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\section*{geometry of physics -- supergeometry and superphysics}

\begin{quote}%
this entry is one chapter of ``[[geometry of physics]]''

previous chapter: \emph{[[geometry of physics -- manifolds and orbifolds|manifolds and orbifolds]]}

next chapter: \emph{[[geometry of physics -- BPS charges|BPS charges]]}

Presently this entry is under construction. It is being incrementally expanded as this lecture series progresses: \emph{[[schreiber:From the Superpoint to T-Folds]]}.


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

\noindent\hyperlink{Superalgebra}{Superalgebra}\dotfill \pageref*{Superalgebra} \linebreak
\noindent\hyperlink{Supergeometry}{Supergeometry}\dotfill \pageref*{Supergeometry} \linebreak
\noindent\hyperlink{spacetime_supersymmetry}{Spacetime supersymmetry}\dotfill \pageref*{spacetime_supersymmetry} \linebreak
\noindent\hyperlink{fundamental_super_branes}{Fundamental super $p$-branes}\dotfill \pageref*{fundamental_super_branes} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{GeneralReading}{General accounts}\dotfill \pageref*{GeneralReading} \linebreak
\noindent\hyperlink{MoreDetails}{More specialized literature}\dotfill \pageref*{MoreDetails} \linebreak
\noindent\hyperlink{ReferencesHigherSuperCartan}{Higher super Cartan geometry}\dotfill \pageref*{ReferencesHigherSuperCartan} \linebreak


In [[nLab:Klein geometry]] and [[nLab:Cartan geometry]] the fundamental geometric concept is the [[nLab:symmetry group]] $G$ of the local model [[nLab:space]], which is then recovered as some [[nLab:coset space]] $G/H$. These symmetry groups $G$ are reflected in their [[nLab:categories of representations]] $Rep(G)$, which are certain nice [[nLab:tensor categories]]. In terms of [[nLab:physics]] via [[nLab:Wigner classification]], the [[nLab:irreducible objects]] in $Rep(G)$ label the possible [[nLab:fundamental particle]] species on the [[nLab:spacetime]] $G/H$. Hence if we regard the [[nLab:tensor category]] $Rep(G)$ as the actual fundamental concept, then the natural question is that of \emph{[[nLab:Tannaka duality|Tannaka reconstruction]]}: Given any nice [[nLab:tensor category]], is it [[nLab:equivalence of categories|equivalent]] to $Rep(G)$ for some symmetry group $G$? For [[nLab:rigid monoidal category|rigid]] [[nLab:tensor categories]] in [[nLab:characteristic zero]] subject only to a mild size constraint this is answered by \textbf{[[nLab:Deligne's theorem on tensor categories]]}: all of them are, but only if we allow $G$ to be a ``[[nLab:supergroup]]''. This we discuss in the first section \hyperlink{Superalgebra}{below}.

\hypertarget{Superalgebra}{}\subsection*{{Superalgebra}}\label{Superalgebra}

this section is at \emph{[[geometry of physics -- superalgebra]]}

\hypertarget{Supergeometry}{}\subsection*{{Supergeometry}}\label{Supergeometry}

this section is at \emph{[[geometry of physics -- supergeometry]]}

\hypertarget{spacetime_supersymmetry}{}\subsection*{{Spacetime supersymmetry}}\label{spacetime_supersymmetry}

this section is at \emph{[[geometry of physics -- supersymmetry]]}

\hypertarget{fundamental_super_branes}{}\subsection*{{Fundamental super $p$-branes}}\label{fundamental_super_branes}

this section is at \emph{[[geometry of physics -- fundamental super p-branes]]}

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

The following first mentions

\begin{itemize}%
\item \hyperlink{GeneralReading}{General accounts}

\end{itemize}
that might usefully be held on to during the seminar. Then I list

\begin{itemize}%
\item \hyperlink{MoreDetails}{Special literature}

\end{itemize}
with refernces to original results and to reviews of these. Then I list pointers to my own work with collaborators on

\begin{itemize}%
\item \hyperlink{ReferencesHigherSuperCartan}{Higher super Cartan geometry}

\end{itemize}
\hypertarget{GeneralReading}{}\subsubsection*{{General accounts}}\label{GeneralReading}

An excellent general textbook for our purposes is

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

\end{itemize}
This is written by physicists in physics style, but the development is careful and thorough, and the ``geometric perspective'' in the title is nothing but the perspective of higher [[nLab:super Cartan geometry]] in slight disguise. See also at \emph{[[nLab:D'Auria-Fré formulation of supergravity]]}.

