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\begin{document}

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\section*{12-dimensional supergravity}

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

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

[[!include gravity contents]]

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

[[!include physicscontents]]

\hypertarget{supergeometry}{}\paragraph*{{Super-Geometry}}\label{supergeometry}

[[!include supergeometry - contents]]

\hypertarget{fields_and_quanta}{}\paragraph*{{Fields and quanta}}\label{fields_and_quanta}

[[!include fields and quanta - table]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{The21Brane}{The $2+1$-brane in $10+2$ dimensions}\dotfill \pageref*{The21Brane} \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{on_the_brane_in__dimensions}{On the $2+1$-brane in $10+2$ dimensions}\dotfill \pageref*{on_the_brane_in__dimensions} \linebreak
\noindent\hyperlink{on_supergravity_in__dimensions}{On supergravity in $9 + 3$ dimensions}\dotfill \pageref*{on_supergravity_in__dimensions} \linebreak


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

There is a sensible [[theory (physics)|theory]] of [[supergravity]] in a total of 12 [[spacetime]] [[dimensions]]. Even though this requires an exotic non-[[Lorentzian signature]] of $(10,2)$ (hence with a ``2-dimensional [[time]]'') it has been argued that this is a better starting point for obtaining low-dimensional [[supergravity]] theory by [[KK-compactification]], since it yields some lower-dimensional theories that are missed when starting with [[11-dimensional supergravity]], notably [[type IIB supergravity]] in 10 dimensions, hence relates to [[F-theory]] as [[11-dimensional supergravity]] relates to [[M-theory]] (e.g. \hyperlink{Nishino97b}{Nishino 97b}, \hyperlink{Hewson97}{Hewson 97}). (A theory in $(9,3)$ signature has also been proposed in (\hyperlink{Kriz05}{Kriz 05}).)

It is an oft-repeated [[folklore]] that the highest number of [[spacetime]] [[dimensions]] for [[supergravity]] to make sense is 11, realized by [[11-dimensional supergravity]]. However, there are some assumptions that go into this conclusion. First of all, the argument goes that after [[KK-compactification]] to 4-dimensions there must not appear [[supermultiplets]] with [[mass]]-less fields of [[spin]] $\gt 2$, since another [[folklore]] argument states that [[quantum field theory]] in $3+1$ dimensions with fields of spin larger than 2 is inconsistent.

(This in turn needs further qualification: Consistent [[quantum field theory]] with an \emph{infinite tower} of higher spin fields \emph{is} consistent, this is called \emph{[[higher spin gauge theory]]} arising as the vanishing [[string tension]]-[[limit of a sequence|limit]] of [[string field theory]]. Ever since this discovery, the modified [[folklore]] is that field theories with a finite number of higher spin fields is inconsistent.)

Since acting with a [[supersymmetry]] generator on elements of a [[supermultiplet]] increases spin by 1/2, this argument requires that there are at most $(2 - (-2)) \times 2 = 8$ super charges in (3+1)d, hence corresponding to [[N=8 d=4 supergravity]].

This, in turn, requires, by the rules of [[KK-compactification]], that

\begin{enumerate}%
\item there be only a single supercharge in dimension $10+1$, since the [[irreducible representation|irreducible]] [[real spin representation]] of $Spin(10,1)$ has real dimension 32, which [[branching rule|branches]] as $\mathbf{32} \mapsto 8 \cdot \mathbf{4}$ under $Spin(3,1) \hookrightarrow Spin(10,1)$;


\item there cannot be any supercharge in dimension $11+1$, since the [[irreducible representation|irreducible]] [[real spin representation]] of $Spin(11,1)$ has real dimension 64, which [[branching rule|branches]] as $\mathbf{64} \mapsto 16 \cdot \mathbf{4}$ under $Spin(3,1) \hookrightarrow Spin(11,1)$.



\end{enumerate}
However, the second conclusion here is evaded by a change of spacetime signature: The [[irreducible representation|irreducible]] [[real spin representation]] of $Spin(10,2)$ still happens to be of dimension 32 and still [[branching rule|branches]] as $\mathbf{32} \mapsto 8 \cdot \mathbf{4}$.

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

\hypertarget{The21Brane}{}\subsubsection*{{The $2+1$-brane in $10+2$ dimensions}}\label{The21Brane}

There is supposed to be a consistent fundamental [[super p-brane]] on $10+2$-dimensional supergravity backgrounds, whose [[double dimensional reduction]] yields the [[M2-brane]] in [[11-dimensional supergravity]] and further the [[superstrings]] not just of [[type IIA supergravity]] but also (?) of [[type IIB supergravity]]. The [[worldvolume]] of this [[p-brane]] has 4 spacetime dimensions with signature $(2,2)$. Therefore some authors refer to this as a ``2+2''-brane, even though this does not mesh well with the naming convention of $p$-branes in Lorentzian signature. Since Lorentzian $p$-branes have $(p+1)$-dimensional worldvolume, the systematic naming here would be ``2+1''-brane.

