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\section*{U-duality}

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

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

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

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_tduality_and_sduality}{Relation to T-duality and S-duality}\dotfill \pageref*{relation_to_tduality_and_sduality} \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{}{$n=3$}\dotfill \pageref*{} \linebreak
\noindent\hyperlink{_2}{$n=4$}\dotfill \pageref*{_2} \linebreak
\noindent\hyperlink{_3}{$n=7$}\dotfill \pageref*{_3} \linebreak
\noindent\hyperlink{_4}{$n=8$}\dotfill \pageref*{_4} \linebreak
\noindent\hyperlink{_5}{$n=9$}\dotfill \pageref*{_5} \linebreak
\noindent\hyperlink{Referencesn10}{$n=10$}\dotfill \pageref*{Referencesn10} \linebreak
\noindent\hyperlink{_7}{$n=11$}\dotfill \pageref*{_7} \linebreak
\noindent\hyperlink{further_details}{Further details}\dotfill \pageref*{further_details} \linebreak
\noindent\hyperlink{relation_to_automorphic_forms}{Relation to automorphic forms}\dotfill \pageref*{relation_to_automorphic_forms} \linebreak


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

U-duality is a kind of [[duality in string theory]].

The [[KK-compactifications]] of [[11-dimensional supergravity]] to lower dimensional [[gauged supergravity]] theories have global/local [[gauge groups]] given by [[split real forms]] of the $E$-series of the [[exceptional Lie groups]].

Here the [[compact space|compact]] [[exceptional Lie groups]] form a series [[E8]],[[E7]], [[E6]]

\begin{displaymath}
E_8, E_7, E_6
\end{displaymath}
which is usefully thought of to continue as

\begin{displaymath}
E_5 := Spin(10), E_4 := SU(5), E_3 := SU(3) \times SU(2)
  \,.
\end{displaymath}
(Notice that $E_4$, $E_5$ and $E_6$ are also the traditional choices for [[phenomenology|phenomenologically]] realistic [[grand unified theories]], see there for more.)

The [[split real forms]] of this are traditionally written

\begin{displaymath}
E_{8(8)}, E_{7(7)}, E_{6(6)}
\end{displaymath}
and one sets

\begin{displaymath}
E_{5(5)} := Spin(5,5), E_{4(4)} := SL(5, \mathbb{R}), 
  E_{3(3)} := SL(3, \mathbb{R}) \times SL(2, \mathbb{R})
  \,.
\end{displaymath}
For instance the [[scalar fields]] in the field [[supermultiplet]] of $3 \leq d \leq 11$-dimensional supergravity have [[moduli spaces]] parameterized by the [[homogeneous spaces]]

\begin{displaymath}
E_{n(n)}/ K_n
\end{displaymath}
for

\begin{displaymath}
n = 11 - d
  \,,
\end{displaymath}
where $K_n$ is the [[maximal compact subgroup]] of $E_{n(n)}$:

\begin{displaymath}
K_8 \simeq Spin(16), K_7 \simeq SU(8), K_6 \simeq Sp(4)
\end{displaymath}
\begin{displaymath}
K_5 \simeq Spin(5) \times Spin(5), 
  K_4 \simeq Spin(5),
  K_3 \simeq SU(2) \times SO(2)
  \,.
\end{displaymath}
Therefore $E_{n(n)}$ acts as a [[global symmetry]] on the supergravity fields and more generally certain [[subgroups]] of it are ``gauged'' (have [[gauge fields]]) in [[gauged supergravity]] version.

So for instance maximal [[3d supergravity]] has global (and in fact also local, see there) gauge group given by (the [[split real form]] of) [[E8]].

This is no longer verbatim true for their [[UV-completion]] by the corresponding [[Kaluza-Klein mechanism|compactifications]] of [[string theory]] (e.g. [[type II string theory]] for [[type II supergravity]], etc.). Instead, on these a [[discrete group|discrete subgroup]]

\begin{displaymath}
E_{n(n)}(\mathbb{Z}) \hookrightarrow E_{n(n)}
\end{displaymath}
acts as global symmetry. This is called the \textbf{U-duality} group of the supergravity theory.

It has been argued that this pattern should continue in some way further to the remaining values $0 \leq d \lt 3$, with ``[[Kac-Moody groups]]'' [[E9]], [[E10]], [[E11]] corresponding to the [[Kac-Moody algebras]]

\begin{displaymath}
\mathfrak{e}_9, \mathfrak{e}_10, \mathfrak{e}_{11}
  \,.
\end{displaymath}
Continuing in the other direction to $d = 10$ ($n = 1$) connects to the [[T-duality]] group $O(d,d,\mathbb{Z})$ of [[type II string theory]].