Lecture notes closely related to the seminar are in

\begin{itemize}%
\item [[nLab:Urs Schreiber]], \emph{[[schreiber:Structure Theory for Higher WZW Terms]]}, lectures at \emph{\href{http://www.esi.ac.at/activities/events/2015/higher-structures-in-string-theory-and-quantum-field-theory}{Higher Structures in String Theory and Quantum Field Theory}}, ESI Vienna, November 30-December 4, 2015

\end{itemize}
\hypertarget{MoreDetails}{}\subsubsection*{{More specialized literature}}\label{MoreDetails}

[[nLab:Deligne's theorem on tensor categories]] is due to

\begin{itemize}%
\item [[nLab:Pierre Deligne]], \emph{Cat\'e{}gorie Tensorielle}, Moscow Math. Journal 2 (2002) no. 2, 227-248. (\href{https://www.math.ias.edu/files/deligne/Tensorielles.pdf}{pdf})

\end{itemize}
building on his general work on [[nLab:Tannakian categories]]

\begin{itemize}%
\item [[nLab:Pierre Deligne]], \emph{[[nLab:Catégories Tannakiennes]]}, Grothendieck Festschrift, vol. II, Birkh\"a{}user Progress in Math. 87 (1990) pp.111-195. (\href{https://publications.ias.edu/sites/default/files/60_categoriestanna.pdf}{pdf})

\end{itemize}
A brief survey is in

\begin{itemize}%
\item [[nLab:Victor Ostrik]], \emph{Tensor categories (after P. Deligne)} (\href{http://arxiv.org/abs/math/0401347}{arXiv:math/0401347})

\end{itemize}
and a more comprehensive texbook account is in chapter 9.11 of

\begin{itemize}%
\item [[nLab:Pavel Etingof]], Shlomo Gelaki, Dmitri Nikshych, [[nLab:Victor Ostrik]], \emph{Tensor categories}, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 (\href{http://www-math.mit.edu/~etingof/egnobookfinal.pdf
}{pdf})

\end{itemize}
The observation that [[nLab:supergeometry]] is naturally regarded as ordinary geometry inside the [[nLab:sheaf topos]] over [[nLab:superpoints]] is due to

\begin{itemize}%
\item [[nLab:Albert Schwarz]], \emph{On the definition of superspace}, Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 37--42, (\href{http://www.mathnet.ru/links/b12306f831b8c37d32d5ba8511d60c93/tmf5111.pdf}{russian original pdf})


\item Anatoly Konechny, [[nLab:Albert Schwarz]], \emph{On $(k \oplus l|q)$-dimensional supermanifolds} in \emph{Supersymmetry and Quantum Field Theory} ([[Dmitry Volkov]] memorial volume) Springer-Verlag, 1998, Lecture Notes in Physics, 509 , J. Wess and V. Akulov (editors)(\href{http://arxiv.org/abs/hep-th/9706003}{arXiv:hep-th/9706003})

\emph{Theory of $(k \oplus l|q)$-dimensional supermanifolds} Sel. math., New ser. 6 (2000) 471 - 486



\end{itemize}
A nice account is in

\begin{itemize}%
\item [[nLab:Christoph Sachse]], \emph{A Categorical Formulation of Superalgebra and Supergeometry} (\href{http://arxiv.org/abs/0802.4067}{arXiv:0802.4067})

