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\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Supergravity}
\setcounter{equation}{0}
\setcounter{footnote}{0}




\myhead{Gravity: on-shell}
\label{Chap15:sugra:sec:onshell}

\noindent Einstein gravity $\leftrightarrow$  gauge theory of local Lorentz/translation symmetry 
\newline generators
$M_{ab}$, 
$P_a$ $\leftrightarrow$ ``gauge fields" $\omega^{ab}_\mu$, spin connection, 
$e^a_\mu$  {\em vierbein}  where $a,b=0,\ldots,3$ are Lorentz gauge group indices \newline  $\mu,\nu=0,\ldots,3$ are spacetime
 indices\newline $e^a_\mu$ and $\omega^{ab}_\mu$ transform  as collections of vectors
\newline  gauge fields $\leftrightarrow$ feld strengths $R^{ab}_{\mu\nu}$,
(Riemann curvature), $C^a_{\mu\nu}$,  (torsion)\newline  $C^a_{\mu\nu}=0$, solve for $\omega^{ab}_\mu$ in terms of $e^a_\mu$\newline 

\noindent counting:  $e^a_\mu$ 16 components 
\newline subtract 4 (equations of motion)
\newline subtract 4 (local translation invariance)
 \newline subtract 6 (local Lorentz invariance)

\vspace{24pt}
\noindent leaves 2 degrees of freedom: massless spin-2 particle


\myhead{Gravity: on-shell}
   Couplings to matter:
\beq
\nabla_\mu= \partial_\mu - e^a_\mu P_a - \omega^{ab}_\mu M_{ab}~
\eeq
feld strengths can be obtained from
\beq
\nabla_\mu \nabla_\nu - \nabla_\nu \nabla_\mu~
\eeq
Writing 
\beq
e=|{\rm det} e^m_\mu|~
\eeq
 invariant action with only two derivatives is linear in the field strength: 
\beq
S_{\rm GR}=  {{M^2_{\rm Pl}}\over{2}} \int d^4x\,  e \,\epsilon^{\mu\nu\rho\lambda}\epsilon_{abcd}\,
e^a_\mu e^b_\nu\, R^{cd}_{\rho\lambda}  ={{M^2_{\rm Pl}}\over{2}} \int d^4x\,  e \,R ~
\eeq
where $R$ is the curvature scalar
%The metric can be written in terms of the vierbein as
%\beq
%g_{\mu\nu} = \eta_{\mu\nu} +{{1}\over{M_{\rm Pl}}} e^a_\mu e^b_\nu \eta_{ab}~.
%\eeq


\myhead{Supergravity: on-shell}
$e^a_\mu$ $\leftrightarrow$ helicity 2 particle,  ${\mathcal N}=1$ SUSY requires  helicity $3/2$  $\psi_{\nu}^{\alpha}$
(gravitino)\newline on-shell
each has  two degrees of freedom\newline 
gravitino is gauge field $\leftrightarrow$
$Q_\alpha$ $\leftrightarrow$  field strength $D_{\mu\nu\alpha}$
\newline $C^a_{\mu\nu}=0$,  solve for $\omega^{ab}_\mu$   find
 on-shell supergravity action:
\beq
S= {{M^2_{\rm Pl}}\over{2}} \int d^4x\,  e \,R +{{i}\over{4}} \int d^4x \,
\epsilon^{\mu\nu\rho\sigma}\,\overline{\psi}_\mu \gamma_5 \gamma_\nu D_{\rho \sigma}~
\label{Chap15:Lgravitino}
\eeq
call  second term
$S_{\rm gravitino}$

metric:
\beq
g_{\mu\nu} = e_\mu^a e_\nu^b \eta_{ab}~
\eeq
in terms of a local inertial coordinate system $\xi^a$ at the point $X$ 
\beq
e_\mu^a(X) = \frac{\partial \xi^a}{\partial x^\mu}~
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Brans--Dicke Gravity}
first consider 
 toy example, 
scale-invariant Brans--Dicke  theory:
\beq
S_{\rm BD} = \int d^4x\, \left[ \frac{e}{2}\,\sigma^2 R + {{e}\over{12}} \partial^\mu \sigma 
\partial_\mu \sigma \right]~
\eeq
treat scalar $\sigma$ as a spurion field and set
\beq
\sigma = M_{\rm Pl}~
\eeq
 break local conformal invariance to local \Poincare invariance\newline $\Rightarrow$ Einstein gravity

