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\begin{document}

\begin{titlepage}
\vskip 2em
\begin{center}
{\Large \bf Group manifold approach to supergravity} \\[3em]

\vskip 0.5cm

{\bf
Leonardo Castellani}
\medskip

\vskip 0.5cm

{\sl Dipartimento di Scienze e Innovazione Tecnologica
\\Universit\`a del Piemonte Orientale, viale T. Michel 11, 15121 Alessandria, Italy\\ [.5em] INFN, Sezione di
Torino, via P. Giuria 1, 10125 Torino, Italy\\ [.5em]
Arnold-Regge Center, via P. Giuria 1, 10125 Torino, Italy
}\\ [4em]
\end{center}

\begin{abstract}
\vskip .4cm

We present a short review of the group-geometric approach to supergravity theories, from the point of view
of recent developments. The central idea is the unification of usual diffeomorphisms, gauge symmetries and supersymmetries into superdiffeomorphisms in a supergroup manifold.  The example of $N=1$ supergravity in $d=4$ is discussed in detail, and used to illustrate all the steps in the construction of a group manifold action.  In the Appendices we summarize basic notions of group manifold geometry, and of integration on supermanifolds.
\end{abstract}

\vskip 4cm\vskip .4cm\sk
\noindent {\small Invited chapter for the ``Handbook of Quantum Gravity", Eds. C. Bambi, L. Modesto and I.L. Shapiro, Springer, expected in 2023.}
\vskip 2cm
 \noindent \hrule \vskip .2cm \noindent {\small
leonardo.castellani@uniupo.it}

\end{titlepage}

\newpage
\setcounter{page}{1}

\tableofcontents

\sect{Introduction}

Fundamental interactions are described by field theories with local invariances: the actions
that govern their dynamics are invariant under field transformations involving parameters that are (arbitrary) functions of spacetime. This holds true both for gravity and gauge theories, where the local symmetries are general coordinate, and gauge transformations, respectively.

The essential difference between these two types of local transformations, in their infinitesimal versions, is that diffeomorphisms always contain a derivative of the field,  which is absent in gauge transformations. As well known, this is due to the fact that general coordinate transformations relate fields at different spacetime points, whereas gauge transformations relate fields at the same spacetime point.

Nonetheless, it is possible to give a unified description of diffeomorphisms and gauge transformations.
This we achieve in a group geometrical framework.

The main idea is to consider as basic fields of the theory the {\sl components of the vielbein} one-form $\sigma^A=\sigma(z)^A_{~\Lambda} dz^\Lambda$ on the manifold of a Lie (super)group $G$,  {\small {\it A}} being an index in the $G$ Lie (super)algebra, and $z^\Lambda$ the coordinates of the group manifold.
 This vielbein satisfies the Cartan-Maurer (CM) equations
 \begin{equation}
d \sigma^A + {1 \over 2} C^A_{BC} ~\sigma^B \wedge \sigma^C =0 \label{CM}
\end{equation}
where $C^A_{BC}$ are the structure constants of the $G$ Lie algebra. A brief account of group manifold geometry is given in Appendix A.

The $G$ vielbein $\sigma^A (z)$  has a fixed dependence on the coordinates $z$, and therefore cannot
be a dynamical object. We must consider a ``soft" group manifold,
diffeomorphic to $G$ and denoted by ${\tilde G}$,
with a vielbein $\sigma^A$ not satisfying anymore the CM equations. The amount of deformation from the original ``rigid" group manifold is measured by the {\sl curvature} two-form:
\begin{equation}
R^A \equiv d \sigma^A + {1 \over 2} C^A_{BC} ~\sigma^B \wedge \sigma^C  \label{Gcurvature}
\end{equation}
Tangent vectors
on $\tilde G$, dual to the vielbein $\sigma^A$, are denoted by $t_B$, so that $\sigma^A (t_B)=\delta^A_B$.
\vskip .4cm
Diffeomorphisms along tangent vectors $\varepsilon=\varepsilon^A t_A$ on $\tilde G$ are generated by the Lie derivative $\ell_\varepsilon$.  When applied to the $\tilde G$ vielbein, the variation under diffeomorphisms takes the  form:
\begin{equation}
\ell_\varepsilon \sigma^A = d \varepsilon^A + C^A_{BC} \sigma^B \varepsilon^C + \iota_\varepsilon R^A \label{Lieder}
\end{equation}
where $\iota_\varepsilon$ is the contraction operator, see Appendix A.  On the right-hand side one recognizes the $G$-covariant derivative of the infinitesimal parameter
 $\varepsilon^A$ plus a curvature term. When the curvature term vanishes, i.e. when
 $\iota_\varepsilon R^A=0$, the diffeomorphism
 takes the form of a {\it gauge transformation},  and the
 curvature is said to be {\it horizontal}
 along the $t_A$'s entering the sum in $\varepsilon=\varepsilon^A t_A$.
 Thus in group manifold geometry {\it gauge transformations} can be interpreted as {\it particular
 diffeomorphisms}, along the directions on which the curvatures are horizontal.

 This group geometric setting
 is particularly suited to supergravity theories, where local supersymmetry variations can be interpreted as diffeomorphisms in the super Poincar\'e group manifold, along the fermionic directions. It is then clear how to proceed to find theories invariant under local supersymmetry transformations: we must devise a procedure
 that yields actions, invariant under superdiffeomorphisms. This is very similar in spirit to the superspace approach \cite{GGRS,WB}, where  supergravity multiplets of dynamical (and auxiliary) fields are contained into a single superfield, depending on superspace coordinates. However the group manifold approach has important differences, as we explain in the coming Sections.

The action is obtained with an algorithmic procedure, as the integral of a $d$-form, ``living" on the whole supergroup (soft) manifold $\tilde G$, but integrated on a $d$-dimensional bosonic {\it submanifold} of $\tilde G$. This leads to an ordinary spacetime action containing the dynamical fields (and possibly also the auxiliary fields) of a $d$-dimensional supergravity theory. This algorithm will be discussed in detail and
applied to obtain the action of $N=1$, $d=4$ supergravity.

The original references, where this approach was first proposed,
 are given in \cite{gm11}-\cite{gm14}. Reviews can be found
 in \cite{gm21}-\cite{gm25}, and \cite{EGH} is a standard reference for the use of differential forms
 in gravity and gauge theories.

 The paper is organized as follows. In Section 2 we recall the algebraic basis of $d=4$, $N=1$ supergravity as a theory on the (soft) superPoincar\'e manifold, and the passage to a spacetime action. Section 3 deals with the symmetries of the spacetime action, as inherited from the diffeomorphism invariances of the group manifold action. In Section 4 the variational principle is formulated for the group manifold action, and
 equations of motion are derived. The building rules for (super)group manifold actions are
 discussed in Section 5, and applied to arrive unambiguously at the group manifold Lagrangian for
 $N=1$, $d=4$ supergravity.  Some conclusions, and a selected list of applications and advantages of the group-geometric approach are discussed in Section 6. Finally, the Appendices contain brief accounts of group manifold geometry, integration on supermanifolds and gamma matrix properties.

 \sect{Supergravity from superPoincaré geometry}

 \subsection{Soft superPoincar\'e manifold}

Supergravity in first order vierbein formalism can be recast in a supergroup geometric setting as follows.
Consider $G$ = superPoincar\'e group, and denote the vielbein on the $\tilde G$ manifold as
$\sigma^A = (V^a,\omega^{ab},\psi^\alpha)$. The index $A=(a,ab,\alpha)$ runs on the translations $P_a$, Lorentz rotations $M_{ab}$ and supersymmetry charges $\overline{Q}_\alpha$ of the superPoincar\'e Lie algebra:
 \eqa
 & &  [P_a,P_b] =0  \label{PoincarePP} \\
 & & [M_{ab},M_{cd}]= -{1 \over 2} (\eta_{ad} M_{bc} + \eta_{bc} M_{ad} -\eta_{ac} M_{bd} -\eta_{bd} M_{ac}) \label{PoincareMM} \\
 & & [M_{ab},P_c]= -{1 \over 2} ( \eta_{bc} P_{a} - \eta_{ac} P_{b}) \label{PoincareMP} \\
 & & [P_a,\overline{Q}_\alpha]=0 \\
 & & [M_{ab},\overline{Q}_\beta]= -{1 \over 4} \overline{Q}_\alpha (\gamma_{ab})^\alpha_{~\beta}   \label{sPoincareMQ} \\
 & & \{ \overline{Q}_\alpha,\overline{Q}_\beta \} = -i (C\gamma^a)_{\alpha \beta}  P_{a}, \label{sPoincareQQ}
  \end{eqnarray}
$\eta$ being the flat Minkowski metric, and $C_{\alpha\beta} $ the charge conjugation matrix.  The spinorial generator $\overline{Q}_\alpha  \equiv Q^\beta
C_{\beta\alpha}$ is a Majorana spinor, i.e. $Q^\beta C_{\beta\alpha} = Q^\dagger_\beta (\gamma_0)^\beta_{~\alpha}$.
Thus the super-Poincar\'e manifold has 10 bosonic directions with coordinates $x^a$, $y^{ab}$, parametrizing  translations and Lorentz rotations, and 4 fermionic directions with Grassmann coordinates $\theta^\alpha$, corresponding to the 4 supercharges $\overline{Q}_\alpha, \alpha=1,..4$.

The components of the supervielbein of the $\tilde G$ =(soft) superPoincar\'e manifold are the vierbein $V^a$, the spin connection $\omega^{ab}$ and the
gravitino $\psi^\alpha$.
 corresponding respectively to the generators $P_a$, $M_{ab}$ and $\overline{Q}_\alpha$,

 Using the structure constants of the Lie superalgebra, the curvature (\ref{Gcurvature})
becomes :
\eqa
& & R^{a}= dV^{a} - \omega^{a}_{~c} V^{c} - \frac{i }{2} \bar\psi\gamma^{a} \psi \equiv {\cal D} V^a  - \frac{i }{2} \bar\psi\gamma^{a} \psi\label{RasuperPoincare}\\
& & R^{ab}=d \omega^{ab} - \omega^{a}_{~c} ~\omega^{cb} \label{RabsuperPoincare}\\
& & \rho= d \psi- \frac{1}{ 4} \omega^{ab} \gamma_{ab} \psi \equiv {\cal D} \psi \label{rhosuperPoincare}
\end{eqnarray}
defining respectively the supertorsion, the
Lorentz curvature and the gravitino field strength. ${\cal D}$ is the Lorentz covariant exterior derivative.
Wedge products between forms are understood when omitted.

