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\section*{D=5 supergravity}

[[!redirects 5-dimensional supergravity]] [[!redirects 5-dimensional supergravity]]

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

\hypertarget{gravity}{}\paragraph*{{Gravity}}\label{gravity}

[[!include gravity contents]]

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

[[!include string theory - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{5dChernSimonsTerm}{5d Chern-Simons term}\dotfill \pageref*{5dChernSimonsTerm} \linebreak
\noindent\hyperlink{black_holes_and_black_rings}{Black holes and black rings}\dotfill \pageref*{black_holes_and_black_rings} \linebreak
\noindent\hyperlink{ViaCYCompactificationOf11dSupergravity}{Via Calabi-Yau compactification of 11d supergravity}\dotfill \pageref*{ViaCYCompactificationOf11dSupergravity} \linebreak
\noindent\hyperlink{uduality}{U-duality}\dotfill \pageref*{uduality} \linebreak
\noindent\hyperlink{_supersymmetry}{$N = 2$ supersymmetry}\dotfill \pageref*{_supersymmetry} \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{via_mtheory_on_calabiyau_3folds}{Via M-theory on Calabi-Yau 3-folds}\dotfill \pageref*{via_mtheory_on_calabiyau_3folds} \linebreak
\noindent\hyperlink{via_type_iib_theory}{Via type IIB theory}\dotfill \pageref*{via_type_iib_theory} \linebreak
\noindent\hyperlink{ReferencesGauged}{Gauged sugra}\dotfill \pageref*{ReferencesGauged} \linebreak
\noindent\hyperlink{ReferencesHWCompactification}{Horava-Witten compactification}\dotfill \pageref*{ReferencesHWCompactification} \linebreak
\noindent\hyperlink{black_hole_solutions}{Black hole solutions}\dotfill \pageref*{black_hole_solutions} \linebreak


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

[[supergravity]] in [[dimension]] 5. For $N = 1$ this arises from [[11-dimensional supergravity]] by [[KK-compactification]] on a [[Calabi-Yau manifold]] of complex dimension 3 (see at \emph{[[M-theory on Calabi-Yau manifolds]]}), hence serves as the low-energy [[effective field theory]] of the strong-coupling version of Calabi-Yau compactifications of [[type IIA string theory]] (see [[supersymmetry and Calabi-Yau manifolds]])

\begin{displaymath}
\itexarray{
    11d \; SuGra\; on\; S^1 \times Y_6 \times X_4
      &\longrightarrow&
    5d \; SuGra\; on\; S^1 \times X_4 && strong \; coupling
    \\
    \downarrow && \downarrow
    \\
    10d\; tpye \;IIA\; Sugra\; on \; Y_6 \times X_4
      &\longrightarrow&
    10d\; Sugra\; on \; X_4 && weak \; coupling    
  }
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{5dChernSimonsTerm}{}\subsubsection*{{5d Chern-Simons term}}\label{5dChernSimonsTerm}

This theory has a 2-form field strength $F_2$, locally $F_2 = d A$, with a [[5d Chern-Simons theory]] [[action functional]] locally of the form $\propto \int_X F_2 \wedge F_2 \wedge A$ (e.g. \hyperlink{CastellaniDAuriaFre}{Castellani-D'Auria-Fre (III.5.70)}, \hyperlink{GauntlettMyersTownsend98}{Gauntlet-Myers-Townsend 98, p. 3}, \hyperlink{GGHPR02}{GGHPR 02 (2.1)}, \hyperlink{BonettiGrimmHohenegger13}{Bonetti-Grimm-Hohenegger 13}). Hence its [[equation of motion]] is of the non-linear form

\begin{displaymath}
d F_3 = F_2 \wedge F_2
\end{displaymath}
with $F_3 \coloneqq \star F_2$ the [[Hodge dual]] of $F_2$ (\hyperlink{GGHPR02}{GGHPR 02 (2.2)}).

