, Cambridge University Press 2012


\item [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], Section 9.4 and 10.6 of: \emph{Basic Concepts of String Theory} Part of the series Theoretical and Mathematical Physics, Springer 2013



\end{itemize}
Relation to [[M-theory]] (via [[Horava-Witten theory]]):

\begin{itemize}%
\item [[Petr Hořava]], [[Edward Witten]], \emph{Heterotic and Type I string dynamics from eleven dimensions}, Nucl. Phys. B460 (1996) 506 (\href{http://arxiv.org/abs/hep-th/9510209}{arXiv:hep-th/9510209})


\item [[Petr Hořava]], [[Edward Witten]], \emph{Eleven dimensional supergravity on a manifold with boundary}, Nucl. Phys. B475 (1996) 94 (\href{http://arxiv.org/abs/hep-th/9603142}{arXiv:hep-th/9603142})



\end{itemize}
A comprehensive discussion of the ([[differential cohomology|differential]]) [[cohomology|cohomological]] nature of general type II/type I [[orientifold]] backgrounds is in

\begin{itemize}%
\item [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], \emph{Orientifold Pr\'e{}cis} in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]} Proceedings of Symposia in Pure Mathematics, AMS (2011) (\href{http://arxiv.org/abs/0906.0795}{arXiv:0906.0795}, \href{http://www.ma.utexas.edu/users/dafr/bilbao.pdf}{slides})

\end{itemize}
with details in

\begin{itemize}%
\item [[Daniel Freed]], \emph{Lectures on twisted K-theory and orientifolds} ([[FreedESI2012.pdf:file]])


\item [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], \emph{Spin structures and superstrings}, Surveys in Differential Geometry, Volume 15 (2010) (\href{http://arxiv.org/abs/1007.4581}{arXiv:1007.4581}, \href{http://dx.doi.org/10.4310/SDG.2010.v15.n1.a4}{doi:10.4310/SDG.2010.v15.n1.a4})



\end{itemize}
Related lecture notes / slides include

\begin{itemize}%
\item [[Jacques Distler]], \emph{Orientifolds and Twisted KR-Theory} (2008) (\href{http://www.perimeterinstitute.ca/pdf/files/731c5f3a-928f-453a-b569-db5c574d2a6c.pdf}{pdf})


\item [[Daniel Freed]], \emph{Dirac charge quantiation, K-theory, and orientifolds}, talk at a workshop \emph{Mathematical methods in general relativity and quantum field theories}, November, 2009 (\href{http://www.ma.utexas.edu/users/dafr/paris_nt.pdf}{pdf})


\item [[Greg Moore]], \emph{The RR-charge of an orientifold}, Oberwolfach talk 2010 (\href{https://www.physics.rutgers.edu/~gmoore/Oberwolfach_June2010_FINAL.pdf}{pdf}, [[MooreOrientifold2010.pdf:file]], \href{http://www.physics.rutgers.edu/~gmoore/AnnArbor_Feb2010_FINAL.ppt}{ppt})



\end{itemize}
\hypertarget{type_i}{}\subsubsection*{{Type I'}}\label{type_i}

Original articles on [[type I' string theory]]:

\begin{itemize}%
\item [[John Schwarz]], \emph{Some Properties of Type I' String Theory}, in: [[Mikhail Shifman]] (ed.), \emph{[[The Many Faces of the Superworld]]}, pp. 388-397 (2000) (\href{https://arxiv.org/abs/hep-th/9907061}{arXiv:hep-th/9907061}, \href{https://doi.org/10.1142/9789812793850_0023}{doi:10.1142/9789812793850\_0023})


\item Justin R. David, Avinash Dhar, Gautam Mandal, \emph{Probing Type I' String Theory Using D0 and D4-Branes}, Phys. Lett. B415 (1997) 135-143 (\href{https://arxiv.org/abs/hep-th/9707132}{arXiv:hep-th/9707132})



\end{itemize}
Type I' on [[toroidal orbifold|toroidal]] [[orientifolds]] with [[ADE-singularities]] ([[duality in string theory|dual]] to [[heterotic M-theory on ADE-orbifolds]]):

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

\end{itemize}
\hypertarget{Phenomenology}{}\subsubsection*{{Phenomenology}}\label{Phenomenology}

Type I [[string phenomenology]] and discussion of [[GUT]]s based on [[SO(32)]] type I strings (see also at \href{https://ncatlab.org/nlab/show/heterotic+string+theory#ReferencesPhenomenology}{heterotic phenomenology}):

\begin{itemize}%
\item H.S. Mani, A. Mukherjee, R. Ramachandran, A.P. Balachandran, \emph{Embedding of $SU(5)$ GUT in $SO(32)$ superstring theories}, Nuclear Physics B Volume 263, Issues 3–4, 27 January 1986, Pages 621-628 ()


\item [[Luis Ibáñez]], C. Muñoz, S. Rigolin, \emph{Aspects of Type I String Phenomenology}, Nucl.Phys. B553 (1999) 43-80 (\href{https://arxiv.org/abs/hep-ph/9812397}{arXiv:hep-ph/9812397})


\item Emilian Dudas, \emph{Theory and Phenomenology of Type I strings and M-theory}, Class. Quant. Grav.17:R41-R116, 2000 (\href{https://arxiv.org/abs/hep-ph/0006190}{arXiv:hep-ph/0006190})


\item Naoki Yamatsu, \emph{String-Inspired Special Grand Unification}, Progress of Theoretical and Experimental Physics, Volume 2017, Issue 10, 1 (\href{https://arxiv.org/abs/1708.02078}{arXiv:1708.02078}, \href{https://doi.org/10.1093/ptep/ptx135}{doi:10.1093/ptep/ptx135})



\end{itemize}
\hypertarget{duality}{}\subsubsection*{{Duality}}\label{duality}

Discussion of [[duality in string theory|duality]] with [[heterotic string theory]] includes the following.

The original conjecture is due to

\begin{itemize}%
\item [[Edward Witten]], section 5 of \emph{[[String Theory Dynamics In Various Dimensions]]}, Nucl.Phys.B443:85-126 (1995) (\href{http://arxiv.org/abs/hep-th/9503124}{arXiv:hep-th/9503124})

\end{itemize}
More details are then in

\begin{itemize}%
\item [[Joseph Polchinski]], [[Edward Witten]], \emph{Evidence for Heterotic - Type I String Duality}, Nucl.Phys.B460:525-540,1996 (\href{http://arxiv.org/abs/hep-th/9510169}{arXiv:hep-th/9510169})

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

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]], \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})



\end{itemize}
[[!redirects type I superstring]] [[!redirects type I superstring theory]]

[[!redirects type I' string theory]]



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