Taking the exterior derivative of these definitions yields the Bianchi identities:
\eqa
& & dR^a -\omega^a_{~b} R^b + R^a_{~b} V^b - i \bar \psi \gamma^a \rho \equiv  {\cal D} R^a + R^a_{~b} V^b - i \bar \psi \gamma^a \rho= 0 \label{BianchiRasuperPoincare}\\
& & dR^{ab} - \omega^a_{~c} R^{cb} +  \omega^b_{~c} R^{ca} \equiv {\cal D} R^{ab}=0 \label{BianchiRabsuperPoincare}\\
& & d\rho - {1 \over 4} \omega^{ab} \gamma_{ab} \rho + {1 \over 4} R^{ab} \gamma_{ab} \psi \equiv {\cal D} \rho
+ {1 \over 4} R^{ab} \gamma_{ab} \psi =0 \label{BianchirhosuperPoincare}
\end{eqnarray}
At this stage all the fields depend on all $\tilde G$ manifold coordinates, corresponding
to the generators of the Lie superalgebra: thus $V^a=V^a (x,y,\theta)$, $\omega^{ab} = \omega^{ab}(x,y,\theta)$, $\psi^\alpha (x,y,\theta)$, where the coordinates $x^a$, corresponding to the translations $P_a$, describe usual spacetime. Moreover the one-forms $V^a$, $\omega^{ab}, \psi$ live on the whole
$\tilde G$, and therefore can be expanded as:
 \eqa
  & & V^a = V^a_\mu (x,y,\theta) dx^\mu + V^a_{\mu\nu} (x,y,\theta) dy^{\mu\nu} + V^a_\alpha (x,y,\theta) d\theta^\alpha \\
  & & \omega^{ab} = \omega^{ab}_\mu (x,y,\theta) dx^\mu + \omega^{ab}_{\mu\nu} (x,y,\theta)dy^{\mu\nu}
  +\omega^{ab}_{\alpha} (x,y,\theta)d\theta^{\alpha}\\
  & & \psi^\alpha=\psi^\alpha_\mu (x,y,\theta) dx^\mu + \psi^\alpha_{\mu\nu} (x,y,\theta) dy^{\mu\nu} +\psi^\alpha_\beta (x,y,\theta) d\theta^\beta
  \end{eqnarray}

\subsection{Group manifold action}

The overabundance of field components, and their dependence on $y$ and $\theta$ coordinates
can be tamed by defining an appropriate action principle. To end up with a geometrical theory in
four spacetime dimensions, we first construct a 4-form Lagrangian $L$ made out of the
$\tilde G$ vielbein $\sigma^A$ and its curvature $R^A$, according to a few
building rules to be discussed in Section 5. The resulting Lagrangian for superPoincar\'e supergravity is given by:
\begin{equation}
L =
R^{ab} V^{c} V^{d} \epsilon_{abcd} + 4
\bar\psi\gamma_{5} \gamma_{a} \rho V^{a}
\end{equation}
We then define an action by integrating this Lagrangian on a 4-dimensional submanifold $M^4$
of the $\tilde G$ manifold, spanned by the $x$ coordinates.

Integration on submanifolds $M^d$ of a $d$-form $L$ that lives on a $g$-dimensional bigger space $\tilde G$ is carried out as follows: we multiply $L$ by the {\it Poincar\'e dual} of $M^d$, a (singular) closed  ({\it g-d})-form $\eta_{M^d}$ that localizes the Lagrangian on the submanifold $M^d$, and integrate the resulting $g$-form on the whole $\tilde G$. Thus the group manifold action has the general expression
\begin{equation}
 S= \int_{\tilde G} L \wedge  \eta_{M^d}    \label{Gintegral}
\end{equation}

The fields of the theory are those contained in $L$, i.e. the $\tilde G$ vielbein components, and the
embedding functions that define the $M^d$ submanifold of $\tilde G$, present in $\eta_{M^d}$. We will see
later that the embedding functions do not enter the field equations obtained from the variation of (\ref{Gintegral}). This program makes use of standard integration theory when $\tilde G$ is a bosonic space, but requires some new ingredients when $\tilde G$ is a supermanifold, discussed in Appendix B.

In our $d=4$ supergravity example the group manifold action is the integral on the 14-dimensional  $\tilde G$ = soft superPoincar\'e manifold:
\begin{equation}
S=  \int_{\tilde G} (R^{ab}V^c V^d \epsilon_{abcd}  + 4
\bar\psi\gamma_{5} \gamma_{a} \rho V^{a})~  \eta_{M^4}    \label{GintegralSP}
\end{equation}

\subsection{Spacetime action}

A spacetime action, i.e. an action that is the integral on $M^4$ of a Lagrangian containing fields depending only on $x$, is obtained from  (\ref{GintegralSP}) with a {\it particular choice} of  $\eta_{M^4}$. Integration on
$y$ and $\theta$ coordinates produces then the spacetime action. This particular Poincaré dual is the product of two pieces: $\eta_{M^4}= \eta_y  \wedge \eta_\theta$, where $\eta_y$ is a (singular) 10-form that  localizes
the Lagrangian on the $y=0$ hypersurface:
 \begin{equation}
  \eta_{y} =  \delta (y^{12}) \delta (y^{13}) \cdots \delta (y^{34}) dy^{12} \wedge dy^{13} \wedge \cdots \wedge dy^{34}  \label{eta6form}
  \end{equation}
 Integration on the $y$ coordinates reduces (\ref{GintegralSP}) to an integral on the superspace $M^{4|4}$ spanned by $x$ and $\theta$. The $y$
 dependence of all fields in $L$ disappears because of the delta functions in $\eta_y$, and
 the ``legs" of $L$ along $dy$ differentials are killed by the product of all independent $dy^{\mu\nu}$ in $\eta$. The other piece of $\eta_{M^4}$ (see Appendix B), after integration on $\theta$ coordinates, produces
 an integral on $M^4$ of a Lagrangian $4$-form, not depending any more on the $\theta$ and on the $d\theta$ differentials.
 Thus the action
 \eqa
 & & S_{spacetime}= \int_{\tilde G} L \wedge \eta_{M^4} = \int_{M^4} L_{y=0,dy=0,\theta=0,d\theta=0} \nonumber \\
 & & ~~~~~~~~~~~~ = \int_{M^4} R^{ab}V^c V^d \epsilon_{abcd}  + 4
\bar\psi\gamma_{5} \gamma_{a} \rho V^{a} \label{M4integral}
 \end{eqnarray}
contains only the usual fields $V^a_\mu(x)$ and $\omega^{ab}_\mu (x)$ and $\psi(x)$ of $N=1$ supergravity, and reproduces the first order supergravity action.
 \vskip .4cm
  \noindent {\bf Note 1:} $\eta_y$ is closed (because it contains ``functions" depending on $y$ multiplied by all the $dy$ differentials) and not exact (because of the Dirac deltas $\delta(y)$), and thus belongs  to a nontrivial de Rahm cohomology class. In general deformations of
the $M^4$ surface generated by diffeomorphisms leave the Poincar\'e dual $\eta$ in the
same cohomology class, since the Lie derivative commutes with the exterior derivative.
  \vskip .4cm
    \noindent {\bf Note 2:}
    We will always assume that integration on the Lorentz coordinates
has been carried out, so that all fields depend only on $x$ and $\theta$ coordinates. Moreover all curvatures
are taken to be horizontal in the Lorentz directions. As a consequence  the theory lives in a superspace $M^{4|4}$ spanned by four bosonic coordinates $x^a$ and four fermionic coordinates $\theta^\alpha$.
\vskip .4cm
\noindent {\bf Note 3:} the spacetime action (\ref{M4integral}) and its invariance under the supersymmetry transformations (\ref{susy1})-(\ref{susy3}) were first found in ref. \cite{FFvN} in second
order formalism and in \cite{DZ} in first order formalism, see also the standard references
 \cite{PvNreport,FVP} on supergravity.

  \sect{Symmetries}

  The action (\ref{GintegralSP}) is the integral on $\tilde G$ of a top form: it is clearly invariant
  under diffeomorphisms on $\tilde G$. But what we are really interested in are the symmetries
  of the spacetime action as given in (\ref{M4integral}), where the variations are carried out only in the $x$-dependent fields in $L|_{y=dy=0,\theta= d\theta=0}$. The only symmetries guaranteed a priori
   are the 4-dimensional spacetime diffeomorphisms, the spacetime action being an integral of a 4-form on $M^4$.

  Here resides most of the power of the group manifold formalism: if one considers
  the ``mother"  action (\ref{Gintegral}) on $\tilde G$, the guaranteed symmetries are {\it all} the diff.s on $\tilde G$, generated by the Lie derivative $\ell_\varepsilon$ along the tangent vectors  $\varepsilon = \varepsilon^A t_A$ of $\tilde G$.
  But how do these symmetries transfer to the spacetime action ?

