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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*{BFSS 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{OpenProblems}{Open problems}\dotfill \pageref*{OpenProblems} \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{ReferencesRelationToSuperMembranes}{Relation to super-membranes}\dotfill \pageref*{ReferencesRelationToSuperMembranes} \linebreak \noindent\hyperlink{ReferencesGroundStateProblem}{Ground state problem}\dotfill \pageref*{ReferencesGroundStateProblem} \linebreak \noindent\hyperlink{ReferencesGravitonScattering}{Graviton scattering}\dotfill \pageref*{ReferencesGravitonScattering} \linebreak \noindent\hyperlink{black_holes}{Black holes}\dotfill \pageref*{black_holes} \linebreak \noindent\hyperlink{relation_to_lattice_gauge_theory}{Relation to lattice gauge theory}\dotfill \pageref*{relation_to_lattice_gauge_theory} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The \emph{BFSS matrix model} (\hyperlink{BanksFischlerShenkerSusskind96}{Banks-Fischler-Shenker-Susskind 96}, \hyperlink{Seiberg97}{Seiberg 97}) is the description of the [[worldline]] dynamics of interacting [[D0-branes]]. In the [[large N limit]] of a large number of [[D0-branes]] this is supposed to encode the [[non-perturbative quantum field theory|strong coupling limit]] of [[type IIA string theory]] known as \emph{[[M-theory]]}. The BFSS matrix model was argued to arise in several equivalent ways: \begin{enumerate}% \item as the [[worldline]] theory of a large number of [[D0-branes]] in [[type IIA string theory]], \item as the [[Kaluza-Klein compactification]] of [[10d super Yang-Mills theory]] to 1+0 space dimensions, \item as a certain non-commutative regularization of the [[Green-Schwarz sigma-model]] for the [[M2-brane]] (\hyperlink{NicolaiHelling98}{Nicolai-Helling 98}, \hyperlink{DasguptaNicolaiPlefka02}{Dasgupta-Nicolai-Plefka 02}). In this picture matrix blocks around the diagonal correspond to blobs of [[membrane]], while off-diagonal matrix elements correspond to thin tubes of membrane connecting these blobs. \end{enumerate} \begin{quote}% graphics grabbed from \hyperlink{DasguptaNicolaiPlefka02}{Dasgupta-Nicolai-Plefka 02} \end{quote} In any case, the BFSS matrix model ends up being a [[quantum mechanics|quantum mechanical]] system whose bosonic degrees of freedom are a set of 9+1 large [[matrices]]. These play the role of would-be [[coordinate functions]] and their [[eigenvalues]] may be in interpreted as points in a [[non-commutative geometry|non-commutative]] [[spacetime]] thus defined. There is also the [[IKKT matrix model]], which takes this one step further by reducing one dimension further down to [[D(-1)-branes]] in [[type IIB string theory]]. See also at \emph{[[membrane matrix model]]}. \hypertarget{OpenProblems}{}\subsection*{{Open problems}}\label{OpenProblems} In the 90s there was much excitement about the BFSS model, as people hoped it might provide a definition of [[M-theory]]. It is from these times that [[Edward Witten]] changed the original suggestion that ``M'' is for ``magic, mystery and membrane'' to the suggestion that it is for ``magic, mystery and matrix''. (See \href{M-theory#Witten14}{Witten's 2014 Kyoto prize speech}, last paragraph.) However, while the BFSS matrix model clearly sees something M-theoretic, just as clearly it is not the full answer. Notably it needs for its definition an ambient asympototic Minkowski background, a light cone limit and a peculiar scaling of [[string coupling]] over [[string length]], all of which means that it pertains to a particular corner of a full theory. From \hyperlink{NicolaiHelling98}{Nicolai-Helling 98, p. 2}: \begin{quote}% Despite the recent excitement, however, we do not think that M(atrix) theory and the $d= 11$ supermembrane in their present incarnation are already the final answer in the search for M-Theory, even though they probably are important pieces of the puzzle. There are still too many ingredients missing that we would expect the final theory to possess. For one thing, we would expect a true theory of quantum gravity to exhibit certain pregeometrical features corresponding to a “dissolution” of space-time and the emergence of some kind of non-commutative