This is reflected in the corresponding cochains on [[super Minkowski spacetime]]

\begin{displaymath}
\mu_{D0,5d} = \overline{\psi}_A \psi_A
  \phantom{AAA} 
  \mu_{string,5d} = \overline{\psi}_A\Gamma_a \psi_A  \wedge e^a
\end{displaymath}
satisfying

\begin{displaymath}
d \mu_{string,5d} = \mu_{D0,5d} \wedge \mu_{D0,5d}
  \,.
\end{displaymath}
due to the [[Fierz identity]] in \hyperlink{CastellaniDAuriaFre}{Castellani-D'Auria-Fré 91 (III.5.50a)}, \href{Fierz+identity#QuadraticFierzIdentitiesIn5d}{this example}:

$\backslash$begin\{imagefromfile\} ``file\_name'': ``FierzIdentitiesForBraneCocyclesIn5d.png'', ``width'': 600 $\backslash$end\{imagefromfile\}

(the other Fierz identity (III.5.50a) gives the [[membrane]] cocycle $\mu_{membrane,5d} \coloneqq \frac{i}{2}\overline{\psi}_A \Gamma_{a b}  \psi \wedge e^a \wedge e^b$, $d \mu_{membrane,5d} = 0$, that appears already in the old [[brane scan]]. )

This is a lower dimensional analogue to the situation for the [[C-field]] $G_4$ (locally $G_4 = d C$) in [[11-dimensional supergravity]], which has a Chern-Simons term locally of the form $\propto \int G_4 \wedge G_4 \wedge C$ and hence the equation of motion

\begin{displaymath}
d G_7 
  \;=\; 
  -\tfrac{1}{2}G_4 \wedge G_4
\end{displaymath}
with $G_7 = \star G_4$.

\hypertarget{black_holes_and_black_rings}{}\subsubsection*{{Black holes and black rings}}\label{black_holes_and_black_rings}

The first [[black ring]] solution in 5d sugra was found in (\hyperlink{ElvangEmparanMateosReall04}{Elvang-Emparan-Mateos-Reall 04}, \hyperlink{ElvangEmparanMateosReall05}{Elvang-Emparan-Mateos-Reall 05}).

Supersymmetric black holes exist precisely only in dimensions 4 and 5 (\hyperlink{GauntlettMyersTownsend98}{Gauntlett-Myers-Townsend 98}). These play a key role in the discussion of [[black holes in string theory]].

(There are supersymmetric particle-like solutions of $d \gt 5$ supergravity theories that are sometimes called black holes, but these are always singular. There are also supersymmetric black holes in $d = 3$, but the spacetime in that case is asymptotically [[anti-de Sitter spacetime]] rather than asymptotically flat. Of course, there are non-singular supersymmetric [[black brane]] solutions in various $d \geq 4$ supergravity theories but these are neither `particle-like' nor, strictly speaking, asymptotically flat.)

\hypertarget{ViaCYCompactificationOf11dSupergravity}{}\subsubsection*{{Via Calabi-Yau compactification of 11d supergravity}}\label{ViaCYCompactificationOf11dSupergravity}

Discussion of 5d supegravity as a [[KK-compactification]] of [[11-dimensional supergravity]] on a [[Calabi-Yau manifold]] of complex dimension 3 ([[M-theory on Calabi-Yau manifolds]]) is discussed in

(\hyperlink{HullTownsend95}{Hull-Townsend 95, p.30-31}, \hyperlink{CadavidCeresoleDAuriaFerrara95}{Cadavid-Ceresole-D'Auria-Ferrara 95} \hyperlink{FerraraKhuriaMinasian96}{Ferrara-Khuria-Minasian 96}, \hyperlink{FerraraMinasianSagnotti96}{Ferrara-Minasian-Sagnotti 96}). See also (\hyperlink{MizoguchiOhta98}{Mizoguchi-Ohta 98}).

\hypertarget{uduality}{}\subsubsection*{{U-duality}}\label{uduality}

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

\hypertarget{_supersymmetry}{}\subsubsection*{{$N = 2$ supersymmetry}}\label{_supersymmetry}

Consider super Lie algebra cocoycles on $N =2$ 5d [[super-Minkowski spacetime]] (as in the [[brane scan]]).

With the notation as used at \emph{\href{super%20Minkowski%20spacetime#CanonicalCoordinates}{super Minkowski spacetime -- Canonical coordinates}}, there are now two copies of spinor-valued 1-forms, denoted $\psi_1$ and $\psi_2$. We use indices of the form $A,B, \cdots$ for these. Then the non-trivial bit of the [[Chevalley-Eilenberg algebra]] differential for $N = 2$, $d = 5$ [[super Minkowski spacetime]] is

\begin{displaymath}
d_{CE} e^a = - \tfrac{i}{2} \overline{\psi}_A \wedge \Gamma^a \psi_A
\end{displaymath}
where summation over repeated indices is understood.

