\documentclass[12pt,titlepage]{article} \usepackage{amsmath} \usepackage{mathrsfs} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{mathtools} \usepackage{graphicx} \usepackage{color} \usepackage{ucs} \usepackage[utf8x]{inputenc} \usepackage{xparse} \usepackage{hyperref} %----Macros---------- % % Unresolved issues: % % \righttoleftarrow % \lefttorightarrow % % \color{} with HTML colorspec % \bgcolor % \array with options (without options, it's equivalent to the matrix environment) % Of the standard HTML named colors, white, black, red, green, blue and yellow % are predefined in the color package. Here are the rest. \definecolor{aqua}{rgb}{0, 1.0, 1.0} \definecolor{fuschia}{rgb}{1.0, 0, 1.0} \definecolor{gray}{rgb}{0.502, 0.502, 0.502} \definecolor{lime}{rgb}{0, 1.0, 0} \definecolor{maroon}{rgb}{0.502, 0, 0} \definecolor{navy}{rgb}{0, 0, 0.502} \definecolor{olive}{rgb}{0.502, 0.502, 0} \definecolor{purple}{rgb}{0.502, 0, 0.502} \definecolor{silver}{rgb}{0.753, 0.753, 0.753} \definecolor{teal}{rgb}{0, 0.502, 0.502} % Because of conflicts, \space and \mathop are converted to % \itexspace and \operatorname during preprocessing. % itex: \space{ht}{dp}{wd} % % Height and baseline depth measurements are in units of tenths of an ex while % the width is measured in tenths of an em. \makeatletter \newdimen\itex@wd% \newdimen\itex@dp% \newdimen\itex@thd% \def\itexspace#1#2#3{\itex@wd=#3em% \itex@wd=0.1\itex@wd% \itex@dp=#2ex% \itex@dp=0.1\itex@dp% \itex@thd=#1ex% \itex@thd=0.1\itex@thd% \advance\itex@thd\the\itex@dp% \makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}} \makeatother % \tensor and \multiscript \makeatletter \newif\if@sup \newtoks\@sups \def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}% \def\reset@sup{\@supfalse\@sups={}}% \def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else% \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}% \else \append@sup#2 \@suptrue \fi% \expandafter\mk@scripts\fi} \def\tensor#1#2{\reset@sup#1\mk@scripts#2_/} \def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2% \reset@sup\mk@scripts#3_/} \makeatother % \slash \makeatletter \newbox\slashbox \setbox\slashbox=\hbox{$/$} \def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$} \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa \copy\slashbox \kern-\@tempdima \box\@tempboxa} \def\slash{\protect\itex@pslash} \makeatother % math-mode versions of \rlap, etc % from Alexander Perlis, "A complement to \smash, \llap, and lap" % http://math.arizona.edu/~aprl/publications/mathclap/ \def\clap#1{\hbox to 0pt{\hss#1\hss}} \def\mathllap{\mathpalette\mathllapinternal} \def\mathrlap{\mathpalette\mathrlapinternal} \def\mathclap{\mathpalette\mathclapinternal} \def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}} \def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}} \def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}} % Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2} \let\oldroot\root \def\root#1#2{\oldroot #1 \of{#2}} \renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}} % Manually declare the txfonts symbolsC font \DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n} \SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n} \DeclareFontSubstitution{U}{txsyc}{m}{n} % Manually declare the stmaryrd font \DeclareSymbolFont{stmry}{U}{stmry}{m}{n} \SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n} % Manually declare the MnSymbolE font \DeclareFontFamily{OMX}{MnSymbolE}{} \DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n} \SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n} \DeclareFontShape{OMX}{MnSymbolE}{m}{n}{ <-6> MnSymbolE5 <6-7> MnSymbolE6 <7-8> MnSymbolE7 <8-9> MnSymbolE8 <9-10> MnSymbolE9 <10-12> MnSymbolE10 <12-> MnSymbolE12}{} % Declare specific arrows from txfonts without loading the full package \makeatletter \def\re@DeclareMathSymbol#1#2#3#4{% \let#1=\undefined \DeclareMathSymbol{#1}{#2}{#3}{#4}} \re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46} \re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12} \re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64} \re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6} \re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77} \re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77} \makeatother % \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE \makeatletter \def\Decl@Mn@Delim#1#2#3#4{% \if\relax\noexpand#1% \let#1\undefined \fi \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}} \def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}} \def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}} \Decl@Mn@Open{\llangle}{mnomx}{'164} \Decl@Mn@Close{\rrangle}{mnomx}{'171} \Decl@Mn@Open{\lmoustache}{mnomx}{'245} \Decl@Mn@Close{\rmoustache}{mnomx}{'244} \makeatother % Widecheck \makeatletter \DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}} \def\@widecheck#1#2{% \setbox\z@\hbox{\m@th$#1#2$}% \setbox\tw@\hbox{\m@th$#1% \widehat{% \vrule\@width\z@\@height\ht\z@ \vrule\@height\z@\@width\wd\z@}$}% \dp\tw@-\ht\z@ \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@ \setbox\tw@\hbox{% \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box \tw@}}}% {\ooalign{\box\tw@ \cr \box\z@}}} \makeatother % \mathraisebox{voffset}[height][depth]{something} \makeatletter \NewDocumentCommand\mathraisebox{moom}{% \IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{% \IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}% }{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}% \mathpalette\@temp{#4}} \makeatletter % udots (taken from yhmath) \makeatletter \def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.} \mkern2mu\raise4\p@\hbox{.}\mkern1mu \raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}} \makeatother %% Fix array \newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}} %% \itexnum is a noop \newcommand{\itexnum}[1]{#1} %% Renaming existing commands \newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}} \newcommand{\widevec}{\overrightarrow} \newcommand{\darr}{\downarrow} \newcommand{\nearr}{\nearrow} \newcommand{\nwarr}{\nwarrow} \newcommand{\searr}{\searrow} \newcommand{\swarr}{\swarrow} \newcommand{\curvearrowbotright}{\curvearrowright} \newcommand{\uparr}{\uparrow} \newcommand{\downuparrow}{\updownarrow} \newcommand{\duparr}{\updownarrow} \newcommand{\updarr}{\updownarrow} \newcommand{\gt}{>} \newcommand{\lt}{<} \newcommand{\map}{\mapsto} \newcommand{\embedsin}{\hookrightarrow} \newcommand{\Alpha}{A} \newcommand{\Beta}{B} \newcommand{\Zeta}{Z} \newcommand{\Eta}{H} \newcommand{\Iota}{I} \newcommand{\Kappa}{K} \newcommand{\Mu}{M} \newcommand{\Nu}{N} \newcommand{\Rho}{P} \newcommand{\Tau}{T} \newcommand{\Upsi}{\Upsilon} \newcommand{\omicron}{o} \newcommand{\lang}{\langle} \newcommand{\rang}{\rangle} \newcommand{\Union}{\bigcup} \newcommand{\Intersection}{\bigcap} \newcommand{\Oplus}{\bigoplus} \newcommand{\Otimes}{\bigotimes} \newcommand{\Wedge}{\bigwedge} \newcommand{\Vee}{\bigvee} \newcommand{\coproduct}{\coprod} \newcommand{\product}{\prod} \newcommand{\closure}{\overline} \newcommand{\integral}{\int} \newcommand{\doubleintegral}{\iint} \newcommand{\tripleintegral}{\iiint} \newcommand{\quadrupleintegral}{\iiiint} \newcommand{\conint}{\oint} \newcommand{\contourintegral}{\oint} \newcommand{\infinity}{\infty} \newcommand{\bottom}{\bot} \newcommand{\minusb}{\boxminus} \newcommand{\plusb}{\boxplus} \newcommand{\timesb}{\boxtimes} \newcommand{\intersection}{\cap} \newcommand{\union}{\cup} \newcommand{\Del}{\nabla} \newcommand{\odash}{\circleddash} \newcommand{\negspace}{\!} \newcommand{\widebar}{\overline} \newcommand{\textsize}{\normalsize} \renewcommand{\scriptsize}{\scriptstyle} \newcommand{\scriptscriptsize}{\scriptscriptstyle} \newcommand{\mathfr}{\mathfrak} \newcommand{\statusline}[2]{#2} \newcommand{\tooltip}[2]{#2} \newcommand{\toggle}[2]{#2} % Theorem Environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{torsion constraints in supergravity} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{gravity}{}\paragraph*{{Gravity}}\label{gravity} [[!include gravity contents]] \hypertarget{supergeometry}{}\paragraph*{{Super-Geometry}}\label{supergeometry} [[!include supergeometry - contents]] \hypertarget{chernweil_theory}{}\paragraph*{{Chern-Weil theory}}\label{chernweil_theory} [[!include infinity-Chern-Weil theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{CanonicalTorsionOfSuperMinkowskiSpacetime}{From the canonical torsion of super-Minkowski spacetime}\dotfill \pageref*{CanonicalTorsionOfSuperMinkowskiSpacetime} \linebreak \noindent\hyperlink{InTermsOfTorsionTwistedGStructure}{In terms of torsion-twisted $G$-structure}\dotfill \pageref*{InTermsOfTorsionTwistedGStructure} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{relation_to_supergravity_equations_of_motion}{Relation to supergravity equations of motion}\dotfill \pageref*{relation_to_supergravity_equations_of_motion} \linebreak \noindent\hyperlink{relation_to_crgeometry}{Relation to CR-geometry}\dotfill \pageref*{relation_to_crgeometry} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{Examples11dSuGra}{11d supergravity}\dotfill \pageref*{Examples11dSuGra} \linebreak \noindent\hyperlink{HeteroticSupergravity}{10d Heterotic supergravity}\dotfill \pageref*{HeteroticSupergravity} \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{for_11d_supergravity}{For 11d supergravity}\dotfill \pageref*{for_11d_supergravity} \linebreak \noindent\hyperlink{for_10d_heterotic_supergravity}{For 10d heterotic supergravity}\dotfill \pageref*{for_10d_heterotic_supergravity} \linebreak \noindent\hyperlink{ReferencesFor4d}{For 4d supergravity}\dotfill \pageref*{ReferencesFor4d} \linebreak \noindent\hyperlink{for_2d_supergravity__superstring_worldsheets__super_riemann_surfaces}{For 2d supergravity / superstring worldsheets / super Riemann surfaces}\dotfill \pageref*{for_2d_supergravity__superstring_worldsheets__super_riemann_surfaces} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The [[equations of motion]] of [[supergravity]] typically imply -- or are even equivalent to (\hyperlink{CandielloLechner93}{Candiello-Lechner 93}, \hyperlink{Howe97}{Howe 97}), that the [[supertorsion|super-]][[torsion of a Cartan connection|torsion]] of the super-[[vielbein fields]] vanishes. At least in some cases these \emph{supergravity torsion constraints} may naturally be understood as saying that