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\section*{exceptional generalized geometry}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{exceptional_structures}{}\paragraph*{{Exceptional structures}}\label{exceptional_structures}

[[!include exceptional structures -- contents]]

\hypertarget{duality_in_string_theory}{}\paragraph*{{Duality in string theory}}\label{duality_in_string_theory}

[[!include duality in string theory -- contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{HigherSupersymmetry}{Higher supersymmetry}\dotfill \pageref*{HigherSupersymmetry} \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{SuperExceptionalGeometryReferences}{Super-exceptional generalized geometry}\dotfill \pageref*{SuperExceptionalGeometryReferences} \linebreak
\noindent\hyperlink{__}{$E_6$, $E_7$, $E_8$}\dotfill \pageref*{__} \linebreak
\noindent\hyperlink{}{$E_{10}$}\dotfill \pageref*{} \linebreak
\noindent\hyperlink{_2}{$E_{11}$}\dotfill \pageref*{_2} \linebreak


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

A variant of the idea of [[generalized complex geometry]] given by passing from generalization of [[complex geometry]] to generalization of [[exceptional geometry]]. Instead of by [[reduction of structure groups]] along inclusions like $O(d)\times O(d) \to O(d,d)$ it is controled by inclusions into [[split real forms]] of [[exceptional Lie groups]].

This serves to neatly encode [[U-duality]] groups in [[supergravity]] as well as higher [[supersymmetry]] of supergravity [[Kaluza-Klein mechanism|compactifications]]. See also at \emph{[[exceptional field theory]]} for more on this.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{HigherSupersymmetry}{}\subsubsection*{{Higher supersymmetry}}\label{HigherSupersymmetry}

Compactification of [[11-dimensional supergravity]] on a [[manifold]] of [[dimension]] 7 preserves $N = 1$ [[supersymmetry]] precisely if its [[generalized tangent bundle]] has [[G-structure]] for the inclusion

\begin{displaymath}
SU(7) \hookrightarrow E_{7(7)}
\end{displaymath}
of the [[special unitary group]] in dimension 7 into the [[split real form]] of [[E7]]. This is shown in (\hyperlink{PachecoWaldram08}{Pacheco-Waldram 08}).

One dimension down, compactification of 10-dimensional [[type II supergravity]] on a 6-manifold $X$ preserves $N = 2$ supersymmetry precisely if the [[generalized tangent bundle]] $T X \otimes T^* X$ in the NS-NS sector admits [[G-structure]] for the inclusion

\begin{displaymath}
SU(3) \times SU(3) \hookrightarrow O(6,6)
  \,.
\end{displaymath}
This is reviewed in (\hyperlink{GLSW}{GLSW, section 2}).

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

\begin{itemize}%
\item [[non-geometric string theory vacua]]


\item [[generalized complex geometry]], [[exceptional geometry]]


\item [[generalized tangent bundle]], [[exceptional tangent bundle]]


\item [[generalized Calabi-Yau manifold]], [[generalized G2-manifold]]


\item [[Bn-geometry]], [[differential T-duality]]


\item [[universal exceptionalism]]



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

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

Original articles include

\begin{itemize}%
\item K. Koepsell, [[Hermann Nicolai]], [[Henning Samtleben]], \emph{An exceptional geometry for d=11 supergravity?}, Class. Quant. Grav.17:3689-3702, 2000 (\href{http://arxiv.org/abs/hep-th/0006034}{arXiv:hep-th/0006034})


\item [[Chris Hull]], \emph{Generalised Geometry for M-Theory}, JHEP 0707:079 (2007) (\href{http://arxiv.org/abs/hep-th/0701203}{arXiv:hep-th/0701203})


\item Paulo Pires Pacheco, [[Daniel Waldram]], \emph{M-theory, exceptional generalised geometry and superpotentials}, JHEP 0809:123, 2008 (\href{http://arxiv.org/abs/0804.1362}{arXiv:0804.1362})


\item [[Mariana Graña]], [[Jan Louis]], Aaron Sim, [[Daniel Waldram]], \emph{$E_{7(7)}$ formulation of $N=2$ backgrounds} (\href{http://arxiv.org/abs/0904.2333}{arXiv:0904.2333})


\item G. Aldazabala, E. Andr\'e{}sb, P. C\'a{}marac, [[Mariana Graña]], \emph{U-dual fluxes and generalized geometry}, JHEP 1011:083,2010 (\href{http://arxiv.org/abs/1007.5509}{arXiv:1007.5509})


