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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{heterotic string theory} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory} [[!include string theory - contents]] \hypertarget{supergeometry}{}\paragraph*{{Super-Geometry}}\label{supergeometry} [[!include supergeometry - contents]] \hypertarget{physics}{}\paragraph*{{Physics}}\label{physics} [[!include physicscontents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{compactifications}{Compactifications}\dotfill \pageref*{compactifications} \linebreak \noindent\hyperlink{enhanced_supersymmetry}{Enhanced supersymmetry}\dotfill \pageref*{enhanced_supersymmetry} \linebreak \noindent\hyperlink{Dualities}{Dualities}\dotfill \pageref*{Dualities} \linebreak \noindent\hyperlink{duality_with_type_i_string_theory}{Duality with type I string theory}\dotfill \pageref*{duality_with_type_i_string_theory} \linebreak \noindent\hyperlink{duality_with_type_ii_string_theory}{Duality with type II string theory}\dotfill \pageref*{duality_with_type_ii_string_theory} \linebreak \noindent\hyperlink{duality_with_mtheory}{Duality with M-theory}\dotfill \pageref*{duality_with_mtheory} \linebreak \noindent\hyperlink{duality_with_ftheory}{Duality with F-theory}\dotfill \pageref*{duality_with_ftheory} \linebreak \noindent\hyperlink{partition_function_and_witten_genus}{Partition function and Witten genus}\dotfill \pageref*{partition_function_and_witten_genus} \linebreak \noindent\hyperlink{GeneralGaugeBackgroundsAndParameterizedWZWModels}{General gauge backgrounds and parameterized WZW models}\dotfill \pageref*{GeneralGaugeBackgroundsAndParameterizedWZWModels} \linebreak \noindent\hyperlink{superspace_formulation}{Superspace formulation}\dotfill \pageref*{superspace_formulation} \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{orbifold_and_orientifold_compactifications}{Orbifold and orientifold compactifications}\dotfill \pageref*{orbifold_and_orientifold_compactifications} \linebreak \noindent\hyperlink{ReferencesPhenomenology}{Phenomenology}\dotfill \pageref*{ReferencesPhenomenology} \linebreak \noindent\hyperlink{superspace_formulation_of_heterotic_supergravity}{Superspace formulation of Heterotic supergravity}\dotfill \pageref*{superspace_formulation_of_heterotic_supergravity} \linebreak \noindent\hyperlink{in_elliptic_cohomology}{In elliptic cohomology}\dotfill \pageref*{in_elliptic_cohomology} \linebreak \noindent\hyperlink{general_flux_backgrounds_and_parameterized_wzw_models}{General flux backgrounds and parameterized WZW models}\dotfill \pageref*{general_flux_backgrounds_and_parameterized_wzw_models} \linebreak \noindent\hyperlink{on_elliptic_fibrations}{On elliptic fibrations}\dotfill \pageref*{on_elliptic_fibrations} \linebreak \noindent\hyperlink{dualities_2}{Dualities}\dotfill \pageref*{dualities_2} \linebreak \noindent\hyperlink{with_type_i_superstring_theory}{With type I superstring theory}\dotfill \pageref*{with_type_i_superstring_theory} \linebreak \noindent\hyperlink{DualityWithFTheory}{With $F$-theory}\dotfill \pageref*{DualityWithFTheory} \linebreak \noindent\hyperlink{open_heterotic_string}{``Open'' heterotic string}\dotfill \pageref*{open_heterotic_string} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} In [[string theory]] a [[spacetime]] [[landscape of string theory vacua|vacuum]] is encoded by a [[sigma-model]] 2-dimensional [[SCFT]]. In \emph{heterotic string theory} that SCFT is assumed to be the sum of a supersymmetric chiral piece and a non-supersymmetric piece (therefore ``heterotic''). \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{compactifications}{}\subsubsection*{{Compactifications}}\label{compactifications} An