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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*{moduli stabilization} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{physics}{}\paragraph*{{Physics}}\label{physics} [[!include physicscontents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{FluxCompactfications}{Freund-Rubin flux compactifications}\dotfill \pageref*{FluxCompactfications} \linebreak \noindent\hyperlink{in_string_theory}{In string theory}\dotfill \pageref*{in_string_theory} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{in_pure_gravity}{In pure gravity}\dotfill \pageref*{in_pure_gravity} \linebreak \noindent\hyperlink{ReferencesFreundRubinCompactificationo}{Freund-Rubin flux compactifications}\dotfill \pageref*{ReferencesFreundRubinCompactificationo} \linebreak \noindent\hyperlink{in_string_theory_2}{In string theory}\dotfill \pageref*{in_string_theory_2} \linebreak \noindent\hyperlink{in_type_ii_string_theory}{In type II string theory}\dotfill \pageref*{in_type_ii_string_theory} \linebreak \noindent\hyperlink{ReferencesInMTheory}{In M-theory}\dotfill \pageref*{ReferencesInMTheory} \linebreak \noindent\hyperlink{in_heterotic_string_theory}{In heterotic string theory}\dotfill \pageref*{in_heterotic_string_theory} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} In [[physics]], \emph{moduli stabilization} refers to the problem of rendering [[Kaluza-Klein compactifications]] stable. A [[Kaluza-Klein compactification]] is a [[model (physics)|model]] of [[gravity]] where [[spacetime]] is assumed to be a higher dimensional [[fiber bundle]], with [[compact topological space|compact]] [[fibers]] of tiny extension, such that the resulting physics looks effectively lower-dimensional, but inheriting extra [[field (physics)|fields]]. Namely the size and shape of the compactified extra dimension is encoded in the [[Riemannian metric]], hence in the field of gravity, hence are themselves dynamical fields. Since these fields parameterize the [[moduli space]] of the KK-compactification, they are called \emph{[[moduli fields]]}. The problem of \emph{moduli stabilization} is the problem of identifying mechanisms or conditions that ensure that as these fields dynamically evolve, the compact spatial dimensions remain stably so, neither opening up nor collapsing. For [[phenomenology|phenomenologically]] realistic KK-compactifications the compact volume has to stably be a tiny but finite value (``volume stabilization''). Equivalently, since fast varying moduli appear as light or massless [[particles]] in the low-dimensional [[effective field theory]] which would show up in accelerator [[experiments]] (such as the [[LHC]]) but don't, the problem is to identify mechanisms or conditions that would render these moduli fields massive. \hypertarget{FluxCompactfications}{}\subsection*{{Freund-Rubin flux compactifications}}\label{FluxCompactfications} In pure [[classical field theory|classical]] [[gravity]] KK-compactifications are argued (\hyperlink{Penrose03}{Penrose 03, section 10.3}) to generically be unstable by the [[Penrose-Hawking singularity theorem]]. But if extra [[gauge fields]] or [[higher gauge field]] beyond pure gravity are admitted in the higher dimensions, then stable compactifications may exist if there is ``magnetic [[flux]]'' in the compact fiber spaces. These are called \emph{[[Freund-Rubin compactifications]]}, or \emph{[[flux compactifications]]}. A well-studied example is 6-dimensional [[Einstein-Maxwell theory]] with magnetic [[flux]] on a 2-dimensional [[fiber]] spaces over a 4-dimensional base space (\hyperlink{FreundRubin80}{Freund-Rubin 80}, \hyperlink{RDSS83}{RDSS 83}). (On the other hand, Freund-Rubin compactifications usually have fibers the site of the curvature radius of the base, and hence not ``small''.) Similarly, in [[string theory]] it is argued that the extra fields and further string theoretic effects may stabilize the compact dimensions, namely a combination of [[flux compactification]] and [[non-perturbative effect|non-perturbative]] [[brane]] effects (\hyperlink{Acharya02}{Acharya 02}, \hyperlink{KKLT03}{KKLT 03}). However, these arguments typically