  The variation of the group manifold action under diff.s generated by $\ell_\epsilon$ is\footnote{Recall $\ell_\varepsilon =  \iota_\varepsilon d + d \iota_\varepsilon$ so that $\ell_\varepsilon$(top form) = $d(\iota_\varepsilon$ top form)}
  \begin{equation}
   \delta S = \int_{\tilde G}  \ell_\varepsilon (L \wedge \eta )= \int_{\tilde G}  (\ell_\varepsilon L) \wedge \eta + L \wedge \ell_\varepsilon \eta =0
   \end{equation}
  modulo boundary terms. One has to vary the fields\footnote{Since $\ell_\varepsilon$ satisfies the Leibniz rule,   $\ell_\varepsilon L$ can be computed by varying in turn all fields inside $L$.} in $L$ as well as the submanifold embedded in $\tilde G$:  the sum of these two variations gives zero\footnote{In the following the vanishing of action variations will always be understood modulo boundary terms.} on the group manifold action $S$.
    But what we need
   in order to have a {\it spacetime} interpretation of all the symmetries of $S$, is really
   \begin{equation}
   \delta S = \int_{\tilde G} (\ell_\varepsilon L) \wedge \eta =0 \label{spacetimesymm}
   \end{equation}
 If this holds, varying the fields $\phi$ inside $L$ with the Lie derivative $\ell_\epsilon$ as in (\ref{Lieder}), and then projecting on spacetime,  yields spacetime variations
 \begin{equation}
\delta \phi (x) =  \ell_\varepsilon \phi (x,y,\theta) |_{x}
 \end{equation}
  that leave the spacetime action (\ref{M4integral}) invariant. We have denoted by $|_x$ the projection on spacetime due to the integration on $y$ and $\theta$ coordinates in (\ref{spacetimesymm}). We call these
  variations {\it spacetime invariances}, since they leave invariant the spacetime action.  They originate from the diff. invariance of the group manifold action,
 and give rise to symmetries of the spacetime action  (\ref{M4integral}) only when (\ref{spacetimesymm})
 holds. This happens if one of the following conditions is satisfied:
 \vskip .4cm
 \noindent $\bullet$ the Lie derivative on $\eta$ vanishes:
 \begin{equation}
 \ell_\varepsilon \eta = 0 \label{elloneta}
 \end{equation}
 \vskip .4cm
\noindent  $\bullet$ the spacetime projection of the Lie derivative of $L$ is exact:
 \begin{equation}
 ( \ell_\varepsilon L )|_{x} = d \alpha  \label{ellonL}
  \end{equation}
  \noindent In this case the variation (\ref{spacetimesymm})
  \begin{equation}
  \delta S = \int_{\tilde G}  (\ell_\varepsilon L) \wedge \eta =\int_{M^4}    (\ell_\varepsilon L )|_{x}
  \end{equation}
  vanishes after integration by parts.
  The requirement  (\ref{ellonL}) is equivalent to
  \begin{equation}
 ( \iota_\varepsilon dL )|_{x} = d \alpha'  \label{idonL}
  \end{equation}
 since $\l_\varepsilon = \iota_\varepsilon d + d \iota_\varepsilon$.
 \vskip .4cm
 \noindent The Lagrangian $L$ depends on the $\tilde G$-vielbein $\sigma^A$ and its curvature $R^A$, so that also $dL$, after use of Bianchi identities, is expressed in terms of $\sigma^A$ and $R^A$. Then condition (\ref{idonL})  translates
 into a {\it condition on the contractions} $\iota_\varepsilon R^A$, i.e. a condition on the curvature components.

 Let us see how this works for superPoincar\'e supergravity.

  \subsection{Symmetries of $d=4$ supergravity}

  The symmetries of the spacetime action  (spacetime invariances) are those
generated by a Lie derivative $\ell_\varepsilon$ such that $\iota_\varepsilon dL|_{x}= d\alpha'$,
cf. (\ref{idonL}). We need to compute $dL$. Using the Bianchi identities
(\ref{BianchiRabsuperPoincare}) and (\ref{BianchirhosuperPoincare}), and the definition
of the torsion $R^a$ in (\ref{RasuperPoincare}) we find:
\begin{align}
& dL= 2 R^{ab} R^c V^d \varepsilon_{abcd} + i R^{ab} \bar \psi \gamma^c \psi V^d \varepsilon_{abcd}+
4 \bar \rho \gamma_5 \gamma_a \rho V^a + \nonumber \\
&  ~~~~~~~ + \bar \psi \gamma_5 \gamma_c \gamma_{ab} \psi R^{ab} V^c -4 \bar \psi \gamma_5 \gamma_a \rho R^a - 2i \bar \psi \gamma_5 \gamma_a \rho \bar \psi \gamma^a \psi \label{dL1}
\end{align}
The gamma matrix identity
\begin{equation}
\gamma_c \gamma_{ab} = \eta_{ac} \gamma_b - \eta_{bc} \gamma_a +i \varepsilon_{abcd}\gamma_5 \gamma^d
\end{equation}
implies $\bar \psi \gamma_5 \gamma_c \gamma_{ab} \psi =i \varepsilon_{abcd} \bar \psi \gamma^d \psi$, so that the second and the fourth term cancel in
(\ref{dL1}). Moreover from the Fierz identity in Appendix C one deduces
\begin{equation}
\gamma_a  \psi \bar \psi \gamma^a \psi =0 \label{fierz1}
\end{equation}
and since $\bar \psi \gamma_5 \gamma_a \rho = \bar \rho \gamma_5 \gamma_a \psi$ also the last term in
(\ref{dL1}) vanishes due to (\ref{fierz1}). Therefore
\begin{equation}
dL= 2 R^{ab} R^c V^d \varepsilon_{abcd} + 4 \bar \rho \gamma_5 \gamma_a \rho V^a
  - 4 \bar \psi \gamma_5 \gamma_a \rho R^a
\end{equation}
\vskip .4cm
\noindent {\bf Lorentz gauge transformations}
\vskip .4cm
It is immediate to see that if all curvatures are horizontal in the Lorentz directions
(no ``legs" along $\omega$) then indeed $\iota_{\varepsilon^{ab} t_{ab}}dL=0$, and Lorentz
transformations are a spacetime invariance of the supergravity action. This is essentially due to the absence of bare $\omega^{ab}$ in $L$. The general
diffeomorphism formula (\ref{Lieder}) yields then the usual Lorentz transformations
 \eqa
 & & \ell_{\varepsilon^{cd} t_{cd}} V^a =   \varepsilon^a_{~b} V^b  \label{LorentzonV2}\\
 & & \ell_{\varepsilon^{cd} t_{cd}}  \omega^{ab} = d \varepsilon^{ab} - \omega^a_{~c} \varepsilon^{cb} + \omega^b_{~c} \varepsilon^{ca}= {\cal D} \varepsilon^{ab} \label{Lorentzonom2} \\
 & &  \ell_{\varepsilon^{cd} t_{cd}} \psi = {1 \over 4} \varepsilon^{ab} \gamma_{ab} \psi
 \end{eqnarray}
We can check directly the invariance of the action under these variations: all curvatures and vierbeins appearing in (\ref{M4integral}) transform
homogeneously, and Lorentz indices are contracted with Lorentz invariant tensors.
\vskip .4cm
 \noindent {\bf Spacetime diffeomorphisms}
 \vskip .4cm
 \noindent Ordinary diff.s along tangent vectors $\partial_\mu$ dual to $dx^\mu$ are invariances of the spacetime action, since (\ref{M4integral}) is an integral on a 4-dimensional manifold of a 4-form.
 \vskip .4cm
  \noindent {\bf Supersymmetry transformations}
  \vskip .4cm
  \noindent Diff.s along tangent vectors $t_\alpha$ dual to $\psi^\alpha$ are spacetime invariances
  provided $\iota_\epsilon dL|_{x}= total ~derivative$ with $\epsilon = \epsilon^\alpha t_\alpha$, that is to say
  \eqa
 & &   \iota_\epsilon dL = 2 (\iota_\epsilon R^{ab}) R^c V^d \varepsilon_{abcd} + 2 R^{ab} (\iota_\epsilon R^c ) V^d \varepsilon_{abcd} + 8 \bar \rho \gamma_5 \gamma_a (\iota_\epsilon \rho) V^a \nonumber \\
 & & - 4 \bar \epsilon \gamma_5 \gamma_a \rho R^a - 4 \bar \psi \gamma_5 \gamma_a (\iota_\epsilon \rho) R^a
 - 4 \bar \psi \gamma_5 \gamma_a \rho (\iota_\epsilon R^a) = tot.~ der. \label{idL2}
   \end{eqnarray}
once projected on spacetime.  This is a condition for the contractions on the curvatures, and it is satisfied by:
 \eqa
 & & \iota_\epsilon R^a =0 \label{rh1}\\
 & & \iota_\epsilon R^{ab} = - \varepsilon^{abef} \bar \rho_{ef} \gamma_5 \gamma_g \epsilon V^g - \varepsilon^{efg[a} \bar \rho_{ef} \gamma_5 \gamma_g \epsilon V^{b]}  \equiv {\bar \theta}^{ab}_c \epsilon V^c \label{rh2}\\
  & & \iota_\epsilon \rho =0 \label{rh3}
 \end{eqnarray}
Thus we have supersymmetry invariance of the spacetime action if the curvatures have the following parametrization on a basis of 2-forms:
 \eqa
 & & R^a=R^a_{~bc} ~V^b V^c  \label{param1}\\
 & & R^{ab}= R^{ab}_{~~cd} V^c V^d + {\bar \theta}^{ab}_c ~\psi ~V^c  \label{param2}\\
 & & \rho= \rho_{ab}~ V^a V^b \label{param3}
 \end{eqnarray}
 where we have taken into account also horizontality in the Lorentz directions. The conditions (\ref{rh1})-(\ref{rh3}) are called ``rheonomic conditions", and similarly (\ref{param1})-(\ref{param3}) are called
 ``rheonomic parametrizations" of the curvatures.

  The diff.s along $\epsilon=\epsilon^\alpha t_\alpha$ (supersymmetry transformations) act on the fields according to the general formula
   (\ref{Lieder}), where the contractions on the curvatures are given in (\ref{rh1})-(\ref{rh3}):
 \eqa
  & & \ell_\epsilon V^a = i \bar \epsilon \gamma^a \psi \label{susy1}\\
 & & \ell_\epsilon \omega^{ab} =  {\bar \theta}^{ab}_c \epsilon V^c  \label{susy2}\\
 & & \ell_\epsilon \psi = {\cal D} \epsilon \equiv d \epsilon - {1 \over 4} \omega^{ab} \gamma_{ab} \epsilon \label{susy3}
 \end{eqnarray}
with ${\bar \theta}^{ab}_c$ defined in (\ref{rh2}).

