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\begin{document}

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\section*{heterotic string theory on CY3-manifolds}

\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{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


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

The [[KK-compactification]] of [[heterotic string theory]] on [[Calabi-Yau manifolds]] of [[complex numbers|complex]] [[dimension]] 3, hence [[real number|real]] [[dimension]] 6. This choice of compactification means exactly that the resulting [[effective field theory]] on 4-dimension has $N =1$ [[supersymmetry]] (see at [[supersymmetry and Calabi-Yau manifolds]]).

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

[[!include N=1 susy compactifications -- table]]

\begin{itemize}%
\item [[supergravity]], [[supersymmetry]]


\item [[Kaluza-Klein mechanism]]


\item [[spontaneously broken symmetry]]


\item [[hierarchy problem]]


\item \href{string%20theory%20FAQ#WhatDoesItMeanToSayStringTheoryHasALandscapeOfSolutions}{string theory FAQ -- What does it mean to say that string theory has a ``landscape'' of solutions?}



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

The idea originates in

\begin{itemize}%
\item [[Philip Candelas]], [[Gary Horowitz]], [[Andrew Strominger]], and [[Edward Witten]], \emph{Vacuum Configurations for Superstrings} , Nucl. Phys. B 258 (1985), p. 46.

\end{itemize}
where in the introduction it says the following

\begin{quote}%
Recently, the discovery 6 of anomaly cancellation in a modified version of $d = 10$ supergravity and superstring theory with gauge group $O(32)$ or $E_8 \times E_8$ has opened the possibility that these theories might be phenomenologically realistic as well as mathematically consistent. A new string theory with $E_8 \times E_8$ gauge group has recently been constructed 7 along with a second $O(32)$ theory.

For these theories to be realistic, it is necessary that the vacuum state be of the form $M_4 \times K$, where $M_4$ is four-dimensional Minkowski space and K is some compact six-dimensional manifold. (Indeed, Kaluza-Klein theory -- with its now widely accepted interpretation that all dimensions are on the same logical footing -- was first proposed 8 in an effort to make sense out of higher-dimensional string theories). Quantum numbers of quarks and leptons are then determined by topological invariants of $K$ and of an $O(32)$ or $E_8 \times E_8$ gauge field defined on $K$ 9. Such considerations, however, are far from uniquely determining $K$.

In this paper, we will discuss some considerations, which, if valid, come very close to determining $K$ uniquely. We require

(i) The geometry to be of the form $H_4 \times K$, where $H_4$ is a maximally symmetric spacetime.

(ii) There should be an unbroken $N = 1$ supersymmetry in four dimensions. General arguments 10 and explicit demonstrations 11 have shown that supersymmetry may play an essential role in resolving the gauge hierarchy or Dirac large numbers problem. These arguments require that supersymmetry is unbroken at the Planck (or compactification) scale.

(iii) The gauge group and fermion spectrum should be realistic.

These requirements turn out to be extremely restrictive. In previous ten-dimensional supergravity theories, supersymmetric configurations have never given rise to chiral fermions -- let alone to a realistic spectrum. However, the modification introduced by Green and Schwarz to produce an anomaly-free field theory also makes it possible to satisfy these requirements. We will see that unbroken $N = 1$ supersymmetry requires that $K$ have, for perturbatively accessible configurations, $SU(3)$ holonomy and that the four-dimensional cosmological constant vanish. The existence of spaces with $SU(3)$ holonomy was conjectured by Calabi 12 and proved by Yau 13.


\end{quote}
(Of course later it was understood that Calabi-Yau spaces, even those of complex dimension 3, are not ``very close to unique''.)