See (\hyperlink{BlencoweDuff88}{Blencowe-Duff 88, section 7}, \hyperlink{HewsonPerry96}{Hewson-Perry 96}, \hyperlink{Nishino97b}{Nishino 97b})

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

\begin{itemize}%
\item [[F-theory]]

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

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

\begin{itemize}%
\item [[Leonardo Castellani]], [[Pietro Fré]], F. Giani, K. Pilch, [[Peter van Nieuwenhuizen]], \emph{Beyond $d=11$ Supergravity and Cartan Integrable Systems}, Phys.Rev. D26 (1982) 1481 (\href{http://inspirehep.net/record/11999}{spire:11999})


\item [[Itzhak Bars]], \emph{Supersymmetry, p-brane duality and hidden space-time dimensions}, Phys. Rev. D54, 5203 (1996) (\href{https://arxiv.org/abs/hep-th/9604139}{arXiv: hep-th/9604139}).


\item [[Itzhak Bars]], \emph{S-Theory}, Phys.Rev. D55 (1997) 2373-2381 (\href{https://arxiv.org/abs/hep-th/9607112}{arXiv:hep-th/9607112})


\item [[Hitoshi Nishino]], \emph{Supergravity in 10 + 2 Dimensions as Consistent Background for Superstring}, (\href{https://arxiv.org/abs/hep-th/9703214}{arXiv:hep-th/9703214})


\item [[Hitoshi Nishino]], \emph{N=2 Chiral Supergravity in (10 + 2)-Dimensions As Consistent Background for Super (2 + 2)-Brane}, Phys. Lett. B437 (1998) 303-314 (\href{https://arxiv.org/abs/hep-th/9706148}{arXiv:hep-th/9706148})


\item Stephen Hewson, \emph{An approach to F-theory}, Nucl. Phys. B534 (1998) 513-530 (\href{https://arxiv.org/abs/hep-th/9712017}{arXiv:hep-th/9712017})


\item [[Hitoshi Nishino]], \emph{Supergravity Theories in $D \geq 12$ Coupled to Super p-Branes}, Nucl.Phys. B542 (1999) 217-261 (\href{https://arxiv.org/abs/hep-th/9807199}{arXiv:hep-th/9807199})


\item Stephen Hewson, \emph{On supergravity in $(10,2)$} (\href{https://arxiv.org/abs/hep-th/9908209}{arXiv:hep-th/9908209})


\item Tatsuya Ueno, \emph{BPS States in 10+2 Dimensions}, JHEP 0012:006, 2000 (\href{https://arxiv.org/abs/hep-th/9909007}{arXiv:hep-th/9909007})


\item [[Leonardo Castellani]], \emph{A locally supersymmetric SO(10,2) invariant action for D=12 supergravity}, (\href{https://arxiv.org/abs/1705.00638}{arXiv:1705.00638})



\end{itemize}
\hypertarget{on_the_brane_in__dimensions}{}\subsubsection*{{On the $2+1$-brane in $10+2$ dimensions}}\label{on_the_brane_in__dimensions}

\begin{itemize}%
\item [[Miles Blencowe]], [[Mike Duff]], \emph{Supermembranes and the Signature of Space-time}, Nucl. Phys. B310 (1988) 387-404 (\href{inspirehep.net/record/262142}{spire:262142}, , \href{http://inspirehep.net/record/262142/files/cer-000099708.pdf}{pdf})


\item S. F. Hewson, M. J. Perry, \emph{The twelve dimensional super $(2+2)$-brane}, Nucl.Phys. B492 (1997) 249-277 (\href{https://arxiv.org/abs/hep-th/9612008}{arXiv:hep-th/9612008})


\item \hyperlink{Nishino97b}{Nishino 97b}



\end{itemize}
\hypertarget{on_supergravity_in__dimensions}{}\subsubsection*{{On supergravity in $9 + 3$ dimensions}}\label{on_supergravity_in__dimensions}

\begin{itemize}%
\item [[Igor Kriz]], \emph{Some remarks on fundamental physical F-theory}, (\href{https://arxiv.org/abs/hep-th/0508046}{arXiv:hep-th/0508046})

\end{itemize}
[[!redirects 12d supergravity]]



\end{document}