There is a [[Fierz identity]]

\begin{displaymath}
\overline{\psi}_A \wedge \psi_A \wedge \overline{\psi}_B \wedge \psi_B
  \;=\;
  \overline{\psi}_A \wedge \Gamma_a \psi_A
  \wedge \overline{\psi}_B \wedge \Gamma^a \psi_B
  \,.
\end{displaymath}
(\hyperlink{CastellaniDAuriaFre}{Castellani-D'Auria-Fr\'e{} (III.5.50a)})

This implies that

\begin{displaymath}
d_{CE} (\overline{\psi}_A \Gamma^a \psi_A \wedge e_a)
  \;\propto\;
  (\overline{\psi}_A \wedge \Gamma^a \psi_A)
    \wedge
  (\overline{\psi}_B \wedge \Gamma^a \psi_B)
  \,.
\end{displaymath}
There is a 4-cocycle of the form

\begin{displaymath}
\mu_2 
    =
  \epsilon^{A B} \overline{\psi}_A \wedge \Gamma^{a b} \psi_B 
  \wedge e_a \wedge e_b
  \,.
\end{displaymath}
(\hyperlink{CastellaniDAuriaFre}{Castellani-D'Auria-Fr\'e{} (III.5.50b), (III.5.53c)})

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

\begin{itemize}%
\item [[5-dimensional Chern-Simons theory]]


\item [[11-dimensional supergravity]]


\item 10-dimensional [[type II supergravity]], [[heterotic supergravity]]


\item [[7-dimensional supergravity]]


\item \textbf{5-dimensional supergravity}


\item [[4-dimensional supergravity]]


\item [[3-dimensional supergravity]]



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

Construction of 5d [[gauged supergravity]] via [[D'Auria-Fré formulation of supergravity]] is in

\begin{itemize}%
\item Laura Andrianopoli, Francesco Cordaro, [[Pietro Fré]], Leonardo Gualtieri, \emph{Non-Semisimple Gaugings of D=5 N=8 Supergravity and FDA.s}, Class.Quant.Grav. 18 (2001) 395-414 (\href{http://arxiv.org/abs/hep-th/0009048}{arXiv:hep-th/0009048})

\end{itemize}
surveyed in

\begin{itemize}%
\item Laura Andrianopoli, Francesco Cordaro, [[Pietro Fré]], \emph{Non-Semisimple Gaugings of D=5 N=8 Supergravity}, Fortsch.Phys.49:511-518,2001 (\href{http://arxiv.org/abs/hep-th/0012203}{arXiv:hep-th/0012203})

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

General discussion includes

\begin{itemize}%
\item [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], chapter III.5 of \emph{[[Supergravity and Superstrings - A Geometric Perspective]]}, World Scientific (1991)

(in [[D'Auria-Fre formulation of supergravity]])


\item W. D. Linch III, Markus A. Luty, J. Phillips, \emph{Five dimensional supergravity in $N = 1$ superspace}, Phys.Rev.D68:025008,2003 (\href{https://arxiv.org/abs/hep-th/0209060}{arXiv:hep-th/0209060})


\item [[Jerome Gauntlett]], [[Jan Gutowski]], [[Christopher Hull]], Stathis Pakis, [[Harvey Reall]], \emph{All supersymmetric solutions of minimal supergravity in five dimensions}, Class.Quant.Grav. 20 (2003) 4587-4634 (\href{http://arxiv.org/abs/hep-th/0209114}{arXiv:hep-th/0209114})


\item Sorin Cucu, \emph{From M-theory to D=5 supergravity and duality-symmetric theories} (\href{http://arxiv.org/abs/hep-th/0310105}{arXiv:hep-th/0310105})


\item [[Eric Bergshoeff]], Sorin Cucu, Tim de Wit, Jos Gheerardyn, Stefan Vandoren, [[Antoine Van Proeyen]], \emph{$N=2$ supergravity in five dimensions revisited} (\href{http://arxiv.org/abs/hep-th/0403045}{arXiv:hep-th/0403045})


\item [[Katrin Becker]], [[Melanie Becker]], Daniel Butter, William D. Linch III, Stephen Randall, \emph{Five-dimensional Supergravity in N = 1/2 Superspace} (\href{https://arxiv.org/abs/1909.09208}{arXiv:1909.09208})