 \sect{Variational principle and field equations}

The group manifold action (\ref{Gintegral}) is a functional of $L$ and of the embedded submanifold $M$, and
therefore varying the action means varying both $L$ and $M$. Varying $M$
corresponds to varying $\eta_{M}$. Then the variational principle reads:
\begin{equation}
\label{Svariation}{\ \delta S[L, M] =
\int_{\tilde G} ( \delta L\wedge\eta_{M} + L\wedge\delta\eta_{M}) } =0\,.
\end{equation}
Any (continuous) variation of $M$ can be obtained by acting on
$\eta_M$ with a diffeomorphism generated by a Lie derivative $\ell_\xi$.
An arbitrary variation is generated by an arbitrary $\xi$ vector, and
the variational principle becomes
\begin{equation}
\label{Svariation1}{\ \delta S[L, M] =
\int_{\tilde G} ( \delta L\wedge\eta_{M} + L\wedge\ell_\xi \eta_{M}) } =0\,.
\end{equation}
Since field
variations in $L$ and variation of $M$ are independent, the two terms in (\ref{Svariation1})
must vanish separately. From the vanishing of the first one we deduce
\begin{equation}
\int_{\tilde G} (  \delta \phi \wedge {\partial L \over \partial \phi}  +  d \delta \phi \wedge {\partial L \over \partial (d\phi) } ) \wedge \eta_M =0
\end{equation}
where $L=L(\phi,d\phi)$ is considered a function of the 1-form fields $\phi$ and their ``velocities" $d\phi$.  A summation on all fields is understood.
Integrating by parts and recalling $d \eta_M=0$ yields
 \begin{equation}
 \int_{\tilde G} \delta \phi \wedge ( {\partial L \over \partial \phi}  + d {\partial L \over \partial (d\phi) } ) \wedge \eta_M =0
\end{equation}
and since the $\delta \phi$ are arbitrary we find
 \begin{equation}
  ({\partial L \over \partial \phi} + d {\partial L \over \partial (d\phi)}) \wedge  \eta_M=0 \label{ELequations0}
  \end{equation}
This must hold for any $\eta_M$ (i.e. for generic embedding functions): we arrive therefore at equations
that hold on the whole $\tilde G$, and are the form version of
  the Euler-Lagrange equations:
  \begin{equation}
{\partial L \over \partial \phi} + d {\partial L \over \partial (d\phi)}  =0 \label{ELequations1}
\end{equation}
If $L$ is a $d$-form, these equations are $(d-1)$-forms. Their content can be examined by
expanding them along a complete basis of  $(d-1)$-forms in $\tilde G$.

Requiring the vanishing of the second term in the variation (\ref{Svariation1}) does
not imply further equations besides the Euler-Lagrange field equations (\ref{ELequations1}): indeed this term vanishes on the shell of solutions of Euler-Lagrange equations. To prove it, notice that
\begin{equation}
\int_{\tilde G} L \wedge \ell_\xi \eta_M = - \int_{\tilde G} \ell_\xi L \wedge \eta_M =0 ~(on~shell) \label{onshell}
\end{equation}
because $\ell_\xi L$ is just a particular variation of $L$, under which the action remains stationary on-shell.

Thus the group manifold variational principle leads to the field equations (\ref{ELequations1}),
holding as $(d-1)$-form equations on the whole $\tilde G$.
\vskip .4cm
\noindent {\bf Note 1:} The variational principle
{\it does not determine} the embedding of $M$ into $\tilde G$.
\vskip .4cm
\noindent {\bf Note 2:} the field equations (\ref{ELequations1}) are form equations, and therefore
invariant under the action of a Lie derivative. More precisely, if $\phi$ is a solution of
(\ref{ELequations1}), so is $\phi + \ell_\varepsilon \phi$: Lie derivatives generate symmetries
of the field equations.

\vskip .4cm
\noindent Finally, we have the following
\vskip .4cm
\noindent {\bf Theorem:} $dL = 0 ~(on~shell) $
\vskip .4cm
\noindent i.e. the Lagrangian, as a $d$-form on $\tilde G$, is closed on shell. To prove it
recall that
$\eta_M$ is closed , so that  on shell we find, cf. (\ref{onshell}):
\begin{equation}
0 = \int_{\tilde G} L \wedge \ell_\xi \eta_M =  \int_{\tilde G} L \wedge d \iota_\xi \eta_M =- (-)^d  \int_{\tilde G} dL \wedge \iota_\xi \eta_M
\end{equation}
$\xi$ being arbitrary, this implies $dL=0$ (on shell)\footnote{
In fact, this is just Stokes theorem applied to a region of $\tilde G$ bounded by two different hypersurfaces $M$ and $M'$.} \square

Let us apply the preceding discussion to the superPoincar\'e supergravity example.

\subsection{Supergravity field equations}

The variational equations (\ref{ELequations1}) for the group manifold action (\ref{GintegralSP}) read:
\eqa
& & 2 R^{c}  V^{d} \epsilon_{abcd}  = 0   \label{sPoincarefieldeqRa}\\
& & 2 R^{ab}  V^{c} \epsilon_{abcd} + 4 \bar \psi \gamma_5 \gamma_d \rho = 0 \label{sPoincarefieldeqRab} \\
& & 8 \gamma_5 \gamma_a \rho V^a -4 \gamma_5 \gamma_a \psi R^a=0 \label{sPoincarefieldeqrho}
\end{eqnarray}
obtained varying $\omega^{ab}, V^d$ and $\psi$ respectively.
The analysis proceeds as follows: we first expand the curvatures on a basis of 2-forms\footnote{assuming
horizontality in the Lorentz directions. This amounts to consider configurations
satisfying the Lorentz horizontality constraints on the curvatures.}
\eqa
& & R^a = R^a_{~bc} V^b V^c + {\bar \theta}^a_{~c} \psi V^c + \bar \psi K^a \psi \\
& & R^{ab} = R^{ab}_{~~cd} V^c V^d + {\bar \theta}^{ab}_{~~c} \psi V^c + \bar \psi K^{ab} \psi \\
& & \rho= \rho_{ab} V^a V^b + H_c \psi V^c + \Omega_{\alpha\beta} \psi^\alpha \psi^\beta
\end{eqnarray}
and then insert them into the field equations (\ref{sPoincarefieldeqRa})-(\ref{sPoincarefieldeqrho}). These, being 3-form equations, can be expanded on the basis $\psi\psi\psi$,  $\psi\psi V$, $\psi VV$, $VVV$. Their content is given below (the three lines correspond to the three eq.s of motion):
\vskip .4cm
\noindent$\psi\psi\psi$ sector:
\eqa
& &\Omega_{\alpha\beta} =0\\
& & 0=0\\
& & K^a=0
\end{eqnarray}
$\psi\psi V$ sector:
\eqa
& & 2 \bar \psi K^{ab} \psi V^c \varepsilon_{abcd} + 4 \bar \psi \gamma_5 \gamma_d H_c \psi V^c=0 \label{psipsiV1}\\
& &  ~~~~~~~~~~~~~~~~~~0=0\\
& & ~~~~~~~~~~~~~~~~~~{\bar \theta}^a_{~c}=0
\end{eqnarray}
$\psi V V$ sector:
\eqa
& & 2 {\bar \theta}^{ab}_{~~e} \psi V^e V^c \varepsilon_{abcd} + 4 \bar \psi \gamma_5 \gamma_d \rho_{ab} V^a V^b =0 \label{psiVV1}\\
& &  ~~~~~~~~~~~~~~~~~~0=0\\
& & \gamma_5 \gamma_a H_b \psi V^b V^a - 4 \gamma_5 \gamma_c \psi R^c_{~ab} V^a V^b =0 \label{psiVV3}
\end{eqnarray}
$V V V$ sector:
\eqa
& & R^a_{~bc}=0 \\
& & R^{ac} _{~~bc} - {1\over 2} \delta^a_b ~R^{cd}_{~~cd} =0 \\
& & \gamma^a \rho_{ab}=0
\end{eqnarray}
Inserting $R^a_{~bc}=0$ into (\ref{psiVV3}) yields $H_c=0$, which used in (\ref{psipsiV1})
gives $K^{ab}=0$. Thus the only nontrivial relation in the ``outer" projections is (\ref{psiVV1}),
that determines $\theta^{ab}_{~~c}$ to be
\begin{equation}
\theta^{ab}_{~~c}=-\varepsilon^{abef} \bar \rho_{ef} \gamma_5\gamma_c - \delta^{[a}_c \varepsilon^{b]efg} \bar \rho_{ef} \gamma_5\gamma_g  \label{thetabar1}
\end{equation}
in agreement with the $\theta^{ab}_{~~c}$ obtained from the condition (\ref{rh2}). Thus we arrive at the same curvature parametrizations (\ref{param1})-(\ref{param3})
obtained in Sect. 3.1 by requiring spacetime supersymmetry invariance.

Finally, the $VVV$ sector reproduces the (super)torsion equation, and the propagation equations
for the vierbein and the gravitino.

\vskip .4cm
\noindent{\bf Note:} from the torsion equation
\begin{equation}
2 R^a_{\mu\nu} \equiv \partial_\mu V^a_\nu - \partial_\nu V^a_\mu - \omega^a_{~b,\mu} V^b_\nu +
\omega^a_{~b,\nu} V^b_\mu - i \bar \psi_\mu \gamma^a \psi_\nu =0
\end{equation}
we can express the spin connection in terms of $V$ and $\psi$, recovering second order formalism:
\eqa
& & \omega_{ab,\mu} = {1 \over 2} V^\nu_a V^\rho_b \eta_{cd} ~(\partial_{[\mu} V_{\nu]}^c V_\rho^d - \partial_{[\mu} V_{\rho]}^c V_\nu^d + \partial_{[\nu} V_{\rho]}^c V_\mu^d) + \nonumber \\
& &  ~~~~~~~~ + {i \over 4} V_a^\nu V_b^\rho (\bar \psi_\mu \gamma_\nu \psi_\rho+
\bar \psi_\nu \gamma_\rho \psi_\mu-\bar \psi_\rho \gamma_\mu \psi_\nu-(\nu \leftrightarrow \rho)) \label{omsecondorder}
\end{eqnarray}