supergravity solutions are ([[higher Cartan geometry|higher]]) [[super-Cartan geometry]] modeled on [[extended super Minkowski spacetime]] \emph{with} its canonical [[torsion of a G-structure]], due to the fact that the [[left invariant 1-forms]] on super-Minkowski space are not closed. \hypertarget{CanonicalTorsionOfSuperMinkowskiSpacetime}{}\subsubsection*{{From the canonical torsion of super-Minkowski spacetime}}\label{CanonicalTorsionOfSuperMinkowskiSpacetime} The torsion constraint is naturally understood by regarding [[supergravity]] as [[Cartan geometry]] for the inclusion of the [[orthogonal group]] into a [[super Poincare group]] and by noticing that the corresponding local model space, which is [[super-Minkowski spacetime]] $\mathbb{R}^{d|N}$, canonically has non-vanishing torsion. Let $(x^a, \theta^\alpha)$ be the canonical [[coordinates]] on the [[supermanifold]] $\mathbb{R}^{d|N}$ underlying the [[super translation group]]. Then the [[left-invariant 1-forms]] are \begin{itemize}% \item $\psi^\alpha = d \theta^\alpha$. \item $e^a = d x^a + \frac{i}{2} \overline{\theta} \Gamma^a d \theta$. \end{itemize} Here the extra summand in the equation for $e^a$ (necessary to make it left-invariant) causes it to be non-closed: \begin{displaymath} \begin{aligned} d e^a & = d (d x^a + \frac{i}{2} \overline{\theta} \Gamma^a d \theta) \\ & = \frac{i}{2} d \overline{\theta}\Gamma^a d \theta \\ & = \frac{i}{2} \overline{\psi}\Gamma^a \psi \end{aligned} \,. \end{displaymath} Taking the [[spin connection]] $(\omega^a{}_b)$ on $\mathbb{R}^{d|N}$ to vanish, as usual, this means that there is non-vanishing [[torsion]]: \begin{displaymath} \begin{aligned} \tau^a & = \mathbf{d} e^a + \omega^a{}_b \wedge e^b \\ & = \mathbf{d} e^a \\ & = \frac{i}{2} \overline{\psi}\Gamma^a \psi \end{aligned} \end{displaymath} Depending on perspective one might say that it is the [[supertorsion]] that vanishes (see at \emph{[[super-Minkowski spacetime]]} and at \emph{[[D'Auria-Fre formulation of supergravity]]} for this perspective), or, alternatively, that one is dealing with [[Cartan geometry]]/[[G-structure]] whose local model space carries non-vanishing torsion, see \hyperlink{InTermsOfTorsionTwistedGStructure}{below}. Notice that the torison-full but left-invariant forms are of course obtained from the torsion-free but non-left-invartiant forms by a $GL(\mathbb{R}^{d|N})$-valued function: \begin{displaymath} \left( \itexarray{ e^a \\ \psi^\alpha } \right) = \left( \itexarray{ id & \tfrac{i}{2}\Gamma^a{}_{\alpha \beta} \theta^\alpha \\ 0 & id } \right) \left( \itexarray{ \mathbf{d}x^a \\ \mathbf{d}\theta^\alpha } \right) \end{displaymath} \begin{displaymath} \left( \itexarray{ \mathbf{d}x^a \\ \mathbf{d}\theta^\alpha } \right) = \left( \itexarray{ id & -\tfrac{i}{2}\Gamma^a{}_{\alpha \beta} \theta^\alpha \\ 0 & id } \right) \left( \itexarray{ e^a \\ \psi^\alpha } \right) \end{displaymath} This shows that regarding \begin{displaymath} (E^A) \coloneqq (E^a, E^\alpha) \coloneqq (e^a, \Psi^\alpha) \end{displaymath} as a [[super-vielbein]] is consistent: this is indeed a [[homotopy]] in \begin{displaymath} \itexarray{ \mathbb{R}^{d|N} &\to& \ast &\to& \mathbf{B}O(\mathbb{R}^{d|N}) \\ & {}_{\mathllap{\tau_{\mathbb{R}^{d|N}}}}\searrow & \swArrow_{E} & \swarrow_{\mathrlap{O(\mathbb{R}^{d|N})\mathbf{Struc}}} \\ && \mathbf{B}GL(\mathbb{R}^{d|N}) } \end{displaymath} but not the tautological one given by \begin{displaymath} \itexarray{ \mathbb{R}^{d|N} &\to& \ast &\to& \mathbf{B}O(\mathbb{R}^{d|N}) \\ & \searrow & \downarrow & \swarrow \\ && \mathbf{B}GL(\mathbb{R}^{d|N}) } \end{displaymath} where the left triangle is that which exhibits the canonical trivialization of the [[frame bundle]] of $\mathbb{R}^{d|N}$. \hypertarget{InTermsOfTorsionTwistedGStructure}{}\subsubsection*{{In terms of torsion-twisted $G$-structure}}\label{InTermsOfTorsionTwistedGStructure} Given a [[subgroup]] $G\hookrightarrow GL(V)$ of the [[general linear group]] of a linear model space $V$ (e.g. [[super-Minkowski spacetime]] $\mathbb{R}^{d|N}$), then a [[G-structure]] is [[integrability of G-structure|first-order integrable]] if on the first-order [[infinitesimal neighbourhoods]] of any point it is equal to the canonical (trivial) $G$-structure on $V$. Ordinarily the standard [[torsion]] on $V$ vanishs, and if so then so does that of any first-order integrable $G$-structire, which is the reason why for these the [[torsion of a G-structure]] vanishes. But in the situation of $V$ being [[super-Minkowski spacetime]] as \hyperlink{CanonicalTorsionOfSuperMinkowskiSpacetime}{above}, the torsion of the local model space does not vanish, and so accordingly neither does that of a first-order integrable $G$-structure in this case. This perspective on the torsion constraints in supergravity is adopted in (\hyperlink{Lott01}{Lott 01}), see there around (38) of the original article or section 4 of the review on the arXiv. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{relation_to_supergravity_equations_of_motion}{}\subsubsection*{{Relation to supergravity equations of motion}}\label{relation_to_supergravity_equations_of_motion} The [[supergravity]] [[equations of motion]] typically imply the torsion constraints. See at \href{Green-Schwarz+action+functional#OnCurvedSpacetime}{super p-brane -- On curved spacetimes} for more. With enough [[supersymmetry]], the torsion constraints (always together with the [[Bianchi identities]] on the superfields, see at \emph{[[D'Auria-Fre formulation of supergravity]]}) may even become equivalent to the supergravity equations of motion. This is so for [[11-dimensional supergravity]] (\hyperlink{CandielloLechner93}{Candiello-Lechner 93}, \hyperlink{Howe97}{Howe 97}, see \hyperlink{CederwallGranNilssonTsimpis04}{Cederwall-Gran-Nilsson-Tsimpis 04, section 2.4}) and maybe its maximally supersymmetric [[KK-compactifications]]. See at \emph{\hyperlink{Examples11dSuGra}{Examples -- 11d SuGra}}. \hypertarget{relation_to_crgeometry}{}\subsubsection*{{Relation to CR-geometry}}\label{relation_to_crgeometry} A close [[analogy]] between [[CR geometry]] and [[supergravity]] [[superspacetimes]] (as both being [[torsion of a G-structure|torsion-ful]] [[integrable G-structures]]) is pointed out in (\hyperlink{Lott01}{Lott 01 exposition (4.2)}). \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} In accord with the \hyperlink{CanonicalTorsionOfSuperMinkowskiSpacetime}{above}, typically the [[equations of motion]] of a [[supergravity]] theory constrain the spinorial part of the torsion to have components $(\Gamma^a)_{\alpha \beta}$. \hypertarget{Examples11dSuGra}{}\subsubsection*{{11d supergravity}}\label{Examples11dSuGra} The torsion constraint for [[11-dimensional supergravity]] is discussed for instance in (\hyperlink{BergshoeffSezginTownsend87}{Bergshoeff-Sezgin-Townsend 87, equation (14)}). Here something special happens: The articles (\hyperlink{CandielloLechner93}{Candiello-Lechner 93 (5.6)}, \hyperlink{Howe97}{Howe 97}, see \hyperlink{CederwallGranNilssonTsimpis04}{Cederwall-Gran-Nilsson-Tsimpis 04, section 2.4}) show that imposing the torsion constraint (on some chart) $\mathbf{d} E^a + \omega^{a}{}_b \wedge E^b - \bar \psi \Gamma^a \psi = 0$ as well as $(\mathbf{d} \Psi +\tfrac{1}{4}\omega^{a b} \Gamma_{a b}\Psi)_{\theta \theta} = 0$ implies the equations of motion of [[11d supergravity]]. Moreover, setting $\mathbf{d} \Psi +\tfrac{1}{4}\omega^{a b} \Gamma_{a b}\Psi = 0$ generally (not just the component proportional to the wedge product of two fermionic 1-forms, hence requiring the full supertorsion tensor to be that of super-Minkowski spacetime)) then (\hyperlink{CandielloLechner93}{Candiello-Lechner 93, (5.8) with (6.5)}) this in addition puts the [[field strength]] of the [[supergravity C-field]] to 0. Hence this implies solutions to the ordinary vacuum [[Einstein equations]] in 11d. Such solutions are considered notably in the context of [[M-theory on G2-manifolds]] (e.g. \href{M-theory+on+G2-manifolds#Acharya02}{Acharya 02, p. 9}). See also at \emph{\href{M-theory+on+G2-manifolds#VacuumSolutionsAndTorsion}{M-theory on G2-manifolds -- Details -- Vacuum solution and torsion constraints}}. \hypertarget{HeteroticSupergravity}{}\subsubsection*{{10d Heterotic supergravity}}\label{HeteroticSupergravity} For [[heterotic supergravity]] in 10d the [[equations of motion]] are equivalent to the condition that \begin{enumerate}% \item the super-torsion of the bosonic part $\{e^a\}$ of the [[super vielbein]] is a bosonic form \begin{displaymath} \mathcal{D}e^a - \overline{\psi} \Gamma^a \psi = T^a{}_{b c} e^b \wedge e^c \end{displaymath} \item the super-torsion of the odd part $\psi^\alpha$ of the [[super vielbein]] is of the form \begin{displaymath} \mathcal{D} \psi^\alpha = T^\alpha{}_{b c} e^b \wedge e^c + \overline{\psi}^\beta \Gamma_a{}_{\beta \gamma} \phi^{\alpha \gamma} \end{displaymath} for \begin{displaymath} \phi^{\alpha \beta} \propto tr(\chi^\alpha \chi^\beta) - tr(T^\alpha T^\beta) \end{displaymath} proportional to the bispinor formed by tracing the square of the [[gaugino]] field $\chi$ \item the [[curvature]] 2-form of the [[gauge field]] has vanishing bispinorial component: \begin{displaymath} F_{\alpha \beta} = 0 \end{displaymath} (this is the [[10d super Yang-Mills theory]] sector) \end{enumerate} This is due to (\hyperlink{Witten86}{Witten 86 (5)+(27)}), see also (\hyperlink{AtickDharRatra86}{Atick-Dhar-Ratra 86 (4.1)}). These authors do not state explicitly that $\phi^{\alpha \beta} \propto tr(\lambda^\alpha \lambda^\beta) - tr (T T)$. (Among authors using a similar but different parameterization this statement is made explicit in \hyperlink{CandielloLechner93}{Candiello-Lechner 93 (2.5) with (2.29)}). But this follows by taking the differential of the bispinorial part of the 3-form field (which is the cocycle term for the heterotic [[Green-Schwarz superstring]]) \begin{displaymath} d \left( \overline{\psi} \wedge \Gamma_a \psi \wedge e^a \right) \propto \underset{i}{\sum} \underset{= F^i_{(1,1)}}{ \underbrace{ \left( \overline{\psi} \Gamma_a \chi^i \right) \wedge e^a } } \wedge \underset{ = F^i_{(1,1)}}{ \underbrace{ \left( \overline{\chi^i} \Gamma_b \psi \right) \wedge e^b } } - tr(R R)_{(2,2)} \end{displaymath} where we used the relation (\hyperlink{Witten86}{Witten 86 (8)}) (recalled for instance in \hyperlink{BonoraBregolaLechnerPastiTonin87}{Bonora-Bregola-Lechner-Pasti-Tonin 87 (2.28)}, \hyperlink{LechnerTonin08}{Lechner-Tonin 08 (2.13)}). According to (\hyperlink{BonoraBregolaLechnerPastiTonin90}{Bonora-Bregola-Lechner-Pasti-Tonin 90}) in fact all these constraints follow from just $T^a_{\alpha \beta} \propto \Gamma^a_{\alpha \beta}$. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[CR manifold]] \item [[D'Auria-Fre formulation of supergravity]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} The formulation of supergravity equations of motion in terms of constraints on the torsion tensor goes back to \begin{itemize}% \item [[Julius Wess]] [[Bruno Zumino]], \emph{Superspace formulation of supergravity}, Phys. Lett. B66 (1977), 361--364. \end{itemize} A mathematical formulation in terms of [[torsion of a G-structure|torsion-full]] [[first-order integrable G-structure|first-order integrable]] [[G-structures]] on [[supermanifolds]] (for low dimensional supergravity theories) is given in \begin{itemize}% \item [[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} which is followed up in \begin{itemize}% \item [[Michel Egeileh]], [[Fida El Chami]], \emph{Some remarks on the geometry of superspace supergravity}, J.Geom.Phys. 62 (2012) 53-60 (\href{http://inspirehep.net/record/1333125}{spire}) \end{itemize} \hypertarget{for_11d_supergravity}{}\subsubsection*{{For 11d supergravity}}\label{for_11d_supergravity} Discussion of torsion constrains for [[11-dimensional supergravity]] from the point of view of consistency of the [[membrane]] [[Green-Schwarz action functional]] is in \begin{itemize}% \item [[Eric Bergshoeff]], [[Ergin Sezgin]], [[Paul Townsend]], \emph{Supermembranes and eleven dimensional supergravity}, Phys.Lett. B189 (1987) 75-78, In [[Mike Duff]], (ed.), \emph{[[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 claim that this torsion constraint in [[11-dimensional supergravity]] is already equivalent to all of the [[equations of motion]] is due to \begin{itemize}% \item A. Candiello, [[Kurt 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 [[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} concisely reviewed in \begin{itemize}% \item [[Martin Cederwall]], [[Ulf Gran]], [[Bengt Nilsson]], [[Dimitrios Tsimpis]], \emph{Supersymmetric Corrections to Eleven-Dimensional Supergravity}, JHEP 0505:052, 2005 (\href{https://arxiv.org/abs/hep-th/0409107}{arXiv:hep-th/0409107}) \end{itemize} For commentary see also (\hyperlink{Nilsson00}{Nilsson 00, section 2}) and \begin{itemize}% \item [[Martin Cederwall]], [[Ulf Gran]], Mikkel Nielsen, [[Bengt Nilsson]], \emph{Manifestly supersymmetric M-theory}, JHEP 0010 (2000) 041 (\href{http://arxiv.org/abs/hep-th/0007035}{arXiv:hep-th/0007035}) \item [[Paul Howe]], [[Ergin Sezgin]], \emph{The supermembrane revisited}, Class.Quant.Grav. 22 (2005) 2167-2200 (\href{https://arxiv.org/abs/hep-th/0412245}{arXiv:hep-th/0412245}) \end{itemize} also \begin{itemize}% \item [[Lars Brink]], [[Paul Howe]], \emph{Eleven-dimensional supergravity on the mass shell in superspace}, Phys. Lett. , B91:384--386, 1980 \end{itemize} Discussion of possible deformations of the torsion constraint ([[M-theory]] corrections) includes \begin{itemize}% \item [[Martin Cederwall]], [[Ulf Gran]], Mikkel Nielsen, [[Bengt Nilsson]], \emph{Generalised 11-dimensional supergravity}, in A. Semikhatov, M. Vasiliev and V. Zaikin (eds.) Proceedings of ``Quantization, Gauge Theory \& Strings'', Moscow 2000 (\href{http://arxiv.org/abs/hep-th/0010042}{arXiv:hep-th/0010042}) \item [[Paul Howe]], [[Dimitrios Tsimpis]], \emph{On higher-order corrections in M theory}, JHEP 0309 (2003) 038 (\href{http://arxiv.org/abs/hep-th/0305129}{arXiv:hep-th/0305129}) \end{itemize} \hypertarget{for_10d_heterotic_supergravity}{}\subsubsection*{{For 10d heterotic supergravity}}\label{for_10d_heterotic_supergravity} Discussion of torsion constraints for [[heterotic supergravity]] goes back to (\href{Green-Schwarz+action+functional#Nilsson81}{Nilsson 81}) and includes \begin{itemize}% \item [[Paul Howe]], A. Umerski, \emph{On superspace supergravity in ten dimensions}, Phys. Lett. B 177 (1986) 163. \item [[Joseph Atick]], Avinash Dhar, and Bharat Ratra, \emph{Superspace formulation of ten-dimensional N=1 supergravity coupled to N=1 super Yang-Mills theory}, Phys. Rev. D 33, 2824, 1986 (\href{https://doi.org/10.1103/PhysRevD.33.2824}{doi.org/10.1103/PhysRevD.33.2824}) \item [[Edward Witten]], \emph{Twistor-like transform in ten dimensions}, Nuclear Physics B Volume 266, Issue 2, 17 March 1986 \item [[Loriano Bonora]], M. Bregola; [[Kurt Lechner]], [[Paolo Pasti]], [[Mario Tonin]], \emph{Anomaly-free supergravity and super-Yang-Mills theories in ten dimensions}, Nuclear Physics B Volume 296, Issue 4, 25 January 1988 () \item [[Loriano Bonora]], M. Bregola; [[Kurt Lechner]], [[Paolo Pasti]], [[Mario Tonin]], \emph{A discussion of the constraints in $N=1$ SUGRA-SYM in 10-D}, International Journal of Modern Physics A, February 1990, Vol. 05, No. 03 : pp. 461-477 () \item [[Paul Howe]], \emph{Heterotic supergeometry revisited} (\href{http://arxiv.org/abs/0805.2893}{arXiv:0805.2893}) \item [[Bengt Nilsson]], \emph{A superspace approach to branes and supergravity} (\href{http://arxiv.org/abs/hep-th/0007017}{arXiv:hep-th/0007017}) \item [[Kurt Lechner]], [[Mario Tonin]], \emph{Superspace formulations of ten-dimensional supergravity}, JHEP 0806:021,2008 (\href{https://arxiv.org/abs/0802.3869}{arXiv:0802.3869}) \end{itemize} \hypertarget{ReferencesFor4d}{}\subsubsection*{{For 4d supergravity}}\label{ReferencesFor4d} For [[d=4 N=1 supergravity]] the torsion is again constrained to be equal to the left-invariant torsion of super-Minkowski spacetime, see for instance \begin{itemize}% \item [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], volume 2, (III.2.28a), (III.3.66a) of \emph{[[Supergravity and Superstrings - A Geometric Perspective]]}, World Scientific (1991) \item Daniel Patrick Butter, section 2.2.5 of \emph{On conformal superspace and the One-Loop Effective Action in Supergravity}, 2010 (\href{http://digitalassets.lib.berkeley.edu/etd/ucb/text/Butter_berkeley_0028E_10582.pdf}{pdf}) \end{itemize} \hypertarget{for_2d_supergravity__superstring_worldsheets__super_riemann_surfaces}{}\subsubsection*{{For 2d supergravity / superstring worldsheets / super Riemann surfaces}}\label{for_2d_supergravity__superstring_worldsheets__super_riemann_surfaces} \begin{itemize}% \item [[Suresh Govindarajan]], [[Burt Ovrut]], \emph{A geometric interpretation for the torsion constrains of $(2,0)$-heterotic worldhseet supergravity}, Mod. Phys. Lett. A6(1991), 3341. (\href{http://www.physics.iitm.ac.in/~suresh/sgtalk/talk_html/torsion.pdf}{pdf}) \end{itemize} [[!redirects supergravity torsion constraint]] [[!redirects supergravity torsion constraints]] [[!redirects torsion constraint of supergravity]] [[!redirects torsion constraints of supergravity]] \end{document}