[[!include U-duality -- table]]

More generally, there is a ``[[magic pyramid]]'' of super-[[Einstein-Yang-Mills theories]] and their U-duality groups.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_tduality_and_sduality}{}\subsubsection*{{Relation to T-duality and S-duality}}\label{relation_to_tduality_and_sduality}

U-duality may be understood as being the combination of [[T-duality]] for the compactification torus and [[S-duality]] of [[type IIB superstring theory]]. see (\hyperlink{West12}{West 12, section 17.5.4}).

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

\begin{itemize}%
\item [[exceptional field theory]]


\item [[duality in string theory]]

\begin{itemize}%
\item [[T-duality]]


\item [[S-duality]]


\item \textbf{U-duality}



\end{itemize}

\item [[M-theory on G2-manifolds]]


\item [[magic pyramid]]


\item [[mysterious duality]]



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

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

The hidden [[E7]]-symmetry of the [[KK-compactification]] of [[11-dimensional supergravity]] on a 7-dimensional fiber to maximal $N = 8$ [[4d supergravity]] was first noticed in

\begin{itemize}%
\item [[Eugene Cremmer]], [[Bernard Julia]], \emph{The $N = 8$ supergravity theory. I. The Lagrangian}, Phys. Lett. 80B (1978) 48


\item [[Eugene Cremmer]], [[Bernard Julia]], \emph{The $SO(8)$ Supergravity}, Nucl. Phys. B 159 (1979) 141 (\href{http://inspirehep.net/record/140465?ln=en}{spire})



\end{itemize}
and more generally in

\begin{itemize}%
\item [[Bernard de Wit]], [[Hermann Nicolai]], \emph{D = 11 Supergravity With Local SU(8) Invariance}, Nucl. Phys. B 274, 363 (1986) (\href{http://inspirehep.net/record/227409?ln=en}{spire}), \emph{Local SU(8) invariance in $d = 11$ supergravity} (\href{http://inspirehep.net/record/218601?ln=en}{spire})

\end{itemize}
The further U-duality groups for compactifications of 11d SuGra including [[E6]], [[E7]], [[E8]] were identified in

\begin{itemize}%
\item [[Eugene Cremmer]], \emph{Supergravities in 5 dimensions}, in Hawking, Rocek (eds.) \emph{Superspace and Supergravity}, Cambridge University Press 1981


\item [[Bernard Julia]], \emph{Group Disintegrations}, in Hawking, Rocek (eds.) \emph{Superspace and Supergravity}, Cambridge University Press 1981



\end{itemize}
Review includes (\hyperlink{ObersPioline98}{Obers-Pioline 98, section 4.2}), see also

\begin{itemize}%
\item [[Pietro Fré]], Floriana Gargiulo, Ksenya Rulik, Mario Trigiante, \emph{The general pattern of Kac Moody extensions in supergravity and the issue of cosmic billiards}, Nucl.Phys. B741 (2006) 42-82 (\href{http://arxiv.org/abs/hep-th/0507249}{arXiv:hep-th/0507249})

\end{itemize}
The concept and terminology of U-duality in string theory/M-theory originates in

\begin{itemize}%
\item [[Chris Hull]], [[Paul Townsend]], \emph{Unity of Superstring Dualities} Nucl.Phys.B438:109-137 (1995) (\href{http://arxiv.org/abs/hep-th/9410167}{arXiv:hep-th/9410167})

\end{itemize}
A textbook account is in

\begin{itemize}%
\item [[Peter West]], section 17.3 of \emph{[[Introduction to Strings and Branes]]}, Cambridge University Press 2012

\end{itemize}
Systematization of U-duality via the relation between [[supersymmetry and division algebras]] and the [[Freudenthal magic square]] is due to

\begin{itemize}%
\item [[Leron Borsten]], [[Michael Duff]], L. J. Hughes, S. Nagy, \emph{A magic square from Yang-Mills squared} (\href{http://arxiv.org/abs/1301.4176}{arXiv:1301.4176})


\item A. Anastasiou, [[Leron Borsten]], [[Michael Duff]], L. J. Hughes, S. Nagy, \emph{A magic pyramid of supergravities}, \href{http://arxiv.org/abs/1312.6523}{arXiv:1312.6523}



\end{itemize}
Quick surveys include

\begin{itemize}%
\item [[Jacques Distler]], \emph{Split real forms} (\href{http://golem.ph.utexas.edu/~distler/blog/archives/001213.html}{blog post}).