focus on fluctuations that preserve given [[special holonomy]] ([[supersymmetry and Calabi-Yau manifolds|Calabi-Yau 3-folds]] in type II or [[M-theory on G2-manifolds|G2-manifolds in M-theory]]). There is also a more generic argument for volume compactification by string winding modes (``[[Brandenberger-Vafa mechanism]]'' \hyperlink{BrandenbergerVafa89}{Brandenberger-Vafa 89}, \hyperlink{WatsonBrandenberger03}{Watson-Brandenberger 03}) and the claim (\hyperlink{KimNishimuraTsuchiya12}{Kim-Nishimura-Tsuchiya 12}) that in the non-perturbative [[IKKT model]] computer simulations show a spontaneous stable compactification to 3+1 dimensions. \hypertarget{in_string_theory}{}\subsection*{{In string theory}}\label{in_string_theory} The issue of stabilization of compact dimensions arises notably in [[string theory]] [[Kaluza-Klein compactifications]]. In the context of [[type II string theory]] one way to design the [[model (in theoretical physics)|model]] such that the moduli fields are massive is to consider the case where [[higher gauge field|higher]] [[background gauge fields]] [[vacuum expectation value|vacuum expectation values]] (VEVs) $F_p$ are present on the compactification space. Since these fields are characterized by their higher [[field strength]]/[[curvature]] forms which are referred to as ``[[flux]]'' terms in physics, these models are called \textbf{[[flux compactification]]} models (\hyperlink{KKLT03}{KKLT 03}). Because the standard [[kinetic action]] term \begin{displaymath} S_{kin} \propto \int F_p \wedge \star_g F_p \end{displaymath} couples the flux VEV to the metric $g$ (via the [[Hodge star operator]]) and hence to the moduli, it generically induces an effective [[potential energy]] for these, which may stabilize them (when including [[non-perturbative effects]]). Similarly in [[M-theory on G2-manifolds]] the 4-form flux of the [[supergravity C-field]] leads to potentials for the moduli, which is argued to generically stabilize them (\hyperlink{Acharya02}{Acharya 02}). Since for these flux compactifications only the [[periods]] of the form fields on the compact space matter, under a bunch of further assumptions on the nature of the compactification, one can reduce the number of possible such compactifications to a combinatorial problem. The resulting space of possibilities is also known as the \emph{[[landscape of string theory vacua]]}. The moduli stabilization in (\hyperlink{KKLT03}{KKLT 03}) was demonstrated in two steps. First, all moduli were stabilized at a fixed minimum with a negative [[cosmological constant]]. This was achieved by combining fluxes with [[non-perturbative effects]]. Second, the minimum was lifted to a metastable vacuum with a positive cosmological constant. This was accomplished by adding anti D-branes and using previous results, obtained in (\hyperlink{KachruPearsonVerlinde01}{Kachru-Pearson-Verlinde 01}), that the flux-anti D-brane system can form a metastable bound state with positive energy. In (\hyperlink{KKLT03}{KKLT 03}) it was also shown that one can fine tune various parameters to make the value of the [[cosmological constant]] consistent with the observed amount of [[dark energy]]. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[flux compactification]] \item [[G2-MSSM]] \item [[landscape of string theory vacua]] \item \href{string+theory+FAQ#StabilityOfKKCompactification}{String Theory FAQ -- Do the extra dimensions lead to instability of 4 dimensional spacetime?} \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{in_pure_gravity}{}\subsubsection*{{In pure gravity}}\label{in_pure_gravity} The problem of generic in-stability of moduli of pure gravity KK-compactifications is highlighted in \begin{itemize}% \item [[Roger Penrose]], section 10.3 in \emph{On the stability of extra space dimensions} in Gibbons, Shellard, Rankin (eds.) \emph{The Future of Theoretical Physics and Cosmology}, Cambridge (2003) (\href{https://inspirehep.net/record/608935/}{spire}) \end{itemize} \hypertarget{ReferencesFreundRubinCompactificationo}{}\subsubsection*{{Freund-Rubin flux compactifications}}\label{ReferencesFreundRubinCompactificationo} [[Freund-Rubin compactification|Freund-Rubin]] [[flux