 \sect{Building rules}

 The group geometric approach provides a systematic set of building rules \cite{gm21} for
 constructing Lagrangians of supersymmetric theories:
 \vskip .4cm
 1) Choose a Lie (super)algebra $G$, containing generators $P_a$ that can be
 associated to $d$ spacetime directions, and a Lorentz-like subalgebra $H$.
Examples are the superPoincar\'e algebras in $d$ dimensions or their uncontracted versions
 (orthosymplectic superalgebras $OSp(N|2^{[d/2]})$). The fields of the theory
 are the vielbein components of the soft group manifold $\tilde G$.
 \vskip .4cm
 2) Construct the most general $d$-form on $\tilde G$, by multiplying (with exterior products) 1-form  vielbein components $\sigma^A$ and 2-form curvatures $R^A$, without bare Lorentz connection and contracting indices with $H$-invariant tensors, so that the resulting Lagrangian
 is a Lorentz scalar.
 \vskip .4cm
 3) Require that the variational equations admit the ``vacuum solution"  $R^A=0$, described by the
 vielbein of the rigid group manifold $G$.
 \vskip .4cm
4) The construction is greatly helped by scaling properties of the fields, dictated by the structure
 of the Lie (super)algebra $G$, or equivalently by the Cartan-Maurer equations for the $G$ vielbein. Consider for example the superPoincar\'e algebra: it is invariant under the rescalings $P_a \rightarrow \lambda P_a, M_{ab} \rightarrow M_{ab}, \overline{Q}_\alpha \rightarrow \lambda^{1\over 2} \overline{Q}_\alpha$. Then the curvature definitions (\ref{RasuperPoincare})-(\ref{rhosuperPoincare}) are invariant under
 \begin{equation}
 V^a \rightarrow \lambda V^a, ~~~\omega^{ab} \rightarrow \omega^{ab}, ~~~\psi \rightarrow \lambda^{1\over 2} \psi \label{rescalings1}
 \end{equation}
 The field equations must be invariant under these rescalings, and therefore the action must scale homogeneously under (\ref{rescalings1}). Since the Einstein-Hilbert term scales as $\lambda^2$, all terms must scale in the same way, and this restricts the candidate terms in the Lagrangian.
 \vskip .4cm
 5) Finally, requiring
 that all terms have the same parity as the Einstein-Hilbert term further narrows the list of candidates.
 \vskip .4cm

  \subsection{The Lagrangian for $d=4$ supergravity}

 Following the above rules, one arrives at the  $d=4$ supergravity action (\ref{GintegralSP}). We recall here
 the steps of the procedure \cite{gm21}.  The most general lagrangian 4-form satisfying Rule 1 can at most contain two curvatures, and is therefore of the type:
 \begin{equation}
 L = R^A R^B \nu_{AB} + R^A \nu_A + \Lambda
 \end{equation}
 with\footnote{repeated indices are contracted with the Minkowski flat metric.}
 \begin{equation}
 R^A R^B \nu^{(2)}_{AB}=c_1 R^{ab} R^{cd} \varepsilon_{abcd}+c_2 R^{ab}R^{ab}
+c_3R^a R^a +c_4 \bar\rho \rho+c_5 \bar\rho \gamma_5 \rho  \label{quadratic}
\end{equation}

\noindent The first two are total derivatives, and are related to the Euler characteristic and to the
Pontriagyn number of $M^4$. The last three can be
reduced to linear terms in the curvatures plus total derivatives.
Actually scaling invariance  eliminates all the terms in
(\ref{quadratic}) except
$R^a R^a$, since the Einstein term scales as $\lambda^2$.
The torsion-squared term can be reduced to a linear term since

\begin{equation}
R^a R^a = ({\cal D} V^a - {i \over 2} {\bar \psi} \gamma^a \psi)R^a=d(V^a R^a)+
V^a(-R^{ab}V^b+i{\bar \psi}\gamma^a\rho)-{i \over 2} {\bar \psi} \gamma^a \psi R^a
\end{equation}

\noindent in virtue of the Bianchi identity (\ref{BianchiRasuperPoincare}). This leaves us with a
lagrangian of the form:

\begin{equation}
L=\Lambda+\nu_{ab} R^{ab} + \nu_a R^a + {\bar\nu} \rho
\end{equation}

\noindent where

\eqa
 & &\Lambda=\alpha_1 \varepsilon_{abcd} V^aV^bV^cV^d+i\alpha_2\varepsilon_{abcd}{\bar \psi}
\gamma^{ab} \psi V^cV^d + i\alpha_3 {\bar \psi} \gamma^{ab} \psi V^aV^b \\
& &\nu_{ab}= \beta_1 \varepsilon_{abcd} V^c V^d + \beta_2 V^a V^b + i \beta_3
{\bar \psi} \gamma_{ab} \psi + i \beta_4 \varepsilon_{abcd} {\bar \psi} \gamma^{cd} \psi \\
& &\nu_a=i \eta_1 {\bar \psi} \gamma_a \psi\\
& &\nu=\delta_1 \gamma_5 \gamma_a \psi V^a + i \delta_2 \gamma_a \psi V^a
\end{eqnarray}

\noindent are the most general Lorentz covariant terms. Notice that the only
nonvanishing $\psi\psi$ currents are ${\bar \psi} \gamma^a \psi$ and ${\bar \psi} \gamma^
{ab} \psi$. Correct $\lambda^2$
scaling of $L $ drastically reduces the possible
terms:
$\alpha_1=\alpha_2=\alpha_3=\beta_3=\beta_4=0$. Moreover parity
 implies $\beta_2=\eta_1= \delta_2=0$ (all terms must
have the same parity as the Einstein term $R^{ab} V^c V^d \varepsilon_{abcd}$,
i.e. must be pseudoscalars). Thus we finally have:

\begin{equation}
L=\beta_1 \varepsilon_{abcd} R^{ab} V^c V^d + \delta_1 {\bar \psi} \gamma_5 \gamma_a
\rho V^a  \label{Lagrangian2}
\end{equation}

\noindent The requirement 3) that
the vacuum be a solution of the field equations fixes the last parameter
$a=\delta_1/ \beta_1$. Indeed the field equations obtained from the Lagrangian
(\ref{Lagrangian2}) by varying $V^a, \omega^{ab}$ and $\psi$ are respectively:

\eqa
& & 2 R^{ab} V^c \varepsilon_{abcd} + a {\bar \psi} \gamma_5 \gamma_d \rho = 0 \\
& & 2{\cal D} V^c V^d \varepsilon_{abcd} + {1 \over 4} a {\bar \psi} \gamma_5 \gamma_d
\gamma_{ab} \psi V^d =0  \label{fieldeq2} \\
& & 2a\gamma_5\gamma_a \rho V^a - a \gamma_5 \gamma_a \psi R^a = 0
\end{eqnarray}

\noindent To find the first is immediate; for the second
we only have to recall
that varying $\omega^{ab}$ in $R^
{ab}$ yields $\delta  R^{ab}={\cal D}(\delta \omega^{ab})$, and that by
integrating by parts the Lorentz covariant derivative ${\cal D}$
can be transferred on $V^a$. Finally for the gravitino variation we have

\eqa
& & {1 \over a} \delta {\cal L}
= (\delta {\bar \psi}) \gamma_5 \gamma_a {\cal D} \psi V^a +  {\bar \psi} \gamma_5
\gamma_a {\cal D} (\delta \psi) V^a =\\
& & =(\delta {\bar \psi}) \gamma_5 \gamma_a {\cal D} \psi V^a+ {\bar \psi} \gamma_5 \gamma_a \delta\psi
{\cal D} V^a+ \delta{\bar \psi} \gamma_5 \gamma_a {\cal D} \psi V^a=\\
& & = 2 (\delta {\bar \psi}) \gamma_5 \gamma_a {\cal D} \psi V^a - \delta {\bar \psi} \gamma_5 \gamma_a \psi
( R^a + {i \over 2} {\bar \psi} \gamma^a \psi ) =\\
& &=(\delta {\bar \psi}) (2 \gamma_5 \gamma_a {\cal D} \psi V^a-\gamma_5 \gamma_a \psi R^a)\\
\end{eqnarray}

\noindent in virtue of ${\bar \psi} \gamma_5 \gamma_a (\delta\psi)=-(\delta {\bar \psi})
\gamma_5 \gamma_a \psi$
and the Fierz identity

\begin{equation}
\gamma_a \psi {\bar \psi} \gamma^a \psi=0
\end{equation}

\noindent Note that using the gamma-algebra identity:

\begin{equation}
\gamma_5 \gamma_d \gamma_{ab}=2\gamma_5 \delta_{d[a} \gamma_{b]}-i \varepsilon_{abcd} \gamma^c
\end{equation}

\noindent the variational equation (\ref{fieldeq2}) can be recast in the form:

\begin{equation}
2R^cV^d\varepsilon_{abcd}+{i \over 4} (4-a) {\bar \psi} \gamma_5 \gamma_d
\gamma_{ab} \psi V^d =0
\end{equation}
\noindent so that the vacuum, defined by vanishing curvatures, is a solution
of the field equations (5.21) only if $a=4$.
\vskip .4cm
In conclusion: applying  the building rules with $G$ = superPoincaré yields  the $N=1$, $d=4$ supergravity action (\ref{GintegralSP}).

\sect{Conclusions}

In this review we have focused mostly on the logic of the group manifold approach, applied to a single example,
i.e. $N=1$, $d=4$ supergravity. Comprehensive discussions on the applications of the method
for the construction of supergravity theories in diverse dimensions can be found in the recent reviews \cite{gm24,gm25}.
\vskip .4cm
\noindent We list here some of the advantages/motivations:
\vskip .4cm
\noindent - all fields have a group-geometric origin, even if they are not all
 gauge fields.

 \noindent - all symmetries have a common origin as diffeomorphisms on $\tilde G$.

\noindent  - there is a systematic procedure based on group geometry to construct
  actions, invariant under diffeomorphisms, and under gauge symmetries closing
  on a subgroup of $G$.

\noindent   - supersymmetry is formulated in a very natural way as a
  diffeomorphism in Grassmann directions of a supermanifold.

\noindent - closer contact is maintained with the usual component actions, whereas in the superfield formalism the actions looks quite different.
  In fact the group manifold action interpolates between the component and the superfield actions of the same supergravity theory, see \cite{if1}-\cite{if3}.

\noindent - in the group manifold formulation of $d=6$ supergravity \cite{d6SG} and $d=10$ supergravity
\cite{d10SG} the selfdual conditions for the 3-form (in $d=6$) and 5-form (in $d=10$) curvatures are a yield of the field equations in the respective superspaces, and do not need to be imposed as external constraints.
\vskip .4cm
\noindent Finally, we recall some conceptual advances due to the group-geometric treatment of supergravity:
\vskip .4cm
\noindent - the generalization to $p$-form
potentials, necessary to treat supergravity theories with $p$-form fields, in the framework of
Free Differential Algebras (FDA) \cite{sullivan,DFd11,gm21,DFTvN,FDAnew1}, and their dual formulation \cite{FDAdual1} - \cite{FDAdual4}.