\item [[Mariana Graña]], Francesco Orsi, \emph{$N=1$ vacua in Exceptional Generalized Geometry} (\href{http://arxiv.org/abs/1105.4855}{arXiv:1105.4855})



\end{itemize}
\begin{itemize}%
\item [[Mariana Graña]], Francesco Orsi, \emph{N=2 vacua in Generalized Geometry}, (\href{http://arxiv.org/abs/1207.3004}{arXiv:1207.3004})


\item Andr\'e{} Coimbra, Charles Strickland-Constable, [[Daniel Waldram]], \emph{$E_{d(d)} \times \mathbb{R}^+$ Generalised Geometry, Connections and M theory} (\href{http://arxiv.org/abs/1112.3989}{arXiv:1112.3989})


\item [[David Baraglia]], \emph{Leibniz algebroids, twistings and exceptional generalized geometry}, Journal of Geometry and Physics 62 (2012), pp. 903-934 (\href{http://arxiv.org/abs/1101.0856}{arXiv:1101.0856})


\item [[David Baraglia]], \emph{Exceptional generalized geometry and $N = 2$ backgrounds} (\href{http://mitchell.physics.tamu.edu/Conference/GeneralizedGeometries/Mon-1-Grana.pdf}{pdf})



\end{itemize}
Reviews include

\begin{itemize}%
\item Daniel Persson, \emph{Arithmetic and Hyperbolic Structures in String Theory} (\href{http://arxiv.org/abs/1001.3154}{arXiv:1001.3154})


\item Nassiba Tabti, \emph{Kac-Moody algebraic structures in supergravity theories} (\href{http://arxiv.org/abs/0910.1444}{arXiv:0910.1444})



\end{itemize}
Relation to [[Borcherds superalgebras]] is surveyed and discussed in

\begin{itemize}%
\item Jakob Palmkvist, \emph{Exceptional geometry and Borcherds superalgebras} (\href{http://arxiv.org/abs/1507.08828}{arXiv:1507.08828})

\end{itemize}
[[black branes]] in the exotic spacetime are discussed in

\begin{itemize}%
\item Ilya Bakhmatov, [[David Berman]], [[Axel Kleinschmidt]], Edvard Musaev, Ray Otsuki, \emph{\href{https://arxiv.org/abs/1710.09740}{arXiv:1710.09740}}

\end{itemize}
 The [[string]] and [[membrane]] [[sigma-models]] on exceptional spacetime (the ``exceptional sigma models'') are discussed in

\begin{itemize}%
\item Yuho Sakatani, Shozo Uehara, \emph{Branes in Extended Spacetime: Brane Worldvolume Theory Based on Duality Symmetry}, Phys. Rev. Lett. 117, 191601 (2016) (\href{https://arxiv.org/abs/1607.04265}{arXiv:1607.04265})


\item [[Alex Arvanitakis]], Chris D. A. Blair, \emph{Type II strings are Exceptional} (\href{https://arxiv.org/abs/1712.07115}{arXiv:1712.07115})


\item [[Alex Arvanitakis]], Chris Blair, \emph{The Exceptional Sigma Model} (\href{https://arxiv.org/abs/1802.00442}{arXiv:1802.00442})



\end{itemize}
The generalized-U-duality+diffeomorphism invariance in 11d is discussed in

\begin{itemize}%
\item [[David Berman]], [[Martin Cederwall]], [[Axel Kleinschmidt]], Daniel C. Thompson, \emph{The gauge structure of generalised diffeomorphisms} (\href{http://arxiv.org/abs/1208.5884}{arXiv:1208.5884})

\end{itemize}
For the [[worldvolume]] [[physical theory|theory]] of the [[M5-brane]] this is discussed in

\begin{itemize}%
\item Machiko Hatsuda, Kiyoshi Kamimura, \emph{M5 algebra and $SO(5,5)$ duality} (\href{http://arxiv.org/abs/1305.2258}{arXiv:1305.2258})

\end{itemize}
\hypertarget{SuperExceptionalGeometryReferences}{}\subsubsection*{{Super-exceptional generalized geometry}}\label{SuperExceptionalGeometryReferences}

The combination/unification of [[exceptional generalized geometry]] with [[supergeometry]] used to be an open problem:

\begin{itemize}%
\item [[Martin Cederwall]], p. 39 of \emph{Fundamental issues in extended geometry}, 8th Mathematical Physics Meeting, Aug 2014 Belgrade, Serbia (\href{http://inspirehep.net/record/1477275}{spire:1477275})


\item [[Martin Cederwall]], Joakim Edlund, Anna Karlsson, p. 4, 7 of \emph{Exceptional geometry and tensor fields}, J. High Energ. Phys. (2013) 2013: 28 (\href{https://arxiv.org/abs/1302.6736}{arXiv:1302.6736})



\end{itemize}
Plausibility arguments that the bosonic body of the [[superspace]] underlying the [[M-theory Lie algebra]] serves as the unifying exceptional generalized geometry for [[M-theory]] for $n = 11$:

\begin{itemize}%
\item [[Igor Bandos]], \emph{Exceptional field theories, superparticles in an enlarged 11D superspace and higher spin theories}, Nucl. Phys. B925 (2017) 28-62 (\href{https://arxiv.org/abs/1612.01321}{arXiv:1612.01321})

\end{itemize}
Arguments that super-exceptional M-geometry for $n = 11$ is in fact a further fermionic extension of that (to the ``hidden supergroup'' of D'Auria-Fre):

\begin{itemize}%
\item [[Silvia Vaula]], \emph{On the underlying $E_{11}$ symmetry of the $D= 11$ Free Differential Algebra}, JHEP 0703:010, 2007 (\href{http://arxiv.org/abs/hep-th/0612130}{arXiv:hep-th/0612130})


\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Higher T-duality of super M-branes]]} (\href{https://arxiv.org/abs/1803.05634}{arXiv:1803.05634})


\item [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Higher T-duality of super M-branes|Higher T-duality in M-theory via local supersymmetry]]}, Physics Letters B Volume 781 2018 (\href{https://arxiv.org/abs/1805.00233}{arXiv:1805.00233})


\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Super-exceptional embedding construction of the M5-brane|Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5]]} (\href{https://arxiv.org/abs/1908.00042}{arXiv:1908.00042})



\end{itemize}
A super-exceptional geometry for $n = 7$:

\begin{itemize}%
\item Daniel Butter, [[Henning Samtleben]], [[Ergin Sezgin]], \emph{$E_{7(7)}$ Exceptional Field Theory in Superspace}, JHEP01(2019)087 (\href{https://arxiv.org/abs/1811.00038}{arXiv:1811.00038})

\end{itemize}
See also the references on the corresponding [[super-geometry]]-enhancement of [[type II geometry|type II]] [[generalized geometry]] (``[[doubled geometry]]''): \emph{\href{type+II+geometry#ReferencesDoubledSupergeometry}{doubled geometry -- References -- Doubled supergeometry}}.

\hypertarget{__}{}\subsubsection*{{$E_6$, $E_7$, $E_8$}}\label{__}

[[E6]],[[E7]], [[E8]]-geometry is discussed in

\begin{itemize}%
\item Christian Hillmann, \emph{Generalized E(7(7)) coset dynamics and D=11 supergravity}, JHEP 0903 (2009) 135 (\href{http://arxiv.org/abs/0901.1581}{arXiv:0901.1581})


\item Hadi Godazgar, Mahdi Godazgar, Hermann Nicolai, \emph{Generalised geometry from the ground up} (\href{http://arxiv.org/abs/1307.8295}{arXiv:1307.8295})


\item [[Olaf Hohm]], [[Henning Samtleben]], \emph{Exceptional Form of $D=11$ Supergravity}, Phys. Rev. Lett. 111, 231601 (2013) (\href{http://arxiv.org/abs/1308.1673}{arXiv:1308.1673})



\end{itemize}
(see also at \href{3-dimensional+supergravity#PossibleGaugings}{3d supergravity -- possible gaugings}).

\hypertarget{}{}\subsubsection*{{$E_{10}$}}\label{}

The [[E10]]-geometry of [[11-dimensional supergravity]] compactified to the line is discussed in

\begin{itemize}%
\item [[Thibault Damour]], [[Hermann Nicolai]], \emph{Higher order M theory corrections and the Kac-Moody algebra E10} (\href{http://arxiv.org/abs/hep-th/0504153}{arXiv:hep-th/0504153})


\item [[Hermann Nicolai]], \emph{Wonders of $E_{10}$ and $K(E_{10})$} (2008) (\href{http://ipht.cea.fr/Pisp/pierre.vanhove/Paris08/talk_PDF/nicolai.pdf}{pdf})


\item [[Axel Kleinschmidt]], [[Hermann Nicolai]], \emph{Standard model fermions and $K(E_{10})$} (\href{https://arxiv.org/abs/1504.01586}{arXiv:1504.01586})



\end{itemize}
\hypertarget{_2}{}\subsubsection*{{$E_{11}$}}\label{_2}

Literature discussing $E_{11}$ [[U-duality]] and in the context of [[exceptional generalized geometry]] of [[11-dimensional supergravity]].