\end{itemize}
Reviews focusing on [[gauged supergravity]] and the non-discrete duality groups include

\begin{itemize}%
\item [[Henning Samtleben]], \emph{Lectures on Gauged Supergravity and Flux Compactifications} (\href{http://arxiv.org/abs/0808.4076}{arXiv:0808.4076})

\end{itemize}
with slides in

\begin{itemize}%
\item [[Henning Samtleben]], \emph{Gauged supergravity and U-duality}, 2007 (\href{http://www.desy.de/uni-th/stringth/ggfl/talks/Samtleben.pdf}{pdf})

\end{itemize}
Reviews with more [[M-theory]] lore include

\begin{itemize}%
\item N.A. Obers B. Pioline, \emph{U-duality and M-Theory}, Phys.Rept. 318 (1999) 113-225 (\href{http://arxiv.org/abs/hep-th/9809039}{arXiv:hep-th/9809039})


\item Shun'ya Mizoguchi, Germar Schroeder, \emph{On Discrete U-duality in M-theory}, Class.Quant.Grav. 17 (2000) 835-870 (\href{http://arxiv.org/abs/hep-th/9909150}{arXiv:hep-th/9909150})


\item Diederik Roest, \emph{M-theory and Gauged Supergravities}, Fortsch.Phys.53:119-230,2005 (\href{http://arxiv.org/abs/hep-th/0408175}{arXiv:hep-th/0408175})



\end{itemize}
Discussion in line with the [[F-theory]] perspective on the $SL(2,\mathbb{Z})$-[[S-duality]] -- namely ``[[F'-theory]]'' -- is in

\begin{itemize}%
\item Alok Kumar, [[Cumrun Vafa]], \emph{U-Manifolds}, Phys.Lett. B396 (1997) 85-90 (\href{http://arxiv.org/abs/hep-th/9611007}{arXiv:hep-th/9611007})

\end{itemize}
Discussion of [[11-dimensional supergravity]] in a form that exhibits the higher U-duality groups already before [[KK-compactification]], via a kind of [[exceptional generalized geometry]],is in

\begin{itemize}%
\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}
\hypertarget{}{}\subsubsection*{{$n=3$}}\label{}

The case of $SL(3,\mathbb{Z}) \times SL(2,\mathbb{Z})$ in [[8d supergravity]] is discussed in

\begin{itemize}%
\item [[James Liu]], [[Ruben Minasian]], \emph{U-branes and $T^3$ fibrations}, Nucl.Phys. B510 (1998) 538-554 (\href{http://arxiv.org/abs/hep-th/9707125}{arXiv:hep-th/9707125})

\end{itemize}
\hypertarget{_2}{}\subsubsection*{{$n=4$}}\label{_2}

The case of $SL(5,\mathbb{Z})$ in [[7d supergravity]] from [[M-theory]] is discussed in

\begin{itemize}%
\item [[Moshe Rozali]], \emph{Matrix Theory and U-Duality in Seven Dimensions}, Phys.Lett. B400 (1997) 260-264 (\href{http://arxiv.org/abs/hep-th/9702136}{arXiv:hep-th/9702136})

\end{itemize}
\hypertarget{_3}{}\subsubsection*{{$n=7$}}\label{_3}

The $E_{7(7)}$-symmetry was first discussed in

\begin{itemize}%
\item [[Bernard de Wit]], [[Hermann Nicolai]], \emph{$D = 11$ Supergravity With Local $SU(8)$ Invariance}, Nucl. Phys. B 274, 363 (1986)

\end{itemize}
\hypertarget{_4}{}\subsubsection*{{$n=8$}}\label{_4}

The case of $E_{8(8)}$ is discussed in

\begin{itemize}%
\item [[Hermann Nicolai]], \emph{$D = 11$ Supergravity with Local $SO(16)$ Invariance} , Phys. Lett. B 187, 316 (1987).