Lecture notes include

\begin{itemize}%
\item [[Brian Greene]], \emph{String Theory on Calabi-Yau Manifolds}, lectures at TASI96 (\href{https://arxiv.org/abs/hep-th/9702155}{arXiv:hep-th/9702155})

\end{itemize}
Further original references include

\begin{itemize}%
\item [[Tom Banks]], [[Lance Dixon]], [[Dan Friedan]], [[Emil Martinec]], \emph{Phenomenology and Conformal Field Theory or Can String Theory Predict the Weak Mixing Angle?}, Nucl. Phys. B299 (1988) 613. (\href{http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-4377.pdf}{pdf})


\item [[Jacques Distler]], [[Brian Greene]], \emph{Aspects Of $(2,0)$ String Compactifications}, Nucl. Phys. B304 (1988)


\item [[Andrew Strominger]], \emph{Special Geometry}, Comm. Math. Phys. 133 (1990) 163.


\item [[Philip Candelas]] and X. De la Ossa, \emph{Moduli Space of Calabi-Yau Manifolds}, Nucl. Phys. B355 (1991) 455.


\item [[Edward Witten]], \emph{Phases of N=2 Theories in Two Dimensions}, Nucl. Phys. B403 (1993) 159 (\href{http://arxiv.org/abs/hep-th/9301042}{arXiv:hep-th/9301042})



\end{itemize}
and chapters 12 - 16 of

\begin{itemize}%
\item [[Michael Green]], [[John Schwarz]], [[Edward Witten]], \emph{Superstring Theory} , Vol. 2, Cambridge University Press, (1987)

\end{itemize}
A canonical textbook reference for the role of Calabi-Yau manifolds in compactifications of 10-dimensional [[supergravity]] is

\begin{itemize}%
\item [[Andrew Strominger]] (notes by [[John Morgan]]), \emph{Kaluza-Klein compactifications, Supersymmetry and Calabi-Yau spaces} , volume II, starting on page 1091 in

[[Pierre Deligne]], [[Pavel Etingof]], [[Dan Freed]], L. Jeffrey, [[David Kazhdan]], [[John Morgan]], D.R. Morrison and [[Edward Witten]], eds. , \emph{[[Quantum Fields and Strings]], A course for mathematicians}, 2 vols. Amer. Math. Soc. Providence 1999. (\href{http://www.math.ias.edu/qft}{web version})



\end{itemize}
Lecure notes in a more general context of [[string phenomenology]] include

\begin{itemize}%
\item [[Martin Wijnholt]], \emph{String compactification}, \href{https://pitp2014.ias.edu}{PITP 2014} lecture notes ([[WijnholtCompactification14.pdf:file]], \href{https://static.ias.edu/pitp/2014/sites/pitp2014.ias.edu/files/PITP2014_P1_wijnholt.pdf}{slides for lecture 1}, \href{https://static.ias.edu/pitp/2014/sites/pitp2014.ias.edu/files/PITP2014_P2_wijnholt.pdf}{slides for lecture 2}, \href{https://static.ias.edu/pitp/2014/sites/pitp2014.ias.edu/files/PITP2014_P3_wijnholt.pdf}{slides for lecture 3})

\end{itemize}
Discussion of [[generalized Calabi-Yau manifold]] backgrounds is for instance in

\begin{itemize}%
\item [[Mariana Graña]], [[Ruben Minasian]], Michela Petrini, [[Alessandro Tomasiello]], \emph{Generalized structures of $N=1$ vacua} (\href{http://arxiv.org/abs/hep-th/0505212}{arXiv:hep-th/0505212})

\end{itemize}
Discussion of [[duality in string theory|duality]] with [[M-theory on G2-manifolds]]:

\begin{itemize}%
\item [[Andreas Braun]], Sakura Schaefer-Nameki, \emph{Compact, Singular G2-Holonomy Manifolds and M/Heterotic/F-Theory Duality}, JHEP04(2018)126 (\href{https://arxiv.org/abs/1708.07215}{arXiv:1708.07215})

\end{itemize}
[[!redirects hetrotic string theory on Calabi-Yau manifolds]] [[!redirects hetrotic string theory on Calabi-Yau 3-folds]]



\end{document}