Review includes

\begin{itemize}%
\item [[Peter West]], section 17.5 of \emph{[[Introduction to Strings and Branes]]}


\item [[Fabio Riccioni]], \emph{$E_{11}$ and M-theory}, talk at \href{http://www.ift.uam.es/strings07/}{Strings07} (\href{http://member.ipmu.jp/yuji.tachikawa/stringsmirrors/2007/riccioni.pdf}{pdf slides})


\item [[Fabio Riccioni]], [[Peter West]], \emph{The $E_{11}$ origin of all maximal supergravities}, JHEP 0707:063,2007 (\href{http://arxiv.org/abs/0705.0752}{arXiv:0705.0752}, \href{http://inspirehep.net/record/749966/}{spire})


\item [[Paul Cook]], \emph{Connections between Kac-Moody algebras and M-theory} PhD thesis (\href{http://arxiv.org/abs/0711.3498}{arXiv:0711.3498})


\item [[Peter West]], \emph{A brief review of E theory} (\href{http://arxiv.org/abs/1609.06863}{arXiv:1609.06863})



\end{itemize}
Original articles include the following:

The observation that $E_{11}$ seems to neatly organize the structures in [[11-dimensional supergravity]]/[[M-theory]] is due to

\begin{itemize}%
\item [[Peter West]], \emph{$E_{11}$ and M theory}, Class. Quant. Grav., 18:4443--4460, 2001. (\href{http://arxiv.org/abs/hep-th/0104081}{arXiv:hep-th/0104081})

\end{itemize}
A precursor to (\hyperlink{West01}{West 01}) is

\begin{itemize}%
\item [[Bernard Julia]], \emph{Dualities in the classical supergravity limits} (\href{http://arxiv.org/abs/hep-th/9805083}{arXiv:hep-th/9805083})

\end{itemize}
as explained in (\hyperlink{HenneauxJuliaLevie10}{Henneaux-Julia-Levie 10}).

The derivation of the [[equations of motion]] of [[11-dimensional supergravity]] and maximally supersymmetric [[5d supergravity]] from a [[vielbein]] with values in the [[semidirect product]] $E_{11}$ with its [[fundamental representation]] is due to

\begin{itemize}%
\item [[Peter West]], \emph{Generalised geometry, eleven dimensions and $E_{11}$}, J. High Energ. Phys. (2012) 2012: 18 (\href{http://arxiv.org/abs/1111.1642}{arXiv:1111.1642})


\item Alexander G. Tumanov, [[Peter West]], \emph{$E_{11}$ must be a symmetry of strings and branes}, Physics Letters B Volume 759, 10 August 2016, Pages 663--671 (\href{https://arxiv.org/abs/1512.01644}{arXiv:1512.01644})


\item Alexander G. Tumanov, [[Peter West]], \emph{$E_{11}$ in $11d$}, Physics Letters B Volume 758, 10 July 2016, Pages 278--285 (\href{https://arxiv.org/abs/1601.03974}{arXiv:1601.03974})



\end{itemize}
This way that elements of [[cosets]] of the [[semidirect product]] $E_{11}$ with its [[fundamental representation]] may encode [[equations of motion]] of [[11-dimensional supergravity]] follows previous considerations for [[Einstein equations]] in

\begin{itemize}%
\item [[Abdus Salam]], J. Strathdee, \emph{Nonlinear realizations. 1: The Role of Goldstone bosons, Phys. Rev. 184 (1969) 1750,}


\item [[Chris Isham]], [[Abdus Salam]], J. Strathdee, \emph{Spontaneous, breakdown of conformal symmetry}, Phys. Lett. 31B (1970) 300.


\item A. Borisov, V. Ogievetsky, \emph{Theory of dynamical affine and conformal symmetries as the theory of the gravitational field}, Theor. Math. Phys. 21 (1973) 1179-1188 (\href{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tmf&paperid=3902&option_lang=eng}{web})


\item V. Ogievetsky, \emph{Infinite-dimensional algebra of general covariance group as the closure of the finite dimensional algebras of conformal and linear groups}, Nuovo. Cimento, 8 (1973) 988.