geometry at short distances; although the matrix model does achieve that to some extent by replacing commuting coordinates by non-commuting matrices, it seems to us that a still more radical departure from conventional ideas about space and time may be required in order to arrive at a truly background independent formulation (the matrix model “lives” in nine \emph{flat} transverse dimensions only). Furthermore, there should exist some huge and so far completely hidden symmetries generalizing not only the duality symmetries of extended supergravity and string theory, but also the principles underlying general relativity. \end{quote} Then, even assuming that in this corner all the crucial [[generalized (Eilenberg-Steenrod) cohomology|cohomological]] aspects of [[D-brane]] and [[M-brane]] charges (in [[twisted differential K-theory]], [[twisted cohomotopy]] etc.) are secretly encoded in the matrix model, somehow, none of this is manifest, making the matrix model spit out numbers about a conceptually elusive theory in close analogy to how [[lattice QCD]] produces numbers without informing us about the actual conceptual nature of [[confinement|confined]] [[hadron]] physics. Furthermore, there are technical open issues, such as the open question whether the theory has a decent ground state the way it needs to have to make sense (see the references below \hyperlink{ReferencesGroundStateProblem}{below}). A similar assessment has been given by [[Greg Moore]], from pages 43-44 of his \emph{[[Physical Mathematics and the Future]]} (\href{https://ncatlab.org/nlab/show/Physical+Mathematics+and+the+Future#AGoodStartWasGivenByTheMatrixTheory}{here}): \begin{quote}% A good start $[$with defining M-theory$]$ was given by the Matrix theory approach of Banks, Fischler, Shenker and Susskind. We have every reason to expect that this theory produces the correct scattering amplitudes of modes in the 11-dimensional supergravity multiplet in 11-dimensional Minkowski space - even at energies sufficiently large that black holes should be created. (This latter phenomenon has never been explicitly demonstrated). But Matrix theory is only a beginning and does not give us the whole picture of M-theory. The program ran into increasing technical difficulties when more complicated compactifications were investigated. (For example, compactification on a six-dimensional torus is not very well understood at all. $[$\ldots{}$]$). Moreover, to my mind, as it has thus far been practiced it has an important flaw: It has not led to much significant new mathematics. If history is a good guide, then we should expect that anything as profound and far-reaching as a fully satisfactory formulation of M-theory is surely going to lead to new and novel mathematics. Regrettably, it is a problem the community seems to have put aside - temporarily. But, ultimately, Physical Mathematics must return to this grand issue. \end{quote} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[IKKT matrix model]] \item [[membrane matrix model]] \item [[Myers effect]] \end{itemize} \hypertarget{References}{}\subsection*{{References}}\label{References} \hypertarget{general}{}\subsubsection*{{General}}\label{general} The original articles are \begin{itemize}% \item [[Tom Banks]], [[Willy Fischler]], S.H. Shenker and [[Leonard Susskind]], \emph{M Theory As A Matrix Model: A Conjecture} Phys. Rev. D55 (1997). (\href{http://arxiv.org/abs/hep-th/9610043}{arXiv:hep-th/9610043}) \item [[Ashoke Sen]], \emph{D0 Branes on $T^n$ and Matrix Theory}, Adv.Theor.Math.Phys.2:51-59, 1998 (\href{https://arxiv.org/abs/hep-th/9709220}{arXiv:hep-th/9709220}) \item [[Nathan Seiberg]], \emph{Why is the Matrix Model Correct?}, Phys.Rev.Lett.79:3577-3580, 1997 (\href{https://arxiv.org/abs/hep-th/9710009}{arXiv:hep-th/9710009}) \end{itemize} Review includes \begin{itemize}% \item [[Tom Banks]], \emph{Matrix Theory}, Nucl.Phys.Proc.Suppl. 