\end{itemize}
\hypertarget{via_mtheory_on_calabiyau_3folds}{}\subsubsection*{{Via M-theory on Calabi-Yau 3-folds}}\label{via_mtheory_on_calabiyau_3folds}

Discussion via [[KK-compactification]] as [[M-theory on Calabi-Yau manifolds]] includes

\begin{itemize}%
\item [[Chris Hull]], [[Paul Townsend]], pages 30 and 31 of \emph{Unity of Superstring Dualities}, Nucl.Phys.B438:109-137,1995 (\href{https://arxiv.org/abs/hep-th/9410167}{arXiv:hep-th/9410167})


\item \{CadavidCeresoleDAuriaFerrara95\} A.C. Cadavid, A. Ceresole, [[Riccardo D'Auria]], [[Sergio Ferrara]], \emph{11-Dimensional Supergravity Compactified on Calabi-Yau Threefolds} (\href{https://arxiv.org/abs/hep-th/9506144}{arXiv:hep-th/9506144})


\item [[Sergio Ferrara]], Ramzi R. Khuria, [[Ruben Minasian]], \emph{M-theory on a Calabi-Yau manifold}, Phys.Lett.B375:81-88,1996 (\href{https://arxiv.org/abs/hep-th/9602102}{arXiv:hep-th/9602102})


\item [[Sergio Ferrara]], [[Ruben Minasian]], [[Augusto Sagnotti]], \emph{Low-Energy Analysis of M and F Theories on Calabi-Yau Threefolds}, Nucl.Phys. B474 (1996) 323-342 (\href{https://arxiv.org/abs/hep-th/9604097}{arXiv:hep-th/9604097})


\item S. Mizoguchi, N. Ohta, \emph{More on the Similarity between $D=5$ Simple Supergravity and M Theory}, Phys.Lett. B441 (1998) 123-132 (\href{https://arxiv.org/abs/hep-th/9807111}{arXiv:hep-th/9807111})



\end{itemize}
Further discussion of the [[5d Chern-Simons theory|5d Chern-Simons term]] includes

\begin{itemize}%
\item Federico Bonetti, [[Thomas Grimm]], [[Stefan Hohenegger]], \emph{One-loop Chern-Simons terms in five dimensions} (\href{http://arxiv.org/abs/1302.2918}{arXiv:1302.2918})

\end{itemize}
(one-loop corrections).

\hypertarget{via_type_iib_theory}{}\subsubsection*{{Via type IIB theory}}\label{via_type_iib_theory}

\begin{itemize}%
\item Arnaud Baguet, [[Olaf Hohm]], [[Henning Samtleben]], \emph{Consistent Type IIB Reductions to Maximal 5D Supergravity} (\href{https://arxiv.org/abs/1506.01385}{arXiv:1506.01385})

\end{itemize}
\hypertarget{ReferencesGauged}{}\subsubsection*{{Gauged sugra}}\label{ReferencesGauged}

The maximal 5d [[gauged supergravity]] was first constructed in

\begin{itemize}%
\item M. Pernici, K. Pilch, [[Peter van Nieuwenhuizen]], \emph{Gauged $N=8$ $D=5$ Supergravity}, Nucl.Phys. B259 (1985) 460 (\href{https://inspirehep.net/record/16067?ln=en}{spire})


\item M. Gunaydin, L.J. Romans, N.P. Warner, \emph{Compact and Noncompact Gauged Supergravity Theories in Five-Dimensions}, Nucl.Phys. B272 (1986) 598-646 (\href{https://inspirehep.net/record/219727?ln=en}{spire})



\end{itemize}
See (\hyperlink{ACFG01}{ACFG 01}).

\begin{itemize}%
\item Murat Gunaydin, Marco Zagermann, \emph{The Gauging of Five-dimensional, $N=2$ Maxwell-Einstein Supergravity Theories coupled to Tensor Multiplets}, Nucl.Phys.B572:131-150,2000 (\href{https://arxiv.org/abs/hep-th/9912027}{arXiv:hep-th/9912027})


\item Murat Gunaydin, Marco Zagermann, \emph{The Vacua of 5d, $N=2$ Gauged Yang-Mills/Einstein/Tensor Supergravity: Abelian Case}, Phys.Rev.D62:044028,2000 (\href{http://arxiv.org/abs/hep-th/0002228}{arXiv:hep-th/0002228})