\end{itemize}
Useful discussion of [[nLab:Majorana spinors]] and the induced [[nLab:supersymmetry]] algebras includes

\begin{itemize}%
\item \hyperlink{II.7.3}{CDF, II.7.3}


\item [[nLab:José Figueroa-O'Farrill]], \emph{Majorana spinors} (\href{http://www.maths.ed.ac.uk/~jmf/Teaching/Lectures/Majorana.pdf}{pdf})



\end{itemize}
The close relation between [[nLab:supersymmetry and division algebras]] was first observed in

\begin{itemize}%
\item [[Taichiro Kugo]], [[Paul Townsend]], \emph{Supersymmetry and the division algebras}, Nuclear Physics B, Volume 221, Issue 2 (1983) p. 357-380. (\href{http://inspirehep.net/record/181889}{spires}, \href{http://cds.cern.ch/record/140183/files/198301032.pdf}{pdf})

\end{itemize}
A clean survey is in

\begin{itemize}%
\item [[John Baez]], [[John Huerta]], \emph{Division algebras and supersymmetry I}, in R. Doran, G. Friedman, [[nLab:Jonathan Rosenberg]](eds.) \emph{Superstrings, Geometry, Topology, and $C^\ast$-algebras}, Proc. Symp. Pure Math. 81, AMS, Providence, 2010, pp. 65-80 (\href{http://arxiv.org/abs/0909.0551}{arXiv:0909.0551})

\end{itemize}
and the discussion of the spinor bilinear pairings from this perspective is in

\begin{itemize}%
\item [[John Baez]], [[John Huerta]], \emph{Division algebras and supersymmetry II}, Adv. Math. Theor. Phys. 15 (2011), 1373-1410 (\href{http://arxiv.org/abs/1003.3436}{arXiv:1003.34360})

\end{itemize}
The seminal analysis of [[nLab:torsion of G-structures]] is due to

\begin{itemize}%
\item [[Victor Guillemin]], \emph{The integrability problem for $G$-structures}, Trans. Amer. Math. Soc. 116 (1965), 544--560. (\href{http://www.jstor.org/stable/1994134}{JSTOR})

\end{itemize}
Discussion of [[nLab:torsion of G-structures]] in the context of [[nLab:supergeometry]] ([[nLab:supertorsion]]) is in

\begin{itemize}%
\item [[nLab:John Lott]], \emph{The Geometry of Supergravity Torsion Constraints} Comm. Math. Phys. 133 (1990), 563--615, (exposition in \href{http://arxiv.org/abs/math/0108125}{arXiv:0108125})

\end{itemize}
An elegant construction of [[nLab:11-dimensional supergravity]], right in the spirit of [[nLab:super Cartan geometry]], is due to

\begin{itemize}%
\item [[nLab:Riccardo D'Auria]], [[nLab:Pietro Fré]]; \emph{Geometric Superravity in D=11 and its hidden supergroup}, Nuclear Physics B201 (1982) 101-140 (\href{https://ncatlab.org/nlab/files/GeometricSupergravity.pdf}{pdf})

\end{itemize}
This is the main original result on which the [[nLab:D'Auria-Fré formulation of supergravity]] is based, as laid out in \hyperlink{CDF}{CDF}.

The observation that the [[nLab:equations of motion]] of bosonic solutions of [[nLab:11-dimensional supergravity]] are equivalent simply to vanishing of the [[nLab:supertorsion]] is due to

\begin{itemize}%
\item A. Candiello, K. Lechner, \emph{Duality in Supergravity Theories}, Nucl.Phys. B412 (1994) 479-501 (\href{http://arxiv.org/abs/hep-th/9309143}{arXiv:hep-th/9309143})


\item [[nLab:Paul Howe]], \emph{Weyl Superspace}, Physics Letters B Volume 415, Issue 2, 11 December 1997, Pages 149--155 (\href{http://arxiv.org/abs/hep-th/9707184}{arXiv:hep-th/9707184})



