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\section*{Hořava-Witten theory}

\begin{quote}%
This entry is about the conjectured relation ([[duality in string theory|duality]]) between [[M-theory]] at [[MO9]]-planes and [[heterotic string theory]] on these. For the relation of [[M-theory]] [[KK-compactification|KK-compactified]] on a [[K3-surface]] and [[heterotic string theory]] on a [[3-torus]] see instead at \emph{[[duality between M/F-theory and heterotic string theory]]}.


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{duality_in_string_theory}{}\paragraph*{{Duality in string theory}}\label{duality_in_string_theory}

[[!include duality in string theory -- contents]]

\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{duality_between_mtheory_and_heterotic_string_theory}{Duality between M-theory and heterotic string theory}\dotfill \pageref*{duality_between_mtheory_and_heterotic_string_theory} \linebreak
\noindent\hyperlink{duality_between_mtheory_and_type_i_string_theory}{Duality between M-theory and type I string theory}\dotfill \pageref*{duality_between_mtheory_and_type_i_string_theory} \linebreak
\noindent\hyperlink{duality_between_heterotic_and_type_i_string_theory}{Duality between heterotic and type I string theory}\dotfill \pageref*{duality_between_heterotic_and_type_i_string_theory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{BoundaryConditions}{Boundary conditions}\dotfill \pageref*{BoundaryConditions} \linebreak
\noindent\hyperlink{related_concepts_2}{Related concepts}\dotfill \pageref*{related_concepts_2} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesGauginoCondensation}{Gaugino condensation and supersymmetry breaking}\dotfill \pageref*{ReferencesGauginoCondensation} \linebreak
\noindent\hyperlink{generalization_to_mtheory_on_}{Generalization to M-theory on $S^1/G_{HW} \times \mathbb{H}/G_{ADE}$}\dotfill \pageref*{generalization_to_mtheory_on_} \linebreak
\noindent\hyperlink{for_7d_supergravity}{For 7d supergravity}\dotfill \pageref*{for_7d_supergravity} \linebreak


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

There is an observation by \hyperlink{HoravaWitten95}{Hoava--Witten 95}, \hyperlink{HoravaWitten96}{Hoava--Witten 96} which suggests that [[M-theory]] on an [[cyclic group of order 2|Z/2]]-[[orbifold]] (actually a [[higher orientifold]]) of the form $X_{10} \times (S^1 \slash \mathbb{Z}_2))$ in ``[[duality in string theory|dual]]'' to [[heterotic string theory]] on its [[boundary]] ([[fixed point]]) ``[[M9-brane]]''. Therefore one also speaks of ``heterotic [[M-theory]]'' (\hyperlink{Ovrut02}{Ovrut 02}).

\begin{quote}%
from \hyperlink{Kashima00}{Kashima 00}


\end{quote}
In the above the circle factor is taken to be the [[circle]] [[fiber]] over the 10d [[type IIA string theory|type IIA]] [[spacetime]].

If instead one considers another of the spatial dimensions to be compactified on $S^1\sslash \mathbb{Z}_2$, then, after [[T-duality]] ([[F-theory]]) result is supposed to be [[type I string theory]]. In this case the intersection of the [[M2-brane]] with the [[M9-brane]] (the latter now wrapping the M-theory circle fiber) is called the \emph{[[E-string]]}.

More in detail:

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{duality_between_mtheory_and_heterotic_string_theory}{}\subsubsection*{{Duality between M-theory and heterotic string theory}}\label{duality_between_mtheory_and_heterotic_string_theory}

Here each of the two copies of the heterotic gauge theory is a ``[[hidden sector]]'' with respect to the other.

The orbifold equivariance condition of the [[supergravity C-field]] is that discussed at \emph{[[orientifold]]} (there for the [[B-field]]). Therefore it has to vanish at the two fixed fixed points of the $\mathbb{Z}_2$-action. Thereby the quantization condition

\begin{displaymath}
[2G_4] = 2 [c_2] - [\frac{1}{2} p_1]
\end{displaymath}
on the [[supergravity C-field]] becomes the condition for the [[Green-Schwarz mechanism]] of the [[heterotic string theory]] on the ``boundary'' (the orbifold fixed points).