\myhead{Superconformal Gravity}
in addition to the ``gauge`` fields
 $e^a_\mu$ and $\psi_{\nu\alpha}$ we have 
$A_\mu$ $\leftrightarrow$ local $U(1)_R$ symmetry, and $b_\mu$
$\leftrightarrow$ local conformal boosts \newline Counting degrees of freedom off-shell
(subtracting gauge invariances):
\beq \begin{array}{rrll}
{\rm field} &&& d.o.f.\\
e^a_\mu :& 16&-4-6 -1& =5 \\
\psi_{\nu}^{\alpha} :& 16&-4-4 &=8  \\
A_\mu :&4&-1&=3  \\
b_\mu :&4&-4&=0 \\
\end{array}~
\eeq
$e^a_\mu$  subtract 4 (translation), 6 (Lorentz), 1 (dilations)
\newline $\psi_{\nu}^{\alpha} $ subtract
4 (SUSY $Q_\alpha$, $\overline{Q}_{\dot\alpha}$), 4 (conformal SUSY  $S_{\beta}$ and $\overline{S}_{\dot\beta}$)  \newline $A_\mu$ subtract 1(local $R$-symmetry)
\newline $b_\mu$ subtract 4 (four conformal boost generators)


\Red{no  auxiliary fields} for the superconformal graviton multiplet\newline 
 ``gauge" fields, \Red{couple with gauge covariant derivatives}

\myhead{Supergravity: off-shell}
spurion chiral superfield to break the conformal symmetry:
\beq
\Sigma = (\sigma,\chi,{\mathcal F}_\Sigma)~
\eeq
in global ${\mathcal N}=1$, $\Sigma$ is a chiral superfield\newline here it contains
part of the off-shell graviton superfield\newline $\Sigma$ called \Red{conformal compensator} 

\vspace{24pt}
\noindent
assign conformal weight 1 to the lowest component of $\Sigma$ \newline ($x^\mu$ and $\theta$ have conformal
weight $-1$ and $-1/2$)\newline full superconformal gravity action is
\beq
S_{\rm scg}=\int d^4x\, \frac{e }{2} \sigma^* \sigma R + e \int d^4\,\theta \Sigma^\dagger \Sigma
+S_{\rm gravitino}~
\eeq
derivatives are covariant in ``gauge`` fields ($e^a_\mu$, $\psi_{\nu\alpha}$,
$A_\mu, b_\mu$)
\newline\Red{ a superconformal Brans--Dicke theory}

\myhead{Supergravity: off-shell}
Treat $\sigma$, $\chi$, and $b_\mu$ as spurion fields
\beq
\sigma&=& M_{\rm Pl}~,\,\,\,
\chi=0~,\,\,\,
b_\mu=0~
\eeq
 local superconformal invariance $\to$ local super-\Poincare invariance
 \newline
resulting action is:
\beq
S_{\rm sg}=\int d^4x\,
 e \left[ {{ M_{\rm Pl}^2}\over{2}} R + 
{\mathcal F}_\Sigma {\mathcal F}_\Sigma^\dagger -{{ 2 M_{\rm Pl}^2}\over{9}} A_\mu A^\mu \right]
+S_{\rm gravitino}~
\eeq
\Red{${\mathcal F}_\Sigma$ and $A_\mu$ are  auxiliary fields}, 
counting:
\beq
\begin{array}{rrll}
{\rm field} &&& d.o.f.\\
e^a_\mu :& 16&-4-6 & =6 \\
\psi_{\nu}^{\alpha} :& 16&-4 &=12  \\
A_\mu :&4&&=4  \\
{\mathcal F}_\Sigma :&2&&=2 \\
\end{array}~
\eeq
6  bosonic degrees of freedom from  ${\mathcal F}_\Sigma$ and $A_\mu$
are just what is required to have ${\mathcal N}=1$ SUSY manifest off-shell

\myhead{Superspace }
eight-dimensional space $z^M=(x^\mu,\theta^\alpha,{\overline \theta}^{\dot \alpha})$\newline require super-general coordinate invariance
\beq
z^M\rightarrow z^{\prime M} = z^M+\xi^M
\eeq
where $\xi^M(z^M)$
\newline Superspace scalars  transform
\beq
\phi^\prime(z^\prime) = \phi(z)~
\eeq
while fields with a superspace index
\beq
\psi_M= \frac{\partial \phi}{\partial z^M}~
\eeq
transform as
\beq
\psi^\prime_M(z^\prime) = \frac{\partial z^N}{\partial z^{\prime M}}\psi_N(z)~
\eeq

\myhead{Superspace }
construct a \Red{ vielbein}  $E^A_M$
\newline relates the superspace world coordinate to a locally Lorentz covariant 
(tangent space) coordinate\newline contains the off-shell multiplet
$(e_\mu^a, \psi_{\nu\alpha},A_\mu, {\mathcal F}_\Sigma)$