\noindent - the bridge between superspace and group manifold methods provided by
superintegration, developed in ref.s \cite{if1}-\cite{if3}.

\noindent - a covariant hamiltonian formalism, initially proposed in \cite{CCF1}-\cite{CCF3}, based on the definition of field momenta as
derivatives of the Lagrangian with respect to the exterior derivative of the fields, not involving
a preferred direction (time). Recent developments \cite{CD,SGcovariantH} include the construction
of all canonical symmetry generators for $N=1$, $d=4$ supergravity \cite{SGcovariantH}. This
covariant hamiltonian formalism can also be generalized to a noncommutative (twisted)
setting \cite{LCtwistedH}, describing noncommutative twisted (super)gravity \cite{AC1,AC2}.

\section*{Acknowledgements}

In writing this review we have benefited from discussions with Carlo Alberto Cremonini, Riccardo D' Auria and Pietro Antonio Grassi. We acknowledge partial support from INFN, CSN4, Iniziativa Specifica GSS.  This research has a financial support from Universit\`a del Piemonte Orientale.

redpajama_set_name: RedPajamaArXiv


This is a part-time 12-hours per week telecommuting internship. The internship is 4 months and responsible for leading public and political outreach in the state and district assigned to. Must be available Monday's 4:30-6:00 PM PST for The Borgen Project's national conference call.
Meet with members of Congress an/or Congressional staffers in your State and District.
To Apply: To be considered for the Political Affairs Internship, please email your resume here.


This entry was posted on Wednesday, May 27th, 2009 at 9:29 am	and tagged with music, sweet georgia brown, tractor and posted in laugh - it will make you feel better.	You can follow any responses to this entry through the RSS 2.0 feed.
« no stimulus for you!
How come I can't see your video you posted? I didn't see the videos of the girls singing either a couple of posts ago. Do you know how I fix this mister tech man?
That seems to happen quite often to my PC friends. I'll do some research and let you know.
Or, you could just buy a Mac!


text:
Thursday News Rodeo: Pancake Pantry, Sinkholes, and Nashville's Tallest Building
October 7, 2021 October 7, 2021 Stephen Yeargin
The sponsor of the city's mask mandate bill has deferred a vote on it again, saying that pressure from a threatened state legislature special session has several council members rethinking their support of the bill. Case counts in Nashville and the surrounding areas have fallen in recent weeks.
The Nashville Rescue Mission is adding 25 COVID-19 isolation pods to limit the spread of the virus among the city's homeless population staying at the facility. The pods replace the area at Fairgrounds Nashville that had previously been used for that purpose.
Folks riding the Music City (WeGo?) Star will have a bit of a detour after a massive sinkhole opened up underneath the tracks near Omohundro Drive. The train is unable to reach the downtown station until the tracks are repaired.
A suspect has died and an Alcohol, Tobacco & Firearms (ATF) agent is in critical condition at Vanderbilt University Medical Center following a deadly shootout on Murfreesboro Pike near the police precinct. Video shows the suspect and two agents exchanging gunfire through car windshields.
The leader of the Small Business Administration was in Nashville to meet with music venue operators hoping to lobby the federal government for additional grant funding.
Late-night diner Hermitage Cafe is said to be closing after a developer purchased the property they lease near downtown. The cafe opened in 1990.
A local grocery store says they are calling applicants back within hours, not weeks, as well as offering incentives and referral bonuses to try and address staff shortages.
The Downtown YMCA shared renderings of its a new look for their building with a partnership with developer Tony Giarratana that would add a 60 story residential tower adjoining the space. If completed, it would be the tallest building in Nashville, passing the 505 building that opened in 2018.
The state will begin issuing new license plates next year featuring a design that received 42 percent of the vote in an online poll. Motorists will be able to choose whether to include "In God We Trust" on their plate.
Local nursing schools are working to recruit new students into the profession that was already facing a shortage prior to the COVID-19 pandemic, made worse by veteran caregivers burning out of the profession entirely because of it. Many area hospitals are offering sizable signing bonuses for staff.
The newly created Nashville Department of Transportation (NDOT) is already under fire by some members of council for taking way too long to complete approved projects in their districts. NDOT says they have been up front with communicating about the challenges and delays for those projects.
The Fisk University Jubilee Singers are celebrating 150 years as a performing group, with many historians crediting the words of the late British Queen Victoria as creating the "Music City" moniker for Nashville. The group was originally formed to raise money for the struggling historically Black university in North Nashville.
The Pancake Pantry will be opening its second location in November inside the Hyatt Centric Hotel downtown, branching out from the popular (in that there's always a ridiculously long line out front) Hillsboro Village restaurant.
The Nashville Business Journal has an interesting look at what areas of Middle Tennessee folks are moving in and out of, based on their change of address forms.
New party bus regulations will be in effect for the new year if the Metro Council passes a series of ordinances on their third reading October 19. The move comes after a summer of complaints about operators disturbing classes at a downtown high school and receiving harsh words from the city's tourism chief.
The Metro Council again deferred a zoning change that would have led to the eviction of about 20 families from an East Nashville mobile home park.
In other council news, they confirmed several appointments and elected folks to offices to shore up vacancies in city government.
Nashville SC broke ground on a new $9 million training facility in Antioch that features three full-size soccer fields and 30,000 square feet of training and office space.
The Nashville Predators picked up a 3-2 win in overtime on the road to the Carolina Hurricanes in their second-to-last game of the preseason. The team closes out the preseason this Saturday against the Hurricanes at 3 p.m.
A Vanderbilt University alumnus has pledged $10 million to its law school, saying that he owes his success running a private equity firm to his time there.
The new owner of the Red Bicycle location in Woodbine shared his story with WSMV of getting his start as a dishwasher at Chipotle in 2009 before climbing the corporate ladder to become a director of operations. He is also one of the so-called "Dreamers," arriving in the United States as a DACA-eligible immigrant from Mexico when he was seven.
PSA: Adoption fees are half off through this Sunday at Metro Animal Care & Control.
Photo by Phillip Riggins. Want to see your photo featured here?
ATF Carolina Hurricanes change of address covid-19 daca dowtown ymca fisk university fisk university jubilee singers hermitage cafe homelessness hyatt centric hotel license plates macc mask mandate metro council Nashville Predators nashville rescue mission Nashville SC ndot nurse shortage pancake pantry red bicycle sba tony giarratana vanderbilt law school vanderbilt university wsmv
Monday News Rodeo: Cannery Row, Glow in the Dark Crosswalks, and Sidewalk Cafes
Monday News Rodeo: Micro-Housing, AMBER Alerts, and TPAC Water Damage

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: What has Maximize been thinking about for so long? The command
Maximize[{Sqrt[2 x + 13] + (3 y + 5)^(1/3) + (8 z + 12)^(1/4), x + y + z == 3&&{x,y,z} >= 0}, {x, y, z}]

is running without any response on my comp for hours (as well as many other Mathematica commands).
This optimization problem is quite standard and easily solved
by Lagrange multipliers
L=Sqrt[2x+13]+(3y+5)^(1/3)+(8z+12)^(1/4)-t*(x+y+z-3);

For simplicity the conditions {x,y,z}>=0 are not taken into account.
Reduce[D[L, x] == 0 && D[L, y] == 0 && D[L, z] == 0 && D[L, t] == 0, {x, y, z, t}, Reals]

x == 3/2 && y == 1 && z == 1/2 && t == 1/4

The Hessian of L at this point is easily found by
M = {{D[L, x, x], D[L, x, y], D[L, x, z], D[L, x, t]}, {D[L, y, x],
D[L, y, y], D[L, y, z], D[L, y, t]}, {D[L, z, x], D[L, z, y],
D[L, z, z], D[L, z, t]}, {D[L, t, x], D[L, t, y], D[L, t, z],
D[L, t, t]}} /. {x -> 3/2, y -> 1, z -> 1/2, t -> 1/4}

Next, the result of
Resolve[ForAll[{a,b,c,d}, ({a,b,c,d}.M).{a,b,c,d} <= 0], PositiveReals]

True

states this is the maximum point (local and global) since the quadratic form ({a,b,c,d}.M).{a,b,c,d} takes
nonpositive values on the positive reals.
Is there anoter way to symbolically solve this optimization problem?

A: With a little help(substitute the rational powers,  see @MichaelE2 coding the square  ) Maximize solves the problem:
Maximize[{wx + wy + wz ,x + y + z == 3 && {x, y, z} >= 0 && wx^2 == 2 x + 13 &&wy^3 == 3 y + 5 && wz^4 == 8z + 12}, {x, y, z, wx, wy, wz} ]
(*{8, {x -> 3/2, y -> 1, z -> 1/2, wx -> 4, wy -> 2, wz -> 2}}*)

redpajama_set_name: RedPajamaStackExchange


text:
Daniel Londoño may refer to:
 Daniel Londoño (footballer) (born 1995), Colombian footballer
 Daniel Londoño (economist), Colombian economist and academic

redpajama_set_name: RedPajamaWikipedia


We are only 7 days away from our spring break. I love my job, but am ready for a breather. The kids are ready too. You can say we have our own brand of mid-March madness! Today's post is a bit of a hodgepodge of things. It is pretty reflective of my mind these days…all hither and dither. I wanted to show you some of the work going on in class and since the samples are from different subjects, I'm offering a bit of a Pu Pu Platter today. You know, a little appetizer from which you can pick and choose.
In writing, we are using mentor texts like crazy! We are working on adding details, expanding our writing and using various means to write. Sometimes the kids choose paper, sometimes iPad. Here are few samples. The first one is a graphic organizer for writing about the pond. The student used Pic Collage to create the organizer.
In this sample, the student used the app Paper Desk Pro as a Reading Response journal after hearing the story In the Tall, Tall, Grass by Denise Fleming.
This last writing sample is one child's Writer's Workshop for the day. She chose this piece to be published from her writing folder.
In math, we've been working on addition, subtraction, and 3-D shapes. Here are a few pages from one child's math journal created in Book Creator app. This image and the next are from the app Number Pieces.
This app is Number Rack. They are like rekenreks.
This is a screen shot from the math journal created in Book Creator. Students took photographs of 3-D objects in our room and labeled them.
In addition to these samples, the students are working on their own books in Book Creator and projects in Explain Everything. They are planning their own day, choosing apps for demonstrating their learning, reading constantly and using inquiry skills across the curriculum. This is such a fun time of the year with kindergarten. With only 49 days remaining in this school year, I'm soaking it all up and enjoying my time with them. They need me for so little now and I enjoy just sitting and being part of their world.
Giving kids of all ages voice and choice in their learning provides deeper learning opportunities and long-term connections to the world around them.
Hello! I teach at Westminster in Atlanta, and I really enjoy your blog! What you do in your classroom with iPads is fabulous! I was wondering if you would be willing to share a list of some of your mentor texts. I am always looking for more texts to use when I teach writing workshop.
I hope you enjoy your spring break! You certainly deserve it!!