\end{itemize}
Discussion of [[nLab:Fierz identities]] includes

\begin{itemize}%
\item \hyperlink{CDF}{CDF, II.8}

\end{itemize}
The classification of the invariant super Lie algebra cocycles on super-Minkowski spacetime, hence that of [[nLab:super p-branes]] without gauge fields on their worldvolume, is due to

\begin{itemize}%
\item Anna Ach\'u{}carro, [[nLab:Jonathan Evans]], [[nLab:Paul Townsend]], [[nLab:David Wiltshire]], \emph{Super $p$-Branes}, Phys. Lett. B \textbf{198} (1987) 441 (\href{http://inspirehep.net/record/22286?ln=en}{spire})

\end{itemize}
The extension of this classification to [[nLab:D-branes]] and to the [[nLab:M5-brane]] using [[nLab:extended super Minkowski spacetime]] is due to

\begin{itemize}%
\item C. Chryssomalakos, [[nLab:José de Azcárraga]], J. M. Izquierdo and C. P\'e{}rez Bueno, \emph{The geometry of branes and extended superspaces}, Nuclear Physics B Volume 567, Issues 1--2, 14 February 2000, Pages 293--330 (\href{http://arxiv.org/abs/hep-th/9904137}{arXiv:hep-th/9904137})


\item Makoto Sakaguchi, \emph{IIB-Branes and New Spacetime Superalgebras}, JHEP 0004 (2000) 019 (\href{https://arxiv.org/abs/hep-th/9909143}{arXiv:hep-th/9909143})



\end{itemize}
The M5-brane cocycle on the ``M2-brane extended super-Minkowski spacetime'' that appears here has in fact been observed, as a cocycle, all the way back in \hyperlink{DAuriaFre82}{D'Auria-Fr\'e{} 82}. But there it was seen just as a means for constructing [[nlab:11-dimensional supergravity]]. That it indeed gives the [[nLab:higher WZW term]] in the [[nLab:Green-Schwarz action functional|Green-Schwarz type action functional]] that defines the fundamental [[nLab:M5-brane]] has been argued in

\begin{itemize}%
\item [[nLab:Igor Bandos]], [[nLab:Kurt Lechner]], Alexei Nurmagambetov, [[nLab:Paolo Pasti]], [[nLab:Dmitri Sorokin]], Mario Tonin, \emph{Covariant Action for the Super-Five-Brane of M-Theory}, Phys.Rev.Lett. 78 (1997) 4332-4334 (\href{http://arxiv.org/abs/hep-th/9701149}{arXiv:hep-th/9701149})

\end{itemize}
The observation that [[nLab:super p-branes]] on curved [[nLab:super spacetimes]] require [[nLab:definite globalization of WZW term|definite globalization]] of super Lie algebra cocycles from Minkowski spacetime over the supermanifold is due to

\begin{itemize}%
\item [[nLab:Eric Bergshoeff]], [[nLab:Ergin Sezgin]], [[nLab:Paul Townsend]], \emph{Superstring actions in $D = 3, 4, 6, 10$ curved superspace}, Phys.Lett., B169, 191, (1986) (\href{http://inspirehep.net/record/223138/?ln=en}{spire})


\item [[nLab:Eric Bergshoeff]], [[nLab:Ergin Sezgin]], [[nLab:Paul Townsend]], \emph{Supermembranes and eleven dimensional supergravity}, Phys.Lett. B189 (1987) 75-78, In [[nLab:Mike Duff]], (ed.), \emph{[[nLab:The World in Eleven Dimensions]]} 69-72 (\href{http://streaming.ictp.trieste.it/preprints/P/87/010.pdf}{pdf}, \href{http://inspirehep.net/record/248230?ln=en}{spire})



\end{itemize}
The formulation of [[nLab:topological T-duality]] is due to

\begin{itemize}%
\item [[nLab:Peter Bouwknegt]], [[nLab:Jarah Evslin]], [[nLab:Varghese Mathai]], \emph{T-Duality: Topology Change from H-flux}, Commun.Math.Phys.249:383-415,2004 (\href{http://arxiv.org/abs/hep-th/0306062}{hep-th/0306062})