compactifications]] are due to A class of stable compactifications of 6d Einstein-Maxwell theory down to four dimensions is due to \begin{itemize}% \item [[Peter Freund]] and M. A. Rubin, \emph{Dynamics Of Dimensional Reduction}, Phys. Lett. B 97, 233 (1980) (, \href{http://inspirehep.net/record/154579}{spire:154579}) \end{itemize} and the special case of compactifications of 6d Einstein-Maxwell theory to 4d is in \begin{itemize}% \item S. Randjbar-Daemi, [[Abdus Salam]] and J. A. Strathdee, \emph{Spontaneous Compactification In Six-Dimensional Einstein-Maxwell Theory}, Nucl. Phys. B 214, 491 (1983) (, \href{https://inspirehep.net/record/182427/}{spire:182427}) \end{itemize} Further discussion of these models as toy models for [[flux compactifications]] in [[string theory]] is in \begin{itemize}% \item [[Michael Douglas]], [[Shamit Kachru]], section II.D.1 of \emph{Flux compactification}, Rev. Mod. Phys. 79, 733 (2007) (\href{https://arxiv.org/abs/hep-th/0610102}{arXiv:hep-th/0610102}) \item [[Frederik Denef]], [[Michael Douglas]], [[Shamit Kachru]], \emph{Physics of String Flux Compactifications}, Ann. Rev. Nucl. Part. Sci. 57:119-144, 2007, \href{https://arxiv.org/abs/hep-th/0701050}{arXiv:hep-th/0701050} \end{itemize} \hypertarget{in_string_theory_2}{}\subsubsection*{{In string theory}}\label{in_string_theory_2} \hypertarget{in_type_ii_string_theory}{}\paragraph*{{In type II string theory}}\label{in_type_ii_string_theory} A generic argument for stabilization of compact dimensions in [[type II string theory]] via string winding modes at the self-[[T-duality]] radius is the [[Brandenberger-Vafa mechanism]], see e.g. \begin{itemize}% \item [[Robert Brandenberger]], [[Cumrun Vafa]], \emph{Superstrings In The Early Universe}, Nucl. Phys. B 316, 391 (1989) (\href{http://inspirehep.net/record/263348}{spire}) \item Scott Watson, [[Robert Brandenberger]], \emph{Stabilization of Extra Dimensions at Tree Level}, JCAP 0311 (2003) 008 (\href{http://arxiv.org/abs/hep-th/0307044}{arXiv:hep-th/0307044}) \item [[Brian Greene]], Daniel Kabat, Stefanos Marnerides, \emph{On three dimensions as the preferred dimensionality of space via the Brandenberger-Vafa mechanism}, 10.1103/PhysRevD.88.043527 (\href{http://arxiv.org/abs/1212.2115}{arXiv:1212.2115}) \end{itemize} Discussion of moduli stabilization via [[flux compactification]] of and [[non-perturbative effects]] in [[type II string theory]]/[[F-theory]] originates with the influential article (``KKLT'') \begin{itemize}% \item [[Shamit Kachru]], [[Renata Kallosh]], [[Andrei Linde]], [[Sandip Trivedi]], \emph{de Sitter Vacua in String Theory}, Phys. Rev. D68:046005, 2003 (\href{http://arxiv.org/abs/hep-th/0301240}{arXiv:hep-th/0301240}) \end{itemize} which led to a little burst of discussion of the [[landscape of string theory vacua]]. The analysis there relies on \begin{itemize}% \item [[Shamit Kachru]], J. Pearson, [[Herman Verlinde]], \emph{Brane/Flux Annihilation and the String Dual of a Non-Supersymmetric Field Theory}, JHEP 0206 (2002) 021 (\href{http://arxiv.org/abs/hep-th/0112197}{arXiv:hep-th/0112197}) \end{itemize} Further developments include \begin{itemize}% \item [[Frederik Denef]], [[Michael Douglas]], Bogdan Florea, Antonella Grassi, [[Shamit Kachru]], \emph{Fixing All Moduli in a Simple F-Theory Compactification}, Adv.Theor.Math.Phys.9:861-929, 2005 (\href{http://arxiv.org/abs/hep-th/0503124}{arXiv:hep-th/0503124}) \item [[Vijay Balasubramanian]], Per Berglund, [[Joseph Conlon]], [[Fernando Quevedo]], \emph{Systematics of Moduli Stabilisation in Calabi-Yau Flux Compactifications}, JHEP 0503:007,2005 (\href{http://arxiv.org/abs/hep-th/0502058}{arXiv:hep-th/0502058}) \end{itemize} A variant via K\"a{}hler uplifting is \begin{itemize}% \item [[Alexander Westphal]], \emph{de Sitter String Vacua from K\"a{}hler Uplifting}, JHEP 0703:102,2007 (\href{https://arxiv.org/abs/hep-th/0611332}{arXiv:hep-th/0611332}) \end{itemize} Review includes \begin{itemize}% \item [[Renata Kallosh]], \emph{Stabilization of moduli in string theory}, lectures 2005 (\href{http://web.stanford.edu/~rkallosh/Talks/LectureI.pdf}{part I pdf}, \href{http://web.stanford.edu/~rkallosh/Talks/LectureII.pdf}{part