effective [[target space]] [[quantum field theory]] induced from a given heterotic 2d [[CFT]] [[sigma model]] that has a [[spacetime]] of the form $M^4 \times Y^6$ for $M^4$ the 4-dimensional [[Minkowski space]] that is experimentally observed locally (say on the scale of a particle accelerator) has $N= 1$ global [[supersymmetry]] precisely if the remaining 6-dimensional [[Riemannian manifold]] $Y^6$ is a [[Calabi-Yau manifold]]. See the \hyperlink{ReferencesNEqOne}{references below}. Since global $N=1$ supersymmetry for a long time has been considered a promising phenomenological model in high energy physics, this fact has induced a lot of interest in [[heterotic string theory on CY3-manifolds]]. \hypertarget{enhanced_supersymmetry}{}\subsubsection*{{Enhanced supersymmetry}}\label{enhanced_supersymmetry} A priori the [[worldsheet]] [[2d SCFT]] describing the quantum heterotic string has $N=(1,0)$ [[supersymmetry]]. Precisely if the corresponding [[target space]] [[effective field theory]] has $N=1$ supersymmetry does the worldsheet theory enhance to $N=(2,0)$ supersymmetry. See at \emph{[[2d (2,0)-superconformal QFT]]} and at \emph{[[Calabi-Yau manifolds and supersymmetry]]} for more on this. \hypertarget{Dualities}{}\subsubsection*{{Dualities}}\label{Dualities} Some [[duality in string theory]] involving the heterotic string: \hypertarget{duality_with_type_i_string_theory}{}\paragraph*{{Duality with type I string theory}}\label{duality_with_type_i_string_theory} \begin{itemize}% \item [[duality between heterotic and type I string theory]] \end{itemize} \hypertarget{duality_with_type_ii_string_theory}{}\paragraph*{{Duality with type II string theory}}\label{duality_with_type_ii_string_theory} See \emph{[[duality between heterotic and type II string theory]]}. \hypertarget{duality_with_mtheory}{}\paragraph*{{Duality with M-theory}}\label{duality_with_mtheory} See \emph{[[duality between heterotic string theory and M-theory]]} \hypertarget{duality_with_ftheory}{}\paragraph*{{Duality with F-theory}}\label{duality_with_ftheory} See \emph{[[duality between heterotic string theory and F-theory]]} and see \hyperlink{DualityWithFTheory}{references below}. \hypertarget{partition_function_and_witten_genus}{}\subsubsection*{{Partition function and Witten genus}}\label{partition_function_and_witten_genus} [[!include genera and partition functions - table]] \hypertarget{GeneralGaugeBackgroundsAndParameterizedWZWModels}{}\subsubsection*{{General gauge backgrounds and parameterized WZW models}}\label{GeneralGaugeBackgroundsAndParameterizedWZWModels} The traditional construction of the [[worldsheet]] theory of the heterotic string produces via the [[current algebra]] of the left-moving worldsheet [[fermions]] only those [[E8]]-[[background gauge fields]] which are reducible to $Spin(16)/\mathbb{Z}_2$-[[principal connections]] (\hyperlink{DistlerSharpe10}{Distler-Sharpe 10, sections 2-4}). But it is known that, for instance, the [[duality between F-theory and heterotic string theory]] produces more general gauge backgrounds (\hyperlink{DistlerSharpe10}{Distler-Sharpe 10, section 5}). In (\hyperlink{DistlerSharpe10}{Distler-Sharpe 10, section 7}), following (\hyperlink{GatesSiegel88}{Gates-Siegel 88}), it is argued that the way to fix this is to consider [[parameterized WZW models]], parameterized over the [[E8]]-[[principal bundle]] over [[spacetime]]. This does allow the incorporation of all $E_8$-background gauge fields, and the [[Green-Schwarz anomaly]] (and its cancellation) of the heterotic string now comes out as being equivalently the [[obstruction]] (and its lifting) for such