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\section*{IKKT matrix model}

\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
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesComputerSimulation}{Computer simulation}\dotfill \pageref*{ReferencesComputerSimulation} \linebreak


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

The [[KK-compactification]] of [[D=10 super Yang-Mills theory]] all the way to the point yields a [[theory (physics)|theory]] whose [[field (physics)|fields]] are simply elements of the gauge [[Lie algebra]] $\mathfrak{g}$, hence [[matrices]] for a [[matrix Lie algebra]]. This [[model (physics)|physics]] is called the \emph{IKKT matrix model}.

Alternatively this model can be motivated from a certain regularization of the [[worldsheet]] [[action functional]] of the [[superstring]]. This is how it was originally obtained in (\hyperlink{IKKT96}{IKKT 96}).

It has been argued that for $\mathfrak{g} = \mathfrak{su}(N)$ for large $N$, this model captures aspects of [[non-perturbative field theory|non-perturbative]] [[type IIB string theory]] (see also at \emph{[[M-theory]]}). Therefore this is also called the \emph{IIB matrix model} (in contast to the [[BFSS matrix model]] in [[type IIA string theory]]).

Several authors have explored the possibility to lift the derivation of the IKKT model from the [[superstring]] to the [[M2-brane]]. See at \emph{[[membrane matrix model]]} for more on this.

In (\hyperlink{KimNishimuraTsuchiya12}{Kim-Nishimura-Tsuchiya 12}) it is claimed that computer simulation of the IKKT matrix model, regarded as non-perturbative [[type IIB string theory]], shows a spontaneous emerging spacetime of macroscopic dimension 3+1, with 6 microscopic dimensions. (A similar claim results from a very different argument: the \emph{[[Brandenberger-Vafa mechanism]]}.)

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

\begin{itemize}%
\item [[BFSS matrix model]]


\item [[lattice gauge theory]]



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

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

The original articles are

\begin{itemize}%
\item N. Ishibashi, H. Kawai, Y. Kitazawa, A. Tsuchiya, \emph{A Large-N Reduced Model as Superstring}, Nucl.Phys. B498 (1997) 467-491 (\href{http://arxiv.org/abs/hep-th/9612115}{arXiv:hep-th/9612115})


\item H. Aoki, S. Iso, H. Kawai, Y. Kitazawa, T. Tada, A. Tsuchiya, \emph{IIB Matrix Model}, Prog.Theor.Phys.Suppl.134:47-83,1999 (\href{http://arxiv.org/abs/hep-th/9908038}{arXiv:hep-th/9908038})



\end{itemize}
See also

\begin{itemize}%
\item [[Mikhail Movshev]], [[Albert Schwarz]], \emph{On maximally supersymmetric Yang-Mills theories}, Nucl.Phys. B681 (2004) 324-350 (\href{https://arxiv.org/abs/hep-th/0311132}{arXiv:hep-th/0311132})

\end{itemize}
Derivation from [[open string field theory]] is discussed in

\begin{itemize}%
\item [[Taejin Lee]], \emph{Covariant Open String Field Theory on Multiple D$p$-Branes} (\href{https://arxiv.org/abs/1703.06402}{arXiv:1703.06402})

\end{itemize}
Arguments that full [[Yang-Mills theory]] generalized to [[noncommutative geometry]] is recovered as the [[perturbation theory]] around classical solutions of the IKKT model are in

\begin{itemize}%
\item H. Aoki, N. Ishibashi, S. Iso, H. Kawai, Y. Kitazawa, T. Tada, \emph{Noncommutative Yang-Mills in IIB Matrix Model}, Nucl.Phys. B565 (2000) 176-192 (\href{http://arxiv.org/abs/hep-th/9908141}{arXiv:hep-th/9908141})


\item Tatsuo Azeyanagi, Masanori Hanada, Tomoyoshi Hirata, \emph{On Matrix Model Formulations of Noncommutative Yang-Mills Theories}, Phys.Rev.D78:105017,2008 (\href{http://arxiv.org/abs/0806.3252}{arXiv:0806.3252})



\end{itemize}
Arguments that [[closed string field theory]] arises from the [[dynamics]] of [[Wilson loops]] IKKT model are in

\begin{itemize}%
\item M. Fukuma, H. Kawai, Y. Kitazawa, A. Tsuchiya, \emph{String Field Theory from IIB Matrix Model}, Nucl.Phys.B510:158-174,1998 (\href{http://arxiv.org/abs/hep-th/9705128}{arXiv:hep-th/9705128})


\item Daiji Ennyu, Hiroshi Kawabe, Naohito Nakazawa, \emph{Note on a Closed String Field Theory from Bosonic IIB Matrix Model}, JHEP 0301 (2003) 025 (\href{http://arxiv.org/abs/hep-th/0212044}{arXiv:hep-th/0212044})



\end{itemize}
Possibilities of generalizing the IKKT model from [[Lie algebras]] to [[Lie 2-algebras]] in some [[membrane matrix model]] are explored in

\begin{itemize}%
\item [[Patricia Ritter]], [[Christian Saemann]], \emph{Lie 2-algebra models} (\href{http://arxiv.org/abs/1308.4892}{arXiv:1308.4892})

\end{itemize}
Discussion of [[standard model of particle physics|standard model]] [[phenomenology]] within the IKKT model includes

\begin{itemize}%
\item [[Harold Steinacker]], Jochen Zahn, \emph{An extended standard model and its Higgs geometry from the matrix model}, PTEP 2014 (2014) 8, 083B03 (\href{http://arxiv.org/abs/1401.2020}{arXiv:1401.2020})

\end{itemize}
See also

\begin{itemize}%
\item A. Stern, Chuang Xu, \emph{Signature change in matrix model solutions} (\href{https://arxiv.org/abs/1808.07963}{arXiv:1808.07963})

\end{itemize}
\hypertarget{ReferencesComputerSimulation}{}\subsubsection*{{Computer simulation}}\label{ReferencesComputerSimulation}

There are claims that numerical computer simulations (as in [[lattice gauge theory]], see the references \href{lattice+gauge+theory#ForSuperYangMills}{there}) show that the IKKT matrix model predicts a spontanously generated spacetime where exactly 3+1 dimensions become macroscopic (hence effectively predicts \emph{[[moduli stabilization]]} in spintaneous [[KK-compactification]] of [[M-theory]] to $D = 3+1$ macroscopic dimensions ):

\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}).


\item Yuta Ito, Jun Nishimura, Asato Tsuchiya, \emph{Large-scale computation of the exponentially expanding universe in a simplified Lorentzian type IIB matrix model} (\href{https://arxiv.org/abs/1512.01923}{arXiv:1512.01923})


\item Toshihiro Aoki, Mitsuaki Hirasawa, Yuta Ito, Jun Nishimura, Asato Tsuchiya, \emph{On the structure of the emergent 3d expanding space in the Lorentzian type IIB matrix model} (\href{https://arxiv.org/abs/1904.05914}{arXiv:1904.05914})



\end{itemize}
[[!redirects IIB matrix model]]

[[!redirects IKKT model]]



\end{document}