II pdf}) \item [[Joseph Conlon]], \emph{Moduli Stabilisation and Applications in IIB String Theory}, Fortsch.Phys.55:287-422,2007 (\href{http://arxiv.org/abs/hep-th/0611039}{arXiv:hep-th/0611039}) \item Sibasish Banerjee, \emph{Calabi-Yau compactification of type II string theories} (\href{http://arxiv.org/abs/1609.04454}{arXiv:1609.04454}) \end{itemize} Analogous discussion in [[type IIA string theory]] includes (\hyperlink{Acharya02}{Acharya 02}) and \begin{itemize}% \item Oliver DeWolfe, Alexander Giryavets, [[Shamit Kachru]], [[Washington Taylor]], \emph{Type IIA Moduli Stabilization} (\href{http://arxiv.org/abs/hep-th/0505160}{arXiv:hep-th/0505160}) \end{itemize} Discussion of volume stabilization of compact dimensions in the context of [[cosmic inflation]] is in \begin{itemize}% \item Jonathan P. Hsu, [[Renata Kallosh]], \emph{Volume Stabilization and the Origin of the Inflaton Shift Symmetry in String Theory}, JHEP 0404 (2004) 042 (\href{http://arxiv.org/abs/hep-th/0402047}{arXiv:hep-th/0402047}) \end{itemize} In \begin{itemize}% \item S.-W. Kim, J. Nishimura, and A. Tsuchiya, \emph{Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions}, Phys. Rev. Lett. 108, 011601 (2012), (\href{https://arxiv.org/abs/1108.1540}{arXiv:1108.1540}). \item S.-W. Kim, J. Nishimura, and A. Tsuchiya, \emph{Late time behaviors of the expanding universe in the IIB matrix model}, JHEP 10, 147 (2012), (\href{https://arxiv.org/abs/1208.0711}{arXiv:1208.0711}). \end{itemize} it is claimed that computer simulation shows that the [[IKKT matrix model]] description of, supposedly, non-perturbative type II string theory exhibits spontanous decompactification of 3+1 large dimensions, with the other 6 remaining tiny. \hypertarget{ReferencesInMTheory}{}\paragraph*{{In M-theory}}\label{ReferencesInMTheory} Discussion of moduli stabilization in [[M-theory on G2-manifolds]] for stabilization via ``[[flux]]'' (non-vanishing bosonic [[field strength]] of the [[supergravity C-field]]) is in \begin{itemize}% \item [[Bobby Acharya]], \emph{A Moduli Fixing Mechanism in M theory} (\href{http://arxiv.org/abs/hep-th/0212294}{arXiv:hep-th/0212294}) \end{itemize} and [[moduli stabilization]] for fluxless compactifications via [[nonperturbative effects]], claimed to be sufficient and necessary to solve the [[hierarchy problem]], is discussed in \begin{itemize}% \item [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Diana Vaman, \emph{An M theory Solution to the Hierarchy Problem}, Phys.Rev.Lett.97:191601,2006 (\href{http://arxiv.org/abs/hep-th/0606262}{arXiv:hep-th/0606262}) \item [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Jing Shao, \emph{Explaining the Electroweak Scale and Stabilizing Moduli in M Theory}, Phys.Rev.D76:126010,2007 (\href{http://arxiv.org/abs/hep-th/0701034}{arXiv:hep-th/0701034}) \item [[Bobby Acharya]], [[Piyush Kumar]], Konstantin Bobkov, [[Gordon Kane]], Jing Shao, Scott Watson, \emph{Non-thermal Dark Matter and the Moduli Problem in String Frameworks},JHEP 0806:064,2008 (\href{http://arxiv.org/abs/0804.0863}{arXiv:0804.0863}) \end{itemize} and specifically for the [[G2-MSSM]] in \begin{itemize}% \item [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Jing Shao, \emph{The $G_2$-MSSM - An $M$ Theory motivated model of Particle Physics} (\href{http://arxiv.org/abs/0801.0478}{arXiv:0801.0478}) \end{itemize} \hypertarget{in_heterotic_string_theory}{}\paragraph*{{In heterotic string theory}}\label{in_heterotic_string_theory} Discussion of moduli stabilization in [[heterotic string theory]] includes \begin{itemize}% \item [[Evgeny Buchbinder]], [[Burt Ovrut]], \emph{Vacuum Stability in Heterotic M-Theory}, Phys.Rev. D69 (2004) 086010 (\href{http://arxiv.org/abs/hep-th/0310112}{arXiv:hep-th/0310112}) \item [[Sergei Gukov]], [[Shamit Kachru]], Xiao Liu, Liam McAllister, \emph{Heterotic Moduli Stabilization with Fractional Chern-Simons Invariants}, Phys.Rev.D69:086008,2004 (\href{http://arxiv.org/abs/hep-th/0310159}{arXiv:hep-th/0310159}) \end{itemize} [[!redirects moduli stabilizations]] [[!redirects Brandenberger-Vafa mechanism]] [[!redirects Brandenberger-Vafa mechanisms]] \end{document}