67 (1998) 180-224 (\href{https://arxiv.org/abs/hep-th/9710231}{arXiv:hep-th/9710231}) \item [[Washington Taylor]], \emph{M(atrix) Theory: Matrix Quantum Mechanics as a Fundamental Theory}, Rev.Mod.Phys.73:419-462,2001 (\href{https://arxiv.org/abs/hep-th/0101126}{arXiv:hep-th/0101126}) \end{itemize} A review of further developments is in \begin{itemize}% \item [[David Berenstein]], \emph{Classical dynamics and thermalization in holographic matrix models}, talk at Leiden, October 2012 (\href{http://www.lorentzcenter.nl/lc/web/2012/514/presentations/Berenstein.pdf}{pdf}) \end{itemize} See also \begin{itemize}% \item [[Paul Townsend]], \emph{M(embrane) theory on $T^0$}, Nucl.Phys.Proc.Suppl.68:11-16,1998 (\href{http://arxiv.org/abs/hep-th/9708034}{arXiv:hep-th/9708034}) \end{itemize} Discussion as a solution to the open problem of defining [[M-theory]] is in \begin{itemize}% \item [[Gregory Moore]], section 12 p. 43-44 \emph{[[Physical Mathematics and the Future]]} talk at \href{http://physics.princeton.edu/strings2014/}{Strings 2014} (\href{http://physics.princeton.edu/strings2014/slides/Moore.pdf}{talk slides}, \href{http://www.physics.rutgers.edu/~gmoore/PhysicalMathematicsAndFuture.pdf}{companion text pdf}, [[MooreVisionTalk2014.pdf:file]]) \end{itemize} where it says: \begin{quote}% A good start was given by the [[BFSS matrix model|Matrix theory]] approach of [[Tom Banks|Banks]], [[Willy Fischler|Fischler]], Shenker and [[Leonard Susskind|Susskind]]. We have every reason to expect that this theory produces the correct [[scattering amplitudes]] of modes in the [[11-dimensional supergravity]] multiplet in 11-dimensional [[Minkowski space]] - even at energies sufficiently large that [[black holes]] should be created. (This latter phenomenon has never been explicitly demonstrated). But Matrix theory is only a beginning and does not give us the whole picture of [[M-theory]]. The program ran into increasing technical difficulties when more complicated compactifications were investigated. (For example, compactification on a six-dimensional torus is not very well understood at all. $[...]$). Moreover, to my mind, as it has thus far been practiced it has an important flaw: It has not led to much significant new mathematics. If history is a good guide, then we should expect that anything as profound and far-reaching as a fully satisfactory formulation of [[M-theory]] is surely going to lead to new and novel mathematics. Regrettably, it is a problem the community seems to have put aside - temporarily. But, ultimately, Physical Mathematics must return to this grand issue. \end{quote} 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} Relation to the [[6d (2,0)-supersymmetric QFT]]: \begin{itemize}% \item [[Micha Berkooz]], [[Moshe Rozali]], [[Nathan Seiberg]], \emph{Matrix Description of M-theory on $T^3$ and $T^5$} (\href{http://arxiv.org/abs/hep-th/9704089}{arXiv:hep-th/9704089}) \end{itemize} \hypertarget{ReferencesRelationToSuperMembranes}{}\subsubsection*{{Relation to super-membranes}}\label{ReferencesRelationToSuperMembranes} The interpretation of the BFSS model as a regularized [[M2-brane]] [[worldvolume]] theory is discussed in \begin{itemize}% \item [[Hermann Nicolai]], Robert Helling, \emph{Supermembranes and M(atrix) Theory}, In: [[Mike Duff]], [[Ergin Sezgin]], [[Brian Greene]] et. al. (eds.) \emph{\href{https://doi.org/10.1142/9789814527651}{Nonperturbative aspects of strings, branes and supersymmetry}}, World Scientific 1999 (\href{http://arxiv.org/abs/hep-th/9809103}{arXiv:hep-th/9809103}, \href{http://inspirehep.net/record/476366}{spire:476366}) \item Arundhati Dasgupta, [[Hermann Nicolai]], [[Jan Plefka]], \emph{An Introduction to the Quantum Supermembrane}, Grav.Cosmol.8:1,2002; Rev.Mex.Fis.49S1:1-10, 2003 (\href{http://arxiv.org/abs/hep-th/0201182}{arXiv:hep-th/0201182}) \end{itemize} Analogous regularizations lead to matrix model descriptions of [[D-branes]]: \begin{itemize}% \item Qiang Jia, \emph{On matrix description of D-branes} (\href{https://arxiv.org/abs/1907.00142}{arXiv:1907.00142}) \end{itemize} \hypertarget{ReferencesGroundStateProblem}{}\subsubsection*{{Ground state problem}}\label{ReferencesGroundStateProblem} There remains the problem of existence of a sensible ground state of the BFSS model. For a new attempt and pointers to previous attempts see \begin{itemize}% \item L. Boulton, M.P. Garcia del Moral, A. Restuccia, \emph{The ground state of the D=11 supermembrane and matrix models on compact regions}, Nuclear Physics B Volume 910, September 2016, Pages 665-684 (\href{https://arxiv.org/abs/1504.04071}{arXiv:1504.04071}) \item