\hypertarget{duality_between_mtheory_and_type_i_string_theory}{}\subsubsection*{{Duality between M-theory and type I string theory}}\label{duality_between_mtheory_and_type_i_string_theory}

[[duality between M-theory and type I string theory]]

originally suggested in (\hyperlink{HoravaWitten95}{Horava-Witten 95, section 3})

Further evidence is reviewed in \hyperlink{APT98}{APT 98}

\hypertarget{duality_between_heterotic_and_type_i_string_theory}{}\subsubsection*{{Duality between heterotic and type I string theory}}\label{duality_between_heterotic_and_type_i_string_theory}

[[duality between heterotic and type I string theory]]

via [[S-duality]] in the [[F-theory]]-picture\ldots{}

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

\begin{itemize}%
\item [[duality between F-theory and heterotic string theory]]

\end{itemize}
[[!include KK-compactifications of M-theory -- table]]

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

\hypertarget{BoundaryConditions}{}\subsubsection*{{Boundary conditions}}\label{BoundaryConditions}

The [[supergravity C-field]] $\hat G_4$ is supposed to vanish, and \emph{differentially} vanish at the boundary in the HW model, meaning that also the local connection 3-form $C_3$ vanishes there. The argument is roughly as follows (similar for as in \hyperlink{Falkowski}{Falkowski, section 3.1}).

The [[higher dimensional Chern-Simons theory|higher Chern-Simons term]]

\begin{displaymath}
C_3 \mapsto C_3 \wedge G_4 \wedge G_4
\end{displaymath}
in the [[Lagrangian]] of [[11-dimensional supergravity]] is supposed to be well-defined on fields on the [[orbifold]] and hence is to be $\mathbb{Z}_2$-invariant.

Let $\iota_{11}$ be the canonical vector field along the circle factor. Then the component of $G \wedge G$ which is annihilated by the contraction $\iota_{11}$ is necessarily even, so the component $d x^{11}\wedge \iota_11 C_3$ is also even. It follows that also $d x^{11}\wedge \iota_11 G_4$ is even.

Moreover, the kinetic term

\begin{displaymath}
C \mapsto G \wedge \star G
\end{displaymath}
is to be invariant. With the above this now implies that the components of $G$ annihiliated by $\iota_{11}$ is odd, because so is the mixed component of the [[metric]] tensor.

This finally implies that the restriction of $C_3$ to the orbifold fixed points has to be closed.

\hypertarget{related_concepts_2}{}\subsection*{{Related concepts}}\label{related_concepts_2}

\begin{itemize}%
\item [[duality in string theory]]

\begin{itemize}%
\item [[duality between F-theory and heterotic string theory]]

\end{itemize}

\item [[11-dimensional supergravity]]


\item [[M-theory]], [[heterotic string theory]]


\item [[M9-brane]]



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

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

The original articles are

\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}
Review is in

\begin{itemize}%
\item [[Piyush Kumar]], \emph{Hoava-Witten theory} (2004) (\href{http://theory.uchicago.edu/~sethi/Teaching/P484-W2004/hwitten.pdf}{pdf})


\item [[Paul Townsend]], \emph{Four Lectures on M-Theory} (\href{http://arxiv.org/abs/hep-th/9612121}{arXiv:hep-th/9612121}).


\item [[Burt Ovrut]], \emph{Lectures on Heterotic M-Theory} (\href{http://arxiv.org/abs/hep-th/0201032}{arXiv:hep-th/0201032})


\item [[Adam Falkowski]], section 3 of \emph{Five dimensional locally supersymmetric theories with branes}, Master Thesis, Warsaw ([[FalkowskiLecture.pdf:file]])



\end{itemize}
The [[black brane|black]] [[M2-brane]] solution in HW-theory, supposedly yielding the [[black brane|black]] [[heterotic string]] at the intersection with the [[M9-brane]] is discussed in

\begin{itemize}%
\item Zygmunt Lalak, Andr\'e{} Lukas, [[Burt Ovrut]], \emph{Soliton Solutions of M-theory on an Orbifold}, Phys. Lett. B425 (1998) 59-70 (\href{https://arxiv.org/abs/hep-th/9709214}{arXiv:hep-th/9709214})


\item Ken Kashima, \emph{The M2-brane Solution of Heterotic M-theory with the Gauss-Bonnet $R^2$ terms}, Prog.Theor.Phys. 105 (2001) 301-321 (\href{https://arxiv.org/abs/hep-th/0010286}{arXiv:hep-th/0010286})



\end{itemize}
Explicit discussion of [[worldvolume]] [[CFT]] of the [[M2-branes]] ending on the HW fixed points and becoming [[heterotic strings]] is discussed, via the [[BLG model]], in

\begin{itemize}%
\item [[Neil Lambert]], \emph{Heterotic M2-branes}, Physics Letters B Volume 749, 7 October 2015, Pages 363--367 (\href{http://arxiv.org/abs/1507.07931}{arXiv:1507.07931})

\end{itemize}
After [[KK-reduction]] to [[5d supergravity]] there is a corresponding 5d mechanism, see the references \href{5-dimensional+supergravity#ReferencesHWCompactification}{there}.