\item A. Ceresole, [[Gianguido Dall'Agata]], \emph{General matter coupled $N=2$, $D=5$ gauged supergravity}, Nucl.Phys. B585 (2000) 143-170 (\href{http://arxiv.org/abs/hep-th/0004111}{arXiv:hep-th/0004111})


\item [[John Ellis]], Murat Gunaydin, Marco Zagermann, \emph{Options for Gauge Groups in Five-Dimensional Supergravity}, JHEP 0111:024,2001 (\href{http://arxiv.org/abs/hep-th/0108094}{arXiv:hep-th/0108094})



\end{itemize}
\hypertarget{ReferencesHWCompactification}{}\subsubsection*{{Horava-Witten compactification}}\label{ReferencesHWCompactification}

Discussion of [[KK-compactification]] on $S^1/(\mathbb{Z}/2)$-orbifolds (the version of [[Horava-Witten theory]] after dimensional reduction) is discussed in

\begin{itemize}%
\item Filipe Paccetti Correia, Michael G. Schmidt, Zurab Tavartkiladze, \emph{4D Superfield Reduction of 5D Orbifold SUGRA and Heterotic M-theory} (\href{https://arxiv.org/abs/hep-th/0602173}{arXiv:hep-th/0602173})

\end{itemize}
\hypertarget{black_hole_solutions}{}\subsubsection*{{Black hole solutions}}\label{black_hole_solutions}

Discussion of lifts of 4d black holes to 5d black holes and [[black rings]] and embedding as [[black holes in string theory]] includes

\begin{itemize}%
\item [[Jerome Gauntlett]], R.C. Myers, [[Paul Townsend]], \emph{Black Holes of D=5 Supergravity}, Class.Quant.Grav.16:1-21,1999 (\href{http://arxiv.org/abs/hep-th/9810204}{arXiv:hep-th/9810204})


\item Henriette Elvang, Roberto Emparan, David Mateos, [[Harvey Reall]], \emph{A supersymmetric black ring}, Phys.Rev.Lett.93:211302,2004 (\href{http://arxiv.org/abs/hep-th/0407065}{arXiv:hep-th/0407065})


\item Henriette Elvang, Roberto Emparan, David Mateos, [[Harvey Reall]], \emph{Supersymmetric 4D Rotating Black Holes from 5D Black Rings}, JHEP0508:042,2005 (\href{http://arxiv.org/abs/hep-th/0504125}{arXiv:hep-th/0504125})


\item I. Bena, [[Per Kraus]], \emph{Microscopic description of black rings in AdS/CFT} JHEP 12 (2004) 070 (\href{http://arxiv.org/abs/hep-th/0408186}{hep-th/0408186})


\item I. Bena and P. Kraus, \emph{Microstates of the D1-D5-KK system} Phys. Rev. D72 (2005) 025007 (\href{http://arxiv.org/abs/hep-th/0503053}{hep-th/0503053})


\item [[Davide Gaiotto]], [[Andrew Strominger]], and X. Yin, \emph{5D black rings and 4D black holes} JHEP 02 (2006) 023 (\href{http://arxiv.org/abs/hep-th/0504126}{hep-th/0504126})


\item [[Davide Gaiotto]], [[Andrew Strominger]], and X. Yin, \emph{New connections between 4D and 5D black holes}, JHEP 02 (2006) 024 (\href{https://arxiv.org/abs/hep-th/0503217}{hep-th/0503217})


\item Alejandra Castro, Joshua L. Davis, [[Per Kraus]], Finn Larsen, \emph{String Theory Effects on Five-Dimensional Black Hole Physics} (\href{http://arxiv.org/abs/0801.1863}{arXiv:0801.1863})


\item Minkyu Park, Masaki Shigemori, \emph{Codimension-2 Solutions in Five-Dimensional Supergravity}, JHEP 1510 (2015) 011 (\href{http://arxiv.org/abs/1505.05169}{arXiv:1505.05169})



\end{itemize}
Review includes

\begin{itemize}%
\item [[Per Kraus]], \emph{Lectures on black holes and the $AdS_3$ / $CFT_2$ correspondence} (\href{http://arxiv.org/abs/hep-th/0609074}{arXiv:hep-th/0609074})

\emph{Stringy black holes in five dimensions}, 2007 (\href{http://hep.ps.uci.edu/~arajaram/Irvine.07.pdf}{pdf slides})



\end{itemize}
[[!redirects 5d supergravity]]



\end{document}