L. Boulton, M.P. Garcia del Moral, A. Restuccia, \emph{Measure of the potential valleys of the supermembrane theory} (\href{https://arxiv.org/abs/1811.05758}{arXiv:1811.05758}) \end{itemize} \hypertarget{ReferencesGravitonScattering}{}\subsubsection*{{Graviton scattering}}\label{ReferencesGravitonScattering} Computation of [[graviton]] [[scattering amplitudes]]: \begin{itemize}% \item [[Katrin Becker]], [[Melanie Becker]], \emph{A Two-Loop Test of M(atrix) Theory}, Nucl.Phys. B506 (1997) 48-60 (\href{https://arxiv.org/abs/hep-th/9705091}{arXiv:hep-th/9705091}) \item [[Katrin Becker]], [[Melanie Becker]], [[Joseph Polchinski]], [[Arkady Tseytlin]], \emph{Higher Order Graviton Scattering in M(atrix) Theory}, Phys.Rev.D56:3174-3178,1997 (\href{https://arxiv.org/abs/hep-th/9706072}{arXiv:hep-th/9706072}) \item also \hyperlink{KabatTaylor97}{Kabat-Taylor 97} \item M. Fabbrichesi, \emph{Graviton scattering in matrix theory and supergravity}, in: Ceresole A., Kounnas C., [[Dieter Lüst]], [[Stefan Theisen]] (eds.) \emph{Quantum Aspects of Gauge Theories, Supersymmetry and Unification}, Lecture Notes in Physics, vol 525. Springer, Berlin, Heidelberg (\href{https://arxiv.org/abs/hep-th/9811204}{arXiv:hep-th/9811204}) \item Robert Helling, [[Jan Plefka]], Marco Serone, Andrew Waldron, \emph{Three-graviton scattering in M-theory}, Nuclear Physics B Volume 559, Issues 1–2, 18 October 1999, Pages 184-204 (\href{https://arxiv.org/abs/hep-th/9905183}{arXiv:hep-th/9905183}) \item Robert Echols, \emph{M-theory, supergravity and the matrix model: Graviton scattering and non-renormalization theorems}, PhD thesis, 1999 \href{https://web.calpoly.edu/~rechols/phys403/mythesis.pdf}{pdf} \end{itemize} \hypertarget{black_holes}{}\subsubsection*{{Black holes}}\label{black_holes} Relation to [[black holes in string theory]]: \begin{itemize}% \item [[Tom Banks]], [[Willy Fischler]], [[Igor Klebanov]], [[Leonard Susskind]], \emph{Schwarzschild Black Holes from Matrix Theory}, Phys.Rev.Lett.80:226-229,1998 (\href{https://arxiv.org/abs/hep-th/9709091}{arXiv:hep-th/9709091}) \item [[Tom Banks]], [[Willy Fischler]], [[Igor Klebanov]], [[Leonard Susskind]], \emph{Schwarzchild Black Holes in Matrix Theory II}, JHEP 9801:008,1998 (\href{https://arxiv.org/abs/hep-th/9711005}{arXiv:hep-th/9711005}) \item [[Igor Klebanov]], [[Leonard Susskind]], \emph{Schwarzschild Black Holes in Various Dimensions from Matrix Theory}, Phys.Lett.B416:62-66,1998 (\href{https://arxiv.org/abs/hep-th/9709108}{arXiv:hep-th/9709108}) \item Edi Halyo, \emph{Six Dimensional Schwarzschild Black Holes in M(atrix) Theory} (\href{https://arxiv.org/abs/hep-th/9709225}{arXiv:hep-th/9709225}) \item [[Gary Horowitz]], [[Emil Martinec]], \emph{Comments on Black Holes in Matrix Theory}, Phys. Rev. D 57, 4935 (1998) (\href{https://arxiv.org/abs/hep-th/9710217}{arXiv:hep-th/9710217}) \item Daniel Kabat, [[Washington Taylor]], \emph{Spherical membranes in Matrix theory}, Adv.Theor.Math.Phys.2:181-206,1998 (\href{https://arxiv.org/abs/hep-th/9711078}{arXiv:hep-th/9711078}) \item Yoshifumi Hyakutake, \emph{Black Hole and Fuzzy Objects in BFSS Matrix Model} (\href{https://arxiv.org/abs/1801.07869}{arXiv:1801.07869}) \item Haoxing Du, Vatche Sahakian, \emph{Emergent geometry from stochastic dynamics, or Hawking evaporation in M(atrix) theory} (\href{https://arxiv.org/abs/1812.05020}{arXiv:1812.05020}) (combination with [[random matrix theory]]) \end{itemize} \hypertarget{relation_to_lattice_gauge_theory}{}\subsubsection*{{Relation to lattice gauge theory}}\label{relation_to_lattice_gauge_theory} Relation to [[lattice gauge theory]] and numerical tests of [[AdS/CFT]]: \begin{itemize}% \item Anosh Joseph, \emph{Review of Lattice Supersymmetry and Gauge-Gravity Duality} (\href{https://arxiv.org/abs/1509.01440}{arXiv:1509.01440}) \item Veselin G. Filev, Denjoe O'Connor, \emph{The BFSS model on the lattice}, JHEP 1605 (2016) 167 (\href{https://arxiv.org/abs/1506.01366}{arXiv:1506.01366}) \item Masanori Hanada, \emph{What lattice theorists can do for superstring/M-theory}, International Journal of Modern Physics AVol. 31, No. 22, 1643006 (2016) (\href{https://arxiv.org/abs/1604.05421}{arXiv:1604.05421}) \end{itemize} [[!redirects BFSS model]] \end{document}