\end{itemize}
and in an alternative form due to

\begin{itemize}%
\item  [[nLab:Ulrich Bunke]], P. Rumpf, [[nLab:Thomas Schick]], \emph{The topology of $T$-duality for $T^n$-bundles}, Rev. Math. Phys. 18, 1103 (2006). (\href{http://arxiv.org/abs/math.GT/0501487}{arXiv:math.GT/0501487})

\end{itemize}
The suggestion that there ought to be ``[[nLab:T-folds]]'' or ``[[nLab:double field theory|doubled geometry]]'' is due to

\begin{itemize}%
\item [[nLab:Chris Hull]], \emph{A Geometry for Non-Geometric String Backgrounds}, JHEP0510:065,2005 (\href{http://arxiv.org/abs/hep-th/0406102}{arXiv:hep-th/0406102})


\item [[nLab:Chris Hull]], \emph{Doubled geometry and T-folds} JHEP0707:080,2007 (\href{http://arxiv.org/abs/hep-th/0605149}{arXiv:hep-th/0605149})



\end{itemize}
The mathematical formalization of this idea in terms of [[nLab:principal 2-bundles]] for the [[nLab:T-duality 2-group]] was claimed in

\begin{itemize}%
\item [[nLab:Thomas Nikolaus]], \emph{T-Duality in K-theory and elliptic cohomology}, talk at \emph{String Geometry Network Meeting}, Feb 2014, ESI Vienna (\href{http://www.ingvet.kau.se/juerfuch/conf/esi14/esi14_34.html}{website})

\end{itemize}
\hypertarget{ReferencesHigherSuperCartan}{}\subsubsection*{{Higher super Cartan geometry}}\label{ReferencesHigherSuperCartan}

The following articles develop the higher super Cartan geometry that we give an exposition of in the second part of the seminar.

The mathematical foundation of higher supergeometry:

\begin{itemize}%
\item [[nLab:Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]}, Thesis, (\href{http://arxiv.org/abs/1310.7930v1}{v1 arXiv:1310.7930}, \href{https://dl.dropboxusercontent.com/u/12630719/dcct.pdf}{v2})

\end{itemize}
The general idea of [[schreiber:The brane bouquet]] and the general construction of [[nLab:higher WZW terms]] from higher $L_\infty$-cocycles:

\begin{itemize}%
\item [[nLab:Domenico Fiorenza]], [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], \emph{[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]} International Journal of Geometric Methods in Modern Physics Volume 12, Issue 02 (2015) 1550018, (\href{http://arxiv.org/abs/1308.5264}{arXiv:1308.5264})

\end{itemize}
The homotopy-[[nLab:descent]] of the [[nLab:M5-brane]] cocycle and of the type IIA [[nLab:D-brane]] cocycles:

\begin{itemize}%
\item [[nLab:Domenico Fiorenza]], [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], \emph{[[schreiber:Rational sphere valued supercocycles in M-theory]]}, to appear in Journal of Geometry and Physics (\href{http://arxiv.org/abs/1606.03206}{arXiv:1606.03206})

\end{itemize}
The derivation of supersymmetric [[nLab:topological T-duality]], rationally, and of the higher super Cartan geometry for [[nLab:super T-folds]]:

\begin{itemize}%
\item [[nLab:Domenico Fiorenza]], [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], \emph{[[schreiber:T-Duality from super Lie n-algebra cocycles for super p-branes]]} (\href{https://arxiv.org/abs/1611.06536}{arXiv:1611.06536})

\end{itemize}
The derivation of the process of higher invariant extensions that leads from the [[nLab:superpoint]] to [[nLab:11-dimensional supergravity]]:

\begin{itemize}%
\item [[nLab:John Huerta]], [[nLab:Urs Schreiber]], \emph{[[schreiber:M-Theory from the Superpoint]]}

\end{itemize}


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