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



\end{itemize}
\hypertarget{_5}{}\subsubsection*{{$n=9$}}\label{_5}

The case of [[E9]] is discussed in

\begin{itemize}%
\item [[François Englert]], Laurent Houart, [[Axel Kleinschmidt]], [[Hermann Nicolai]], Nassiba Tabti, \emph{An E9 multiplet of BPS states}, JHEP 0705:065,2007 (\href{http://arxiv.org/abs/hep-th/0703285}{arXiv:hep-th/0703285})

\end{itemize}
\hypertarget{Referencesn10}{}\subsubsection*{{$n=10$}}\label{Referencesn10}

The case of [[E10]] is discussed for [[boson|bosonic]] degrees of freedom in

\begin{itemize}%
\item [[Thibault Damour]], [[Marc Henneaux]], [[Hermann Nicolai]], \emph{$E(10)$ and a `small tension expansion' of M theory}, Phys. Rev. Lett. 89, 221601 (2002) (\href{http://arxiv.org/abs/hep-th/0207267}{arXiv:hep-th/0207267});


\item [[Axel Kleinschmidt]], [[Hermann Nicolai]], \emph{$E(10)$ and $SO(9,9)$ invariant supergravity}, JHEP 0407, 041 (2004) (\href{http://arxiv.org/abs/hep-th/0407101}{arXiv:hep-th/0407101})



\end{itemize}
and for fermionic degrees of freedom in [[supergravity|supersymmetric]] [[quantum cosmology]] in

\begin{itemize}%
\item [[Thibault Damour]], \emph{Quantum supersymmetric Bianchi IX Cosmology and its hidden Kac-Moody Structure}, talk at \href{http://www.lestudium-ias.com/#!registration-conference-21-may-gibbons/ci2z}{Gravitation, Solitons and Symmetries} ([[DamourSQC14.pdf:file]])

\end{itemize}
Review includes

\begin{itemize}%
\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 [[Hermann Nicolai]], \emph{On Exceptional Geometry and Supergravity}, talk at \href{http://www.lestudium-ias.com/#!registration-conference-21-may-gibbons/ci2z}{Gravitation, Solitons and Symmetries} ([[NicolaiTalk14.pdf:file]])



\end{itemize}
Discussion of [[phenomenology]]:

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


\item Krzysztof A. Meissner, [[Hermann Nicolai]], \emph{Standard Model Fermions and Infinite-Dimensional R-Symmetries}, Phys. Rev. Lett. 121, 091601 (2018) (\href{https://arxiv.org/abs/1804.09606}{arXiv:1804.09606})


\item Krzysztof A. Meissner, [[Hermann Nicolai]], \emph{Planck Mass Charged Gravitino Dark Matter}, Phys. Rev. D 100, 035001 (2019) (\href{https://arxiv.org/abs/1809.01441}{arXiv:1809.01441})



\end{itemize}
\hypertarget{_7}{}\subsubsection*{{$n=11$}}\label{_7}

The case of of [[E11]] is discussed in

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


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



\end{itemize}
\hypertarget{further_details}{}\subsubsection*{{Further details}}\label{further_details}

A careful discussion of the [[topology]] of the [[Kac-Moody group|Kac-Moody]] U-duality groups is in

\begin{itemize}%
\item [[Arjan Keurentjes]], \emph{The topology of U-duality (sub-)groups} (\href{http://arxiv.org/abs/hep-th/0309106}{arXiv:hep-th/0309106})


\item [[Arjan Keurentjes]], \emph{U-duality (sub-)groups and their topology} (\href{http://arxiv.org/abs/hep-th/0312134}{arXiv:hep-th/0312134})



\end{itemize}
A discussion in the context of [[generalized complex geometry]] / [[exceptional generalized complex geometry]] is in

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


\item Nicholas Houston, \emph{Supergravity and Generalized Geometry} Thesis (2010) (\href{https://workspace.imperial.ac.uk/theoreticalphysics/Public/MSc/Dissertations/2010/Nicholas%20Houston%20Dissertation.pdf}{pdf})



\end{itemize}
General discussion of the [[Kac-Moody groups]] arising in this context is for instance in

\begin{itemize}%
\item Philipp Fleig, [[Axel Kleinschmidt]], \emph{Eisenstein series for infinite-dimensional U-duality groups} (\href{http://arxiv.org/abs/1204.3043}{arXiv:1204.3043}, [[Kleinschmidt12.pdf:file]])

\end{itemize}
\hypertarget{relation_to_automorphic_forms}{}\subsubsection*{{Relation to automorphic forms}}\label{relation_to_automorphic_forms}

String theory [[partition functions]] as [[automorphic forms]] for U-duality groups are discussed in

\begin{itemize}%
\item [[Michael Green]], Jorge G. Russo, Pierre Vanhove, \emph{Automorphic properties of low energy string amplitudes in various dimensions} (\href{http://arxiv.org/abs/1001.2535}{arXiv:1001.2535})

\end{itemize}
[[!redirects U-duality]] [[!redirects U-duality groups]]



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