a parameterized WZW term to exist. Moreover, where the traditional construction only produces level-1 [[current algebras]], this construction accommodates all levels, and it is argued (\hyperlink{DistlerSharpe10}{Distler-Sharpe 10, section 8.5}) that the [[elliptic genus]] of the resulting [[parameterized WZW models]] are the [[equivariant elliptic genera]] found by Liu and Ando (\hyperlink{Ando07}{Ando 07}). However, presently questions remain concerning formulating a [[sigma-model]] for strings propagating on the total space of the bundle, as it is only the chiral part of the geometric WZW model that appears in the heterotic string. (\ldots{}) \hypertarget{superspace_formulation}{}\subsubsection*{{Superspace formulation}}\label{superspace_formulation} The [[gauge field]] [[field strength|strength]]: $F_{\alpha \beta} = 0$ (\hyperlink{Witten86}{Witten 86}, \hyperlink{BonoraBregolaLechnerPastiTonin87}{Bonora-Bregola-Lechner-Pasti-Tonin 87, above (2.7)}, \hyperlink{BonoraBregolaLechnerPastiTonin87}{Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.13)}). $F_{a \alpha} = \Gamma_{a \alpha \beta} \chi^\beta$ (\hyperlink{Witten86}{Witten 86 (8)}, \hyperlink{AtickDharRatra86}{Atick-Dhar-Ratra 86, (4.14)}, \hyperlink{BonoraPastiTonin87}{Bonora-Pasti-Tonin 87, below (11)}, \hyperlink{BonoraBregolaLechnerPastiTonin87}{Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.27)}). Here $\chi^\alpha$ is the [[gaugino]]. $F_{a b} = \tfrac{1}{4} (\Gamma_{a b})_\alpha{}^\beta D_\beta \chi^\alpha$ (\hyperlink{BonoraPastiTonin87}{Bonora-Pasti-Tonin 87, below (11)}, \hyperlink{BonoraBregolaLechnerPastiTonin87}{Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.28)}) [[equations of motion]]: $(D^a \Gamma_a)_{\alpha\beta} \chi^\beta =0$ (\hyperlink{BonoraBregolaLechnerPastiTonin87}{Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.30)}) $D^b F_{b a} + T_a{}^{b c} F_{b c} = - (\Gamma_a)_{\alpha \beta} \chi^\alpha \chi^\beta - \chi^\alpha L_{\alpha a}$ (\hyperlink{BonoraBregolaLechnerPastiTonin87}{Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.31)}) (where $L_{\alpha a}$ is defined by (2.20) there\ldots{}) $\,$ The [[B-field]] [[field strength|strength]]: $H_{\alpha \beta \gamma} = 0$ (\hyperlink{AtickDharRatra86}{Atick-Dhar-Ratra 86, (4.2)}, \hyperlink{BonoraPastiTonin87}{Bonora-Pasti-Tonin 87, (15)}, \hyperlink{BonoraBregolaLechnerPastiTonin87}{Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.14)}) $H_{a \alpha \beta} = \phi \Gamma_{a \alpha \beta}$ (\hyperlink{AtickDharRatra86}{Atick-Dhar-Ratra 86, (4.19)}, \hyperlink{BonoraPastiTonin87}{Bonora-Pasti-Tonin 87, (15)}, \hyperlink{BonoraBregolaLechnerPastiTonin87}{Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.15)}) $\rho \coloneqq D_\alpha \phi$ (\hyperlink{AtickDharRatra86}{Atick-Dhar-Ratra 86, (4.20)}) $H_{a b \alpha} = -\tfrac{1}{2} \Gamma_{a b }_\alpha{}^\beta \rho_\beta$ (\hyperlink{AtickDharRatra86}{Atick-Dhar-Ratra 86, (4.21)}) $H_{a b c} = - \tfrac{3}{2} \phi T_{a b c} + \tfrac{c_1}{4} (\Gamma_{a b c})_{\alpha \beta} tr(\chi^\alpha \chi^\beta)$ (\hyperlink{AtickDharRatra86}{Atick-Dhar-Ratra 86, (4.22)}) According to (\hyperlink{BonoraBregolaLechnerPastiTonin90}{Bonora-Bregola-Lechner-Pasti-Tonin 90}) in fact all these constraints follow from just $T^a_{\alpha \beta} \propto \Gamma^a_{\alpha \beta}$, up to field redefinition. See also at \emph{[[torsion constraints in supergravity]]}. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[SemiSpin(16)]], [[SemiSpin(32)]] \item [[E-string]] \item [[string theory]] \begin{itemize}% \item \textbf{heterotic string theory} \begin{itemize}% \item [[2d SCFT]], \begin{itemize}% \item [[2-spectral triple]], [[Dirac-Ramond operator]] \end{itemize} \item [[2d (2,0)-superconformal QFT]] \item [[Green-Schwarz mechanism]] \item [[dual heterotic string theory]] \item [[heterotic string theory on CY3-manifolds]] \item [[Witten genus]], [[(2,1)-dimensional Euclidean field theories and tmf]] \end{itemize} \item [[type II string theory]] \item [[type 0 string theory]] \item [[landscape of string theory vacua]] \item [[supersymmetry and Calabi-Yau manifolds]] \end{itemize} \item [[string field theory]] \item [[11-dimensional supergravity]] \begin{itemize}% \item [[Hořava-Witten theory]] \item [[M-theory]] \end{itemize} \item [[AdS-CFT correspondence]] \item \href{string+theory+FAQ#DoesSTPredictSupersymmetry}{string theory FAQ -- Does string theory predict supersymmetry?} \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} Heterotic strings were introduced in \begin{itemize}% \item [[David Gross]], J. A. Harvey, E. Martinec and R. Rohm, \emph{Heterotic string theory (I). The free heterotic string} Nucl. Phys. B 256 (1985), 253. \emph{Heterotic string theory (I). The interacting heterotic string} , Nucl. Phys. B 267 (1986), 75. \item [[Philip Candelas]], [[Gary Horowitz]], [[Andrew Strominger]], [[Edward Witten]], \emph{Vacuum configurations for superstrings}, Nuclear Physics B Volume 258, 1985, Pages 46-74 Nucl. Phys. B 258, 46 (1985) () \item [[Bert Schellekens]], \emph{Classification of Ten-Dimensional Heterotic Strings}, Phys.Lett. B277 (1992) 277-284 (\href{http://arxiv.org/abs/hep-th/9112006}{arXiv:hep-th/9112006}) \end{itemize} Textbook accounts: \begin{itemize}% \item [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], vol 3 (which is part 6) of \emph{[[Supergravity and Superstrings - A Geometric Perspective]]}, World Scientific (1991) \item [[Joseph Polchinski]], volume II, section 11 of \emph{[[String theory]]}, \item [[Eric D'Hoker]], \emph{String theory -- lecture 8: Heterotic strings} in part 3 (p. 941 of volume II) of [[Pierre Deligne]], P. 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} Supersymmetric [[perturbative string theory vacua|vacua]]: \begin{itemize}% \item Andrea Fontanella, Tomas Ortin, \emph{On the supersymmetric solutions of the Heterotic Superstring effective action} (\href{https://arxiv.org/abs/1910.08496}{arxiv:1910.08496}) \end{itemize} Relation to [[Donaldson-Thomas theory]] and [[quiver gauge theory]]: \begin{itemize}% \item [[Yang-Hui He]], Seung-Joo Lee, \emph{Quiver Structure of Heterotic Moduli}, J. High Energ. Phys. (2012) 2012: 119 (\href{https://arxiv.org/abs/1208.3004}{arXiv:1208.3004}) \end{itemize} \hypertarget{orbifold_and_orientifold_compactifications}{}\subsubsection*{{Orbifold and orientifold compactifications}}\label{orbifold_and_orientifold_compactifications} Heterotic strings on [[orbifolds]]: \begin{itemize}% \item [[Lance Dixon]], [[Jeff Harvey]], [[Cumrun Vafa]], [[Edward Witten]], \emph{Strings on orbifolds}, Nuclear Physics B Volume 261, 1985, Pages 678-686 () \item [[Lance Dixon]], [[Jeff Harvey]], [[Cumrun Vafa]], [[Edward Witten]], \emph{Strings on orbifolds (II)}, Nuclear Physics B Volume 274, Issue 2, 15 September 1986, Pages 285-314 () \item Joel Giedt, \emph{Heterotic Orbifolds} (\href{https://arxiv.org/abs/hep-ph/0204315}{arXiv:hep-ph/0204315}) \item Kang-Sin Choi, \emph{Spectra of Heterotic Strings on Orbifolds}, Nucl. Phys. B708: 194-214, 2005 (\href{https://arxiv.org/abs/hep-th/0405195}{arXiv:hep-th/0405195}) \end{itemize} Specifically on [[ADE-singularities]]: \begin{itemize}% \item [[Paul Aspinwall]], [[David Morrison]], \emph{Point-like Instantons on K3 Orbifolds}, Nucl. Phys. B503 (1997) 533-564 (\href{https://arxiv.org/abs/hep-th/9705104}{arXiv:hep-th/9705104}) \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{ReferencesPhenomenology}{}\subsubsection*{{Phenomenology}}\label{ReferencesPhenomenology} The historical origin of all [[string phenomenology]] is the [[top-down model building|top-down approach]] in the [[heterotic string]] due to (\hyperlink{CHSW85}{Candelas-Horowitz-Strominger-Witten 85}). A brief review of motivations for [[GUT]] models in [[heterotic string theory]] is in \begin{itemize}% \item [[Edward Witten]], \emph{Quest For Unification}, Heinrich Hertz lecture at \href{http://www.desy.de/susy02/}{SUSY 2002} at DESY, Hamburg (\href{http://arxiv.org/abs/hep-ph/0207124}{arXiv:hep-ph/0207124}) \end{itemize} Specifically phenomenology for the [[SemiSpin(32)]]-heterotic string (see also at \href{type+I+string+theory#Phenomenology}{type I phenomenology}): \begin{itemize}% \item Kang-Sin Choi, Stefan Groot Nibbelink, Michele Trapletti, \emph{Heterotic $SO(32)$ model building in four dimensions}, JHEP 0412:063, 2004 (\href{https://arxiv.org/abs/hep-th/0410232}{arXiv:hep-th/0410232}) \item [[Hans-Peter Nilles]], Saul Ramos-Sanchez, Patrick K.S. Vaudrevange, Akin Wingerter, \emph{Exploring the $SO(32)$ Heterotic String}, JHEP 0604:050, 2006 (\href{https://arxiv.org/abs/hep-th/0603086}{arXiv:hep-th/0603086}) \item Saul Ramos-Sanchez, \emph{Towards Low Energy Physics from the Heterotic String}, Fortsch. Phys. 10:907-1036, 2009 (\href{https://arxiv.org/abs/0812.3560}{arXiv:0812.3560}) ([[heterotic string|heterotic]] [[SemiSpin(32)]] [[R-parity]] [[MSSM]] [[string theory vacua|vacua]]) \end{itemize} The following articles establish the existences of exact realization of the [[gauge group]] and [[matter]]-content of the [[MSSM]] in [[heterotic string theory]] (not yet checking [[Yukawa couplings]]): \begin{itemize}% \item [[Volker Braun]], [[Yang-Hui He]], [[Burt Ovrut]], [[Tony Pantev]], \emph{A Heterotic Standard Model}, Phys. Lett. B618 : 252-258 2005 (\href{http://arxiv.org/abs/hep-th/0501070}{arXiv:hep-th/0501070}) \item [[Volker Braun]], [[Yang-Hui He]], [[Burt Ovrut]], [[Tony Pantev]], \emph{The Exact MSSM Spectrum from String Theory}, JHEP 0605:043,2006 (\href{http://arxiv.org/abs/hep-th/0512177}{arXiv:hep-th/0512177}) \item [[Vincent Bouchard]], [[Ron Donagi]], \emph{An SU(5) Heterotic Standard Model}, Phys. Lett. B633:783-791,2006 (\href{http://arxiv.org/abs/hep-th/0512149}{arXiv:hep-th/0512149}) \end{itemize} A computer search through the ``[[landscape of string theory vacua|landscape]]'' of [[Calabi-Yau varieties]] showed severeal hundreds more such exact heterotic standard models (about one billionth of all CYs searched, and most of them arising as $SU(5)$-[[GUTs]]) \begin{itemize}% \item [[Lara Anderson]], [[Yang-Hui He]], [[Andre Lukas]], \emph{Heterotic Compactification, An Algorithmic Approach}, JHEP 0707:049, 2007 (\href{https://arxiv.org/abs/hep-th/0702210}{arXiv:hep-th/0702210}) \item [[Lara Anderson]], James Gray, [[Andre Lukas]], [[Eran Palti]], \emph{Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds} (\href{http://arxiv.org/abs/1106.4804}{arXiv:1106.4804}) \item [[Lara Anderson]], James Gray, [[Andre Lukas]], [[Eran Palti]], \emph{Heterotic Line Bundle Standard Models} JHEP06(2012)113 (\href{https://arxiv.org/abs/1202.1757}{arXiv:1202.1757}) \item [[Lara Anderson]], Andrei Constantin, James Gray, [[Andre Lukas]], [[Eran Palti]], \emph{A Comprehensive Scan for Heterotic SU(5) GUT models}, JHEP01(2014)047 (\href{https://arxiv.org/abs/1307.4787}{arXiv:1307.4787}) \item [[Yang-Hui He]], Seung-Joo Lee, [[Andre Lukas]], Chuang Sun, \emph{Heterotic Model Building: 16 Special