Welcome to the YABUUCHI Satoshi Museum.
We will be starting a new section in the MUSEUM featuring essays by YABUUCHI Satoshi. We plan to publish a different essay every month and hope that you will enjoy them.


We are very excited to partner with Rocky Mountain Wild to offer new and existing members of 100 Women for the Wild a 20% discount on registration for Women Rising Wild, 2017 and again in 2018.
Adrienne Rich, feminist trailblazer and literary force for social awakening described the connections between and among women as "the most potentially transforming force on the planet." Join forces with Women Rising Wild and 100 Women for the Wild and become a force of nature.
At Women Rising Wild this Labor Day weekend we will reconnect on Mission: Wolf wolf sanctuary in southcentral Colorado to revive our primal nature as wise and nurturing beings with deep and powerful ties in sisterhood with nature. On retreat for 3 days and 2 nights in this remote wilderness setting we will experience direct contact with wolves, earth stewardship, nature-based therapies for recentering and transformation; earth ritual and ceremonial rebirth; soulsounding vocal explorations; mindful movement and meditation; camaraderie with women; outdoor survival skills; wilderness living; catered organic meals; and nourishment for the soul.
And through your membership and service with 100 Women for the Wild you will directly support the professional development of women in science and environmental advocacy, biodiversity conservation, and wildlands protection in the southern Rocky Mountains.


Find all you need to plan and design your party here within our product hire catalogue.
Create your wishlist by selecting your chosen products and send it to us for a free, no obligation quotation.
If you get stuck or need some advice, please do get in touch so we can help with any questions you may have.