Disucussion of the [[duality between heterotic and type I string theory]] includes

\begin{itemize}%
\item I. Antoniadis, H. Partouche, T.R. Taylor, \emph{Lectures on Heterotic-Type I Duality}, Nucl.Phys.Proc.Suppl. 61A (1998) 58-71; Nucl.Phys.Proc.Suppl. 67 (1998) 3-1

\end{itemize}
Discussion of [[string phenomenology]] in Horava-Witten theory:

\begin{itemize}%
\item [[Burt Ovrut]], \emph{Vacuum Constraints for Realistic Heterotic M-Theories} (\href{https://arxiv.org/abs/1811.08892}{arXiv:1811.08892})

\end{itemize}
\hypertarget{ReferencesGauginoCondensation}{}\subsubsection*{{Gaugino condensation and supersymmetry breaking}}\label{ReferencesGauginoCondensation}

Discussion of [[gaugino condensation]] and [[supersymmetry breaking]] in Horava-Witten theory, where the two heterotic [[MO9-branes]] provide a natural mechanism for [[supersymmetry breaking]] on one (the physical brane) by gaugino condensation on the other (the ``dark sector'' brane):

lecture notes:

\begin{itemize}%
\item [[Hans-Peter Nilles]], \emph{Gaugino Condensation and SUSY Breakdown}, Lectures at Cargese School of Physics and Cosmology, Cargese, France, August 2003 (\href{https://arxiv.org/abs/hep-th/0402022}{arXiv:hep-th/0402022})

\end{itemize}
original articles:

\begin{itemize}%
\item I. Antoniadis, M. Quiros, \emph{On the M-theory description of gaugino condensation}, Phys.Lett. B416 (1998) 327-333 (\href{https://arxiv.org/abs/hep-th/9707208}{arXiv:hep-th/9707208})


\item I. Antoniadis, M. Quiros, \emph{Supersymmetry breaking in M-theory and gaugino condensation}, Nucl.Phys. B505 (1997) 109-122 (\href{https://arxiv.org/abs/hep-th/9705037}{arXiv:hep-th/9705037})


\item [[André Lukas]], [[Burt Ovrut]], [[Daniel Waldram]], \emph{Gaugino condensation in M theory on $S^1/Z_2$}, Phys. Rev. D 57, 7529 (1998) (\href{https://arxiv.org/abs/hep-th/9711197}{arXiv:hep-th/9711197}, \href{https://doi.org/10.1103/PhysRevD.57.7529}{doi:10.1103/PhysRevD.57.7529})


\item [[André Lukas]], [[Burt Ovrut]], [[Daniel Waldram]], \emph{Five-Branes and Supersymmetry Breaking in M-Theory}, JHEP 9904:009, 1999 (\href{https://arxiv.org/abs/hep-th/9901017}{arXiv:hep-th/9901017})



\end{itemize}
specifically for [[M-theory on S1/G\_HW times H/G\_ADE]]:

\begin{itemize}%
\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}, )

\end{itemize}
\hypertarget{generalization_to_mtheory_on_}{}\subsubsection*{{Generalization to M-theory on $S^1/G_{HW} \times \mathbb{H}/G_{ADE}$}}\label{generalization_to_mtheory_on_}

Generalization to [[M-theory on S1/G\_HW times H/G\_ADE]]:

\begin{itemize}%
\item Zygmunt Lalak, Steven Thomas, \emph{Gaugino Condensation, Moduli Potentials and Supersymmetry Breaking in M-Theory Models} (\href{https://arxiv.org/abs/hep-th/9707223}{hep-th/9707223})


\item V. Kaplunovsky, J. Sonnenschein,[[Stefan Theisen]], S. Yankielowicz, \emph{On the Duality between Perturbative Heterotic Orbifolds and M-Theory on $T^4/Z_N$} (\href{https://arxiv.org/abs/hep-th/9912144}{arXiv:hep-th/9912144})


\item E. Gorbatov, V.S. 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})


\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 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 [[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})



\end{itemize}
\hypertarget{for_7d_supergravity}{}\subsubsection*{{For 7d supergravity}}\label{for_7d_supergravity}

Discussion for [[7d supergravity]]:

\begin{itemize}%
\item Tony Gherghetta, [[Alex Kehagias]], \emph{Anomaly Cancellation in Seven-Dimensional Supergravity with a Boundary}, Phys.Rev. \textbf{D68} (2003), 065019, (\href{http://arxiv.org/abs/hep-th/0212060}{arXiv:hep-th/0212060})


\item Spyros D. Avramis, [[Alex Kehagias]], \emph{Gauged $D=7$ Supergravity on the $S^1/\mathbb{Z}_2$ Orbifold} (\href{https://arxiv.org/abs/hep-th/0407221}{arXiv:hep-th/0407221})


\item T.G. Pugh, [[Ergin Sezgin]], [[Kellogg Stelle]], \emph{$D=7$ / $D=6$ Heterotic Supergravity with Gauged R-Symmetry} (\href{https://arxiv.org/abs/1008.0726}{arXiv:1008.0726})



\end{itemize}
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