Manifolds} (\href{http://arxiv.org/abs/1309.0223}{arXiv:1309.0223}) \item Andrei Constantin, [[Yang-Hui He]], [[Andre Lukas]], \emph{Counting String Theory Standard Models} (\href{https://arxiv.org/abs/1810.00444}{arXiv:1810.00444}) \item Alon E. Faraggi, Glyn Harries, Benjamin Percival, John Rizos, \emph{Towards machine learning in the classification of $\mathbb{Z}_2 \times \mathbb{Z}_2$ orbifold compactifications (\href{https://arxiv.org/abs/1901.04448}{arXiv:1901.04448})} \end{itemize} The resulting database of compactifications is here: \begin{itemize}% \item [[Lara Anderson]], James Gray, [[Andre Lukas]], [[Eran Palti]], \emph{Heterotic standard model database} (\href{http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/linebundlemodels/index.html.}{web}) \end{itemize} Review includes \begin{itemize}% \item [[Lara Anderson]], \emph{New aspects of heterotic geometry and phenomenology}, talk at \href{http://wwwth.mpp.mpg.de/conf/strings2012/}{Strings2012}, Munich 2012 (\href{http://wwwth.mpp.mpg.de/members/strings/strings2012/strings_files/program/Talks/Wednesday/Anderson.pdf}{pdf}) \item [[Yang-Hui He]], \emph{Deep-learning the landscape}, talk at \emph{\href{https://ims.nus.edu.sg/events/2018/wstring/wk.php}{String and M-Theory: The new geometry of the 21st century}} (\href{https://ims.nus.edu.sg/events/2018/wstring/files/yang.pdf}{pdf slides}, \href{https://www.youtube.com/watch?v=x3ThgBgkPlE}{video recording}) \end{itemize} Computation of [[metrics]] on these Calabi-Yau compactifications (eventually needed for computing their induced [[Yukawa couplings]]) is started in \begin{itemize}% \item [[Volker Braun]], Tamaz Brelidze, [[Michael Douglas]], [[Burt Ovrut]], \emph{Calabi-Yau Metrics for Quotients and Complete Intersections}, JHEP 0805:080, 2008 (\href{https://arxiv.org/abs/0712.3563}{arXiv:0712.3563}) \end{itemize} This ``heterotic standard model'' has a ``hidden sector'' copy of the actual standard model, more details of which are discussed here: \begin{itemize}% \item [[Volker Braun]], [[Yang-Hui He]], [[Burt Ovrut]], \emph{Supersymmetric Hidden Sectors for Heterotic Standard Models} (\href{http://arxiv.org/abs/1301.6767}{arXiv:1301.6767}) \end{itemize} The issue of [[moduli stabilization]] in these kinds of models is discussed in \begin{itemize}% \item Michele Cicoli, Senarath de Alwis, Alexander Westphal, \emph{Heterotic Moduli Stabilization} (\href{http://arxiv.org/abs/1304.1809}{arXiv:1304.1809}) \item [[Lara Anderson]], James Gray, [[Andre Lukas]], [[Burt Ovrut]], \emph{Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories} (\href{http://arxiv.org/abs/1304.2704}{arXiv:1304.2704}) \end{itemize} Principles singling out heterotic models with three generations of fundamental particles are discussed in: \begin{itemize}% \item [[Philip Candelas]], [[Xenia de la Ossa]], [[Yang-Hui He]], Balazs Szendroi, \emph{Triadophilia: A Special Corner in the Landscape}, Adv.Theor.Math.Phys.12:2,2008 (\href{http://arxiv.org/abs/0706.3134}{arXiv:0706.3134}) \end{itemize} See also \begin{itemize}% \item Hajime Otsuka, \emph{$SO(32)$ heterotic line bundle models}, (\href{https://arxiv.org/abs/1801.03684}{arXiv:1801.03684}) \item Carlo Angelantonj, Ioannis Florakis, \emph{GUT Scale Unification in Heterotic Strings} (\href{https://arxiv.org/abs/1812.06915}{arXiv:1812.06915}) \end{itemize} \hypertarget{superspace_formulation_of_heterotic_supergravity}{}\subsubsection*{{Superspace formulation of Heterotic supergravity}}\label{superspace_formulation_of_heterotic_supergravity} Discussion of heterotic supergravity in terms of [[superspace]] includes the following. One solution of the heterotic superspace [[Bianchi identities]] is due to \begin{itemize}% \item [[Joseph Atick]], Avinash Dhar, and Bharat Ratra, \emph{Superspace formulation of ten-dimensional N=1 supergravity coupled to N=1 super Yang-Mills theory}, Phys. Rev. D 33, 2824, 1986 (\href{https://doi.org/10.1103/PhysRevD.33.2824}{doi.org/10.1103/PhysRevD.33.2824}) \item [[Edward Witten]], \emph{Twistor-like transform in ten dimensions}, Nuclear Physics B Volume 266, Issue 2, 17 March 1986 \end{itemize} A second solution is due to [[Bengt Nilsson]], [[Renata Kallosh]] and others \begin{itemize}% \item [[Bengt Nilsson]], \emph{Simple 10-dimensional supergravity in superspace}, Nucl. Phys. B188 (1981) 176 () \end{itemize} These two solutions are supposed to be equivalent under field redefinition. See also at \emph{[[torsion constraints in supergravity]]}. Further references include these: \begin{itemize}% \item [[Loriano Bonora]], [[Paolo Pasti]], [[Mario Tonin]], \emph{Superspace formulation of 10D SUGRA+SYM theory a la Green-Schwarz}, Physics Letters B Volume 188, Issue 3, 16 April 1987, Pages 335--339 () \item [[Loriano Bonora]], M. Bregola, [[Kurt Lechner]], [[Paolo Pasti]], [[Mario Tonin]], \emph{Anomaly-free supergravity and super-Yang-Mills theories in ten dimensions}, Nuclear Physics B Volume 296, Issue 4, 25 January 1988 () \item [[Loriano Bonora]], M. Bregola; [[Kurt Lechner]], [[Paolo Pasti]], [[Mario Tonin]], \emph{A discussion of the constraints in $N=1$ SUGRA-SYM in 10-D}, International Journal of Modern Physics A, February 1990, Vol. 05, No. 03 : pp. 461-477 () \item \hyperlink{CastellaniDAuriaFre91}{Castellani-D'Auria-Fre 91, vol 3, part 6} \item L. Bonora, M. Bregola, R. D'Auria, P. Fr\'e{} [[Kurt Lechner]], [[Paolo Pasti]], I. Pesando, M. Raciti, F. Riva, [[Mario Tonin]] and D. Zanon, \emph{Some remarks on the supersymmetrization of the Lorentz Chern-Simons form in $D = 10$ $N= 1$ supergravity theories}, Physics Letters B 277 (1992) ([[BonoraSuperGS.pdf:file]]) \item [[Kurt Lechner]], [[Mario Tonin]], \emph{Superspace formulations of ten-dimensional supergravity}, JHEP 0806:021,2008 (\href{https://arxiv.org/abs/0802.3869}{arXiv:0802.3869}) \end{itemize} \hypertarget{in_elliptic_cohomology}{}\subsubsection*{{In elliptic cohomology}}\label{in_elliptic_cohomology} For more mathematically precise discussion in the context of [[elliptic cohomology]] and the [[Witten genus]] see also the references at \emph{\href{Witten+genus#TwistedWittenGenus}{Witten genus -- Heterotic (twisted) Witten genus, loop group representations and parameterized WZW models}}. \hypertarget{general_flux_backgrounds_and_parameterized_wzw_models}{}\subsubsection*{{General flux backgrounds and parameterized WZW models}}\label{general_flux_backgrounds_and_parameterized_wzw_models} Discussion of heterotic strings whoe [[current algebra]]-sector is parameterized by a [[principal bundle]] originates with \begin{itemize}% \item [[Jim Gates]], [[Warren Siegel]], \emph{Leftons, Rightons, Nonlinear $\sigma$-Models, and Superstrings}, Phys.Lett. B206 (1988) 631 (\href{https://inspirehep.net/record/251286/}{spire}) \item [[Jim Gates]], \emph{Strings, superstrings, and two-dimensional lagrangian field theory}, pp. 140-184 in Z. Haba, J. Sobczyk (eds.) \emph{Functional integration, geometry, and strings}, proceedings of the XXV Winter School of Theoretical Physics, Karpacz, Poland (Feb. 1989), , Birkh\"a{}user, 1989. \item [[Jim Gates]], S. Ketov, S. Kozenko, O. Solovev, \emph{Lagrangian chiral coset construction of heterotic string theories in $(1,0)$ superspace}, Nucl.Phys. B362 (1991) 199-231 (\href{http://inspirehep.net/record/314337/?ln=en}{spire}) \end{itemize} and is further expanded on in \begin{itemize}% \item [[Jacques Distler]], [[Eric Sharpe]], \emph{Heterotic compactifications with principal bundles for general groups and general levels}, Adv. Theor. Math. Phys. 14:335-398, 2010 (\href{http://arxiv.org/abs/hep-th/0701244}{arXiv:hep-th/0701244}) \end{itemize} reviewed in \begin{itemize}% \item [[Eric Sharpe]], \emph{Recent developments in heterotic compactifications}, in [[Eric Sharpe]], [[Arthur Greenspoon]], \emph{\href{http://www.ams.org/bookstore-getitem/item=AMSIP-44}{Advances in String Theory: The First Sowers Workshop in Theoretical Physics}}, AMS 2008 (\href{http://www.phys.vt.edu/sowers/talks/sharpe-sowers.pdf}{pdf slides (39-49)}) \end{itemize} The relation of this to [[equivariant elliptic cohomology]] is amplified in \begin{itemize}% \item [[Matthew Ando]], \emph{Equivariant elliptic cohomology and the Fibered WZW models of Distler and Sharpe}, \href{http://www.math.ucsb.edu/~drm/GTPseminar/2007-fall.php}{talk 2007} (\href{http://www.math.ucsb.edu/~drm/GTPseminar/notes/20071026-ando/20071026-malmendier.pdf}{lecture notes pdf}) \end{itemize} \hypertarget{on_elliptic_fibrations}{}\subsubsection*{{On elliptic fibrations}}\label{on_elliptic_fibrations} Compactified on an [[elliptic curve]] or, more generally, [[elliptic fibration]], heterotic string compactifictions are controled by a choice holomorphic [[stable bundle]] on the compact space. Dually this is an [[F-theory]] compactification on a [[K3]]-bundles. The basis of this story is discussed in \begin{itemize}% \item Robert Friedman, [[John Morgan]], [[Edward Witten]], \emph{Vector Bundles And F-Theory} (\href{http://arxiv.org/abs/hep-th/9701162}{arXiv:hep-th/9701162}) \end{itemize} A more formal discussion is in \begin{itemize}% \item B. Andreas and D. Hernandez Ruiperez, Adv. Theor. Math. Phys. Volume 7, Number 5 (2003), 751-786 \emph{Comments on N = 1 Heterotic String Vacua} (\href{http://projecteuclid.org/euclid.atmp/1111510429}{project Euclid}) \end{itemize} \hypertarget{dualities_2}{}\subsubsection*{{Dualities}}\label{dualities_2} \hypertarget{with_type_i_superstring_theory}{}\paragraph*{{With type I superstring theory}}\label{with_type_i_superstring_theory} 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{DualityWithFTheory}{}\paragraph*{{With $F$-theory}}\label{DualityWithFTheory} The [[duality between F-theory and heterotic string theory]] originates in \begin{itemize}% \item [[Ashoke Sen]], \emph{F-theory and Orientifolds} (\href{http://arxiv.org/abs/hep-th/9605150}{arXiv:hep-th/9605150}) \item Robert Friedman, [[John Morgan]], [[Edward Witten]], \emph{Vector Bundles And F Theory} (\href{http://arxiv.org/abs/hep-th/9701162}{arXiv:hep-th/9701162}) \end{itemize} Reviews include \begin{itemize}% \item [[Ron Donagi]], \emph{ICMP lecture on heterotic/F-theory duality} (\href{http://arxiv.org/abs/hep-th/9802093}{arXiv:hep-th/9802093}) \item Bj\"o{}rn Andreas, \emph{$N=1$ Heterotic/F-theory duality} PhD thesis (\href{http://edoc.hu-berlin.de/dissertationen/physik/andreas-bjoern/PDF/Andreas.pdf}{pdf}) \end{itemize} \hypertarget{open_heterotic_string}{}\subsubsection*{{``Open'' heterotic string}}\label{open_heterotic_string} A kind of unusual boundary condition for heterotic strings, (analogous to open [[M5-branes]] ending in [[Yang monopoles]] on [[M9-branes]]) is discussed in \begin{itemize}% \item [[Joseph Polchinski]], \emph{Open Heterotic Strings}, JHEP 0609 (2006) 082 (\href{http://arxiv.org/abs/hep-th/0510033}{arXiv:hep-th/0510033}) \end{itemize} [[!redirects heterotic superstring]] [[!redirects heterotic superstrings]] [[!redirects heterotic string]] [[!redirects heterotic strings]] [[!redirects heterotic supergravity]] \end{document}