text:
Board index In the Meadow Walking Around in the Newbie's Shoes
Discrepancies
A discussion of the novella The Short Second Life of Bree Tanner
Moderators: Bronze Haired Girl, una
Location: Trudging Through the Ashes
Re: Discrepancies
Post by Knives » Mon Jul 26, 2010 11:53 am
December wrote: Friendly tone reminder from your Fearless and Frost-clad Mod: This was not actually meant to be a debate thread -- or a fencing match. So not to spoil your fun or anything, but...a less adversarial mode of conversation would be welcome. There are ways of getting your point across without taking a rhetorical bludgeon to the opposing view. Thanks!
But how, O Great and Glorious Moderator? Retaining an advocatus diaboli (me) means, of necessity, the introduction of concepts and viewpoints which are, by their nature, oppositional. I can't make a statement like "Ms. Meyer has idiotically designed a being that invalidates the vermisilitude of her entire setting," (the short version of my side of the debate above) without being adverserial simply by making my point of view known. Naturally, folks who disagree are going to want to try and pick apart my reasoning, and I theirs - and thus does one get a debate.
The question, however, was honest - how can we discuss this without turning it becoming cut-and-thrust?
Openhome wrote: Knives, I believe that..
you are right.
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Post by December » Mon Jul 26, 2010 8:36 pm
Ah, well...now you've started a subject which I could discourse on far longer than anyone here will want to sit still for! (Ladies and Gentlemen, if you wish you may make your way to the exits now...).
I think it's possible to have a collaborative, unadversarial discussion even where participants' views diverge enormously -- it's all about abandoning the goal of establishing which view is right. So (for instance) I find that the tension between Edward's desire to cherish Bella and his desire to kill her gives Stephenie's story a kind of mythopoeic resonance; you find the insipidity of her "vampires" saps that story of any power to invoke myth. I lay out my thinking in a way that -- I hope -- answers the (politely) unspoken question in your mind: "how the hell can she be getting anything out of this banal love story?". You lay out yours in a way that -- I can attest -- makes it much clearer to me why the Twilight phenomenon irritates the hell out of you. Neither of us is, generally speaking, trying refute the other's position or prove the superiority of our own. That doesn't mean we don't query each other's exposition: identify elements we don't understand, or that don't seem to add up. The point is that we're not, on the whole, doing it to demonstrate the errors in each other's arguments but to genuinely try to understand what the other is getting at. In short, it's not a zero-sum game. You don't win points when I acknowledge the force of some of your observations, and vice versa. Which opens the way to a more nuanced discussion.
Now...WHY this is true of the conversation in question is an intriguing puzzle. It's hard to pinpoint just what the linguistic and rhetorical markers are which signal to you that I'm not out to show you up or challenge your arguments. Or precisely how you convey an amicable interest and respect for what I have to say. It's like that famous definition of pornography: one just knows whether or not someone's stance is confrontational -- no doubt (I say, clambering precariously out on a socioevolutionary limb) because recognizing combative behaviour is a really primitive animal capacity, one that goes back to our earliest mammalian ancestors....
Anyway, I think I'm right in saying that our earlier discussion of myth, vampire traditions and realism simply didn't have that confrontational character. And yet, unquestionably, this is a subject about which we disagree in many respects. So when you ask me "how can we discuss opposing viewpoints without it becoming adversarial?", my honest reply is: "I'm not sure. We just did." And how is a really good question! I'd be interested if you have any insights here, because (as you'll have gathered -- I did warn you not to get me started on this!) it's a topic which interests me unduly....
And yeah...this is really getting pretty off-topic. One of the mods will probably be along shortly to remind us to get back in line! (*grin*)
"When did you ever promise to kill yourself falling out of Charlie's tree?"
Openhome
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Post by Openhome » Mon Jul 26, 2010 11:39 pm
So, to bring it back to a non-confrontational idea about vampires and such...
Knives, you are writing a wonderful novel about traditional vamps (that you need to come and update us about because I'm dying to know more). Andy, you like the twist on the vamps that SM created. Rings, you are our literary canon expert as far as the story goes. You may also be the reigning king of gaming, but I'll let you three handle that.
How does the idea of an indestructible predator affect the overall story line of a story. Back to my original question, what about the mythology of Twilight. How does it change things if Aro sees humans as a potential (small though it may be) threat?
I'm asking because it is an interesting character point for me. Knives, you have often said that vamps in traditional works get bored and loose themselves over the years, yet they are destructible. Rings, you wrote a beautiful story about an older vamp who kept up his hope against all odds. Does the presence of a threat make a difference when you are a killer? Can that very threat keep you from becoming insane with the boredom?
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Post by Knives » Tue Jul 27, 2010 1:24 am
Openhome wrote: So, to bring it back to a non-confrontational idea about vampires and such...
An interesting question. Allow me to try to explain.
Let's say we're dealing with one of White Wolf's vampires, which everyone here should be semi-familiar with from previous discussions. However, to recap: These vampires will not age or die of natural causes. They may, however, be killed by (in order from most difficult to least) clubs, guns, cobra venom, animal attacks, slashing instruments, bombs & other explosives, decapitation, fire, and sunlight. They can shift shape, enhance their physical capabilities, develop super-senses (as well as psychometry, telepathy, and astral projection), learn blood-magic, crush minds with a glance, and sway entire crowds to their whims. Powerful being, right?
Inevitably, however, such a being grows bored. A businessman soon has phenomenal money and influence in death that he could never have dreamed of in life. The artist has created his masterpiece and finds himself incapable of true emotional insight any longer; the thrill-seeker, having tested the limits of death, has no more boundries left to push. What, then, do they turn to?
That's where the concept of a threat comes in. In White Wolf's world, they don't provoke humanity, not because humans are an insipid threat to them, but because the masses of the herd are simply too dangerous. In numbers, with knowledge and training, humans are dangerous even at night. During the day, when the Kindred sleep? There is no match. The humans will win.
Instead, they turn to treachery, plotting and scheming against each other for petty gain, personal vendettas, pet causes, or even the sheer vainglorious joy of having the most toys. This Danse Macabre is pointless, an endless spiral leading to change that never truly brings about improvement - but it also allows them to bottom out somewhere below "average moral being" and above "mindless animal". Are they cruel, selfish, and fractious? Certainly. But by championing pet causes, opposing contrary viewpoints, and even making attempts at morality, vampires can improve the world - but only by first corrupting some part of it.
Does that make any kind of sense?
ringswraith
Running with Leah 'cause she thinks I'm hawt
Post by ringswraith » Tue Jul 27, 2010 3:37 am
Ah, Vampire. I'm more partial to Werewolf but the World of Darkness (a setting developed by White Wolf) intrigues me.
A threat is a threat, no matter who you are. Predators in the wild would react if a bigger predator happened upon them. If Aro (or indeed, any vampire) began to see humans as a threat, certainly things would change. The Volturi for one have put a lot of time and effort to get where they are now, and they're not likely to upset the status quo. They will ascertain the nature of the threat and work to counter it immediately. In fact, I believe they have layers upon layers of preventive measures in place in the unlikely event that something came to pass. (WW vamps certainly do, and they're quite more vulnerable than TW vamps.)
ringswraith wrote: Ah, Vampire. I'm more partial to Werewolf but the World of Darkness (a setting developed by White Wolf) intrigues me.
All of that is well and good, but if all the threats have been dealt with, you're right back at square Immortal Boredom, aye?
Post by ringswraith » Tue Jul 27, 2010 2:15 pm
Well, of course. At least WoD got all the scheming and plotting down right.
andypalmer
Learning to Love Green
Post by andypalmer » Wed Jul 28, 2010 3:47 pm
I do agree that the threat level that humans represent to vampires in the World of Darkness is significantly more evident, if not actual, than that of those in Twilight. The addition of other supernatural threats (Twilight has the rare Werewolf and Shapeshifter, but nothing of the magnitude of the magical and supernatural of the World of Darkness) adds to the necessity of intrigue over overt actions.
I find the danger of humans in Twlight to be more subtle. Humans do not have a demonstrable ability to defeat vampires, yet can be argued to have the potential to do so. I think it is this potential that Aro fears. Your typical vampire thinks of humans as nothing more than herd animals; it is the fear of the Volturi that maintains a level of subterfuge in their activities. You could say that, to 99% of vampires in SM's world, humans are not considered to represent a threat. However, as long as Aro perceives that humans are a potential threat, if not only to vampires in general, but to him and the lifestyle he has created specifically, then he will continue to drive the Volturi to enforce his own version of the "Masquerade."
Knives and I may disagree on the degree of potential that humans have to defeat vampires, but I believe that we would both be agreed on Aro's view on that potential, based on his words in the Saga.
Seeker-Mar
Wandering Through Town
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Post by Seeker-Mar » Fri Aug 06, 2010 2:57 am
Knives wrote: Alright, I've said it before, and I'll say it again - in a world containing intelligent twi-pires, "modern earth" would not exist.
Let us assume, just for a moment, that Twi-pires came into existence shortly before the Inquisition, since the Voulturii remember those guys. What medieval weapon could possibly pose a threat to them? None. What force of arms could overcome one? None. Vampires were truly invincible.
This leads to the next statement - it's easier to farm than it is to hunt.
If the Big Vs were to act as intelligent as we're expected to believe they are, vampires would rule the world, openly, and without fear. Humans would be food, slaves, and fodder, not allowed near true technology. Promising mortals can be turned into vampires; rebellious mortals can be hunted down. After all, Twi-pires don't need to rest, drink, or stop for any reason short of their total destruction. Human beings have to eat, drink, sleep, avoid injury and still manage to throw off pursuit by beings with super-senses, confirmed telepaths and confirmed precognitives - beings which have forever with which to chase them, at speeds humans cannot match, with a dozen other advantages that merely add insult to injury.
Which is why, earlier, I was getting annoyed by the various statements that the Volturii were smart to hide. Hiding was the dumbest move they could have made. Why let your enemy get strong when you can conquer them?
Maybe it's as simple as this; vampires like to hunt not herd. When you are the worlds best predator why would you want to farm? Their lives are so static perhaps this is a way for them to feel a little bit alive? Of course this only fits if they take turns gathering up the humans. If only one member does this then it is a little like farming already.
But the wisdom that comes from heaven is first of all pure and full of quiet gentleness. Then it is peace-loving and courteous. It allows discussion and is willing to yield to others; it is full of mercy and good deeds. James 3:17
olorin
Settled in Forks
Post by olorin » Thu Aug 19, 2010 6:16 am
Sorry, I didn't read much of this thread, but I still want to voice my opinion to it.
Humans would be food, slaves, and fodder, not allowed near true technology.
Hiding was the dumbest move they could have made. Why let your enemy get strong when you can conquer them?
You're thinking the vampires would join together and conquer their common enemy, the humans.
But this world doesn't work that way. The humans aren't enemies, just easy inconsequential food.
The enemy is anyone who tries to destroy their long established way of life, like other vampires tying to
take over, throwing the world into chaos and messing with their beloved homes.
Vampires are very independent, territorial, protective of their hunting ground and don't bow down to authority.
Now I suppose after a few more centuries, when the humans become an actual threat, exposing them and even
destroying some of the Volturi, then the vampires would be forced to do something about them.
But they wouldn't enslave humans and become farmers. That's too much work.
Why farm animals, keeping them under constant observation, when they multiply on their own right in front of you.
Instead they would probably destroy all civilization, throwing mankind back into the Stone Age,
and then hide again for a few generations, until they become myth again and slowly be forgotten.
(Maybe they already did that a few times. )
Of course humans have superiority in numbers, but the vampires can quickly change that, by creating
Newborn armies. (while others hide under ground or in the oceans somewhere waiting it out)
The southern wars in Mexico almost wiped out the humans there too, and that were vamps trying to be careful and fighting each other.
So unless the humans develop some miracle disease or radiation, to destroy all vamps around the planet very quickly, I don't think the humans would stand any chance.
Now the time afterwards would be very violent for the vamps too, fighting each other over the new scarcely populated hunting grounds.
So in the end only a fraction of the humans and vamps would survive, and things would start anew.
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My friend and I were sitting in Bryant Park recently, discussing the serious lack of fun in our lives. It dawned on both of us simultaneously that we were having fun at that very moment, but overlooking it in the search for something Really Fun. Given what I know about my love for fun, I was surprised at my own willingness to overlook the Little Fun, the Mundane Fun, and the Regular Fun in a quest for some big kind of Transcendental Fun or SuperGiantAwesome Fun.
And at that moment, in a very uncomfortable chair in a very crowded Bryant Park, I committed myself to a month of fun. Or, in the world of Kate, a Funth!
Every day for a month I am going to do something fun. It can either be wacky -- like a Thai Yoga Massage class (that, sadly, got cancelled on Sunday), or a viewing of Punderdome 3000 (a groan-inducing punning competition) -- or it can be simple, like the surprise of being able to talk to my friend while on the elliptical machine in the basement, or having a quiet brunch full of smoked salmon with a dear friend. There are no rules about the fun that is allowed to count, I just have to be intentional about it. And I have to have some every day.
And this reminds me: fun is a choice. I left the park that afternoon and found myself in my office, working on the same project I was tackling before lunch, but approaching it with a totally new attitude. More levity. More curiosity. More of a song in my heart.
In a way, my Funth is a continuation of the Year of Yes! (well, ok, the 14 months of yes) and in a way, it's a completely different project. During the YoY, I tried to imbue my life with different qualities, to see how they showed up in my actions. During the Funth, I am committing to taking action, to see how those actions impact my thinking. The YoY was inside out; the Funth is outside in.
And you're all invited. Anybody want to have some fun?
I now have the internet at my fingertips 24/7. My adorable little phone (aptly named "Spankie") has ushered me into the 21st century, and now I know how the other 99% live. I can text, tweet, post to facebook, upload photos, and otherwise reach out and touch someone at any hour of the day (except while I'm in the subway).
And I haven't felt this disconnected since I can't remember when.
Texting is a slippery slope for me. It's a great way to passively reach out and drop a funny line or thought on someone's radar without putting too much energy or thought into it. I can share a link, tell a joke, or otherwise flit in and out of a friend's life in a matter of seconds. I sometimes have full conversations with someone over text, and those conversations are well thought out, carefully constructed, and usually ridiculously funny.
And the trend is only going to go up, so I feel like a bit of a Luddite even having an issue with this. But I do.
I don't really feel like I'm getting to know the people on the other end of my text-line any better as I text with them. There's so much less revealed. People have time to think, react, and respond before sharing their impulses or insights. There is also much less clarity in texting. The tone of voice -- so crucial in spoken communication -- is completely gone, and the context -- which usually helps to set the tone of an email -- is also missing. So all I get in a text is the nugget, the core of the message. Which is fine if that message is "I'm running late," or "Just wanted to say hi."
I know I need to get used to it, but for me, true connection comes voice to voice, if not face to face. I want to hear the hemming and hawing. I want to know when there's a sigh. Emotions, mistakes, and non-verbal noises help me feel more human and more connected to others. I've thought about instituting a no texting policy for a month, just to see how it goes, but I think I've gotten so used to texting that I'd miss it if it was gone. The secret is to blend both, and to set boundaries around how much I'm willing to text. But how much is too much? And how much is not enough?
When I figure out what that balance is, I'll let you know. Or if you want to find out, you can always call me.
If I can say this without sounding too self-congratulatory, I will: people find it easy to talk to me. I guess I have one of those faces. They ask me for directions, guys admit things to me on dates that perhaps they shouldn't, and strays think I'm the cat's meow.
In the last few years I've gotten better at actually listening (instead of just waiting for my turn to talk), and I've developed a skill that has really surprised me -- my ability to hold the space for other people.
Now, "holding the space" is a really coachy phrase -- akin to "maximizing your potential" and, let's admit it "fully living" -- so what does it mean? It's the ability to be there with someone else who is experiencing emotions without feeling the need to shut them up, fix them, or get them to be anywhere but where they are right now. It's watching someone cry and waiting for her to ask for the box of tissues instead of passing it to her right away. It's opening your heart to the pain and suffering of someone else, trusting both yourself and the other person to make it through, to recover, to let this moment be what it is and also to let it pass.
I bring this up because I was at a workshop recently that really flexed this muscle, and it's one I don't think enough people are aware of, let alone "maximizing the potential" of.
What does it take to hold the space for someone else? Empathy. Support. Wanting to be there for someone else, to let him know that it hurts right now but it won't hurt forever. Trust. Faith. Acceptance. And it takes a real vulnerability. The emotional flexibility to dig deep, connect with your own strength and your own pain, and just breathe.
The next time you're with someone who's hurting, try this: just sit with him or her. Hug him so he knows you've got him, but not so that he feels he has to stop crying. Absorb just a little of the pain so it's not so intense for her. Rain blessings down on the both of you to make life just a little easier after this moment. And trust that, when the time comes, he or she will be there for you. And see what's different.


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text:
Aymen Dahmen (; born 28 January 1997) is a Tunisian professional footballer who plays as a goalkeeper for the Tunisian club CS Sfaxien.

Club career
Dahmen made his professional debut with CS Sfaxien in a 2–0 Tunisian Ligue Professionnelle 1 win over ES Métlaoui on 16 September 2018.

International career
Dahmen was called up to represent the Tunisia U23 national team for 2019 Africa U-23 Cup of Nations qualification matches.

He made his debut for senior national team on 28 March 2021 in an AFCON 2021 qualifier against Equatorial Guinea.

On November 14, 2022, he was selected by Jalel Kadri to participate in the 2022 FIFA World Cup. He participated in all Tunisia's matches in the World Cup.

Honours
CS Sfaxien

 Tunisian Cup: 2018–19, 2020–21, 2021–22

References

External links

1997 births
Living people
People from Sfax
Tunisian footballers
Tunisia youth international footballers
Tunisia international footballers
Association football goalkeepers
CS Sfaxien players
Tunisian Ligue Professionnelle 1 players
2022 FIFA World Cup players