\end{itemize}
Further developments of the proposed $E_{11}$ formulation of [[M-theory]] include

\begin{itemize}%
\item [[Peter West]], \emph{$E_{11}$, ten forms and supergravity}, JHEP0603:072,2006 (\href{http://arxiv.org/abs/hep-th/0511153}{arXiv:hep-th/0511153})


\item [[Fabio Riccioni]], [[Peter West]], \emph{Dual fields and $E_{11}$}, Phys.Lett.B645:286-292,2007 (\href{http://arxiv.org/abs/hep-th/0612001}{arXiv:hep-th/0612001})


\item [[Fabio Riccioni]], [[Peter West]], \emph{E(11)-extended spacetime and gauged supergravities}, JHEP 0802:039,2008 (\href{http://arxiv.org/abs/0712.1795}{arXiv:0712.1795})


\item [[Fabio Riccioni]], Duncan Steele, [[Peter West]], \emph{The E(11) origin of all maximal supergravities - the hierarchy of field-strengths}, JHEP 0909:095 (2009) (\href{http://arxiv.org/abs/0906.1177}{arXiv:0906.1177})


\item [[Eric Bergshoeff]], I. De Baetselier, T. Nutma, \emph{E(11) and the Embedding Tensor} (\href{http://arxiv.org/abs/0705.1304}{arXiv:0705.1304}, \href{http://mms.technologynetworks.net/posters/0364.pdf}{poster})


\item Guillaume Bossard, [[Axel Kleinschmidt]], Jakob Palmkvist, [[Christopher Pope]], [[Ergin Sezgin]], \emph{Beyond $E_{11}$} (\href{https://arxiv.org/abs/1703.01305}{arXiv:1703.01305})



\end{itemize}
Discussion of the [[semidirect product]] of $E_{11}$ with its $l_1$-[[representation]], and arguments that the [[charges]] of the [[M-theory super Lie algebra]] and in fact further brane charges may be identified inside $l_1$ originate in

\begin{itemize}%
\item [[Peter West]], \emph{$E_{11}$, $SL(32)$ and Central Charges}, Phys.Lett.B575:333- 342,2003 (\href{http://arxiv.org/abs/hep-th/0307098}{arXiv:hep-th/0307098})

\end{itemize}
and was further explored in

\begin{itemize}%
\item [[Axel Kleinschmidt]], [[Peter West]], \emph{Representations of $G^{+++}$ and the role of space-time}, JHEP 0402 (2004) 033 (\href{http://arxiv.org/abs/hep-th/0312247}{arXiv:hep-th/0312247})


\item [[Paul Cook]], [[Peter West]], \emph{Charge multiplets and masses for $E(11)$}, JHEP 11 (2008) 091 (\href{http://arxiv.org/abs/0805.4451}{arXiv:0805.4451})


\item [[Peter West]], \emph{$E_{11}$ origin of Brane charges and U-duality multiplets}, JHEP 0408 (2004) 052 (\href{http://arxiv.org/abs/hep-th/0406150}{arXiv:hep-th/0406150})



\end{itemize}
Relation to [[exceptional field theory]] is discussed in

\begin{itemize}%
\item Alexander G. Tumanov, [[Peter West]], \emph{$E_{11}$ and exceptional field theory} (\href{http://arxiv.org/abs/1507.08912}{arXiv:1507.08912})

\end{itemize}
Relation to [[Borcherds superalgebras]] is discussed in

\begin{itemize}%
\item Pierre Henry-Labordere, [[Bernard Julia]], Louis Paulot, \emph{Borcherds symmetries in M-theory}, JHEP 0204 (2002) 049 (\href{http://arxiv.org/abs/hep-th/0203070}{arXiv:hep-th/0203070})


\item [[Marc Henneaux]], [[Bernard Julia]], J\'e{}r\^o{}me Levie, \emph{$E_{11}$, Borcherds algebras and maximal supergravity} (\href{http://arxiv.org/abs/1007.5241}{arxiv:1007.5241})


\item Jakob Palmkvist, \emph{Tensor hierarchies, Borcherds algebras and $E_{11}$}, JHEP 1202 (2012) 066 (\href{http://arxiv.org/abs/1110.4892}{arXiv:1110.4892})



\end{itemize}
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