\vspace{24pt}\noindent we can choose a
coordinate system where, for  $\theta=0$,
\beq
E^a_\mu= e^a_\mu,\,\,\, E^\alpha_\mu = \sfrac{1}{2} \psi^\alpha_\mu,\,\,\, 
E^{\dot \alpha}_\mu =  \sfrac{1}{2}{\overline \psi}\vphantom{E}^{\dot \alpha}_\mu~
\eeq
%transforms as
%\beq
%\delta E^A_M = \xi^N\partial_N E_M^A+\partial_M\xi^L E_L^A
%\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Coupling to matter}
\label{Chap15:sugra:sec:coupling}

arbitrary global SUSY theory:
\beq
{\mathcal L}_{\rm gl}&=& \int d^4 \theta K(\Phi^\dagger, e^V \Phi) +\int d^2 \theta \,\left( W(\Phi) 
- {{i\tau}\over{16 \pi}} W^\alpha W_\alpha \right) +h.c.~\,\,\,\,\,\,
\eeq
define conformal weight 0 fields and mass parameters by
\beq
\Phi^\prime = \Sigma\, \Phi~ \\
m^\prime = \Sigma \, m~
\eeq
dropping the primes,  \Red{local superconformal-\Poincare invariant Lagrangian}:
\beq
{\mathcal L}&=& \int d^4 \theta f(\Phi^\dagger, e^V \Phi) {{\Sigma^\dagger \Sigma}\over{M_{\rm Pl}^2}} +\int d^2 \theta {{\Sigma^3}\over{M_{\rm Pl}^3}} W(\Phi) 
-\int d^2 \theta {{i\tau}\over{16 \pi}} W^\alpha W_\alpha +h.c. \nonumber \\
&&-{{1}\over{6}} f(\phi^\dagger,\phi) \sigma^* \sigma R  + 
{\mathcal F}_\Sigma {\mathcal F}_\Sigma^\dagger -{{ 2 M_{\rm Pl}^2}\over{9}} A_\mu A^\mu+{\mathcal L}_{\rm gravitino}~
\label{Chap15:genGravLagr}
\eeq
action:
\beq
S &=& \int d^4x\, e\, {\mathcal L}~
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Coupling to matter}

$M_{\rm Pl} \rightarrow \infty$ (global SUSY) limit,  choose
\beq
f(\Phi^\dagger, e^V \Phi) = -3 \,M_{\rm Pl}^2 e^{{-K(\Phi^\dagger, e^V \Phi)}/{3 M_{\rm Pl}^2}}~
\label{Chap15:Kahlerpot}
\eeq
rescaling the vierbein by a Weyl (local scale) transformation
\beq
e^a_\mu \rightarrow e^{{-K}/{12 M_{\rm Pl}^2}} e^a_\mu
\eeq
one finds bosonic piece of the action:
\beq
S_{\rm B}&=&\int d^4x\,e \left[ {{ M_{\rm Pl}^2}\over{2}} R + K^i_j(\phi^\dagger,\phi)
(\nabla^\mu \phi^{i})^\dagger  \nabla_\mu \phi_j \right. \nonumber \\
&& \left. -{\mathcal V}(\phi^\dagger,\phi) +{{i \tau}\over{16 \pi}} (F_{\mu\nu}F^{\mu\nu} + i F_{\mu \nu}
\wtilde{F}^{\mu \nu}) +h.c. \right]
\eeq
where $K^i$ and $K^i_j$ (the \Kahler metric) are given by 
\beq
K^i(\phi^\dagger,\phi) =\frac{\partial K}{ \partial \phi_i}~,
\,\,\,K^i_j(\phi^\dagger,\phi) ={{\partial^2 K}\over{\partial \phi^{j \dagger} \partial \phi_i}}~
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Coupling to matter}
scalar potential:
\beq
{\mathcal V}(\phi^\dagger,\phi) &=& e^{{K}/{M_{\rm Pl}^2}}
\left[\left(K^{-1}\right)^j_i\left(W^i + {{W K^i}\over{M_{\rm Pl}^2}} \right)
\left(W^*_j + {{W^* K_j}\over{M_{\rm Pl}^2}} \right) - {{3 |W|^2}\over{M_{\rm Pl}^2}}
\right] \nonumber \\
&& +{{g^2}\over{2}} \left( K^i T^a \phi_i \right)^2~
\label{Chap15:scalarpot}
\eeq
last term  is just the $D$-term potential


\vspace{24pt}\noindent in supergravity the energy density can be negative\newline usually tune tree-level vacuum energy to zero by adding the appropriate constant to $W$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Coupling to matter}
auxiliary components of  chiral superfields (no fermion bilinear VEVs):
\beq
{\mathcal F}_i = - e^{{K}/{2 M_{\rm Pl}^2}}\left(K^{-1}\right)^j_i
\left(W^*_j + {{W^* K_j}\over{M_{\rm Pl}^2}} \right)~~~~~~~~~(*)
\label{Chap15:Fcomp}
\eeq

\noindent from fermionic piece of Lagrangian,  $\nabla_\mu \wtilde \phi_i$ contains a gravitino term
\beq
\frac{1}{M_{\rm Pl}} \psi_{\mu}^\alpha Q_\alpha \wtilde{\phi}_i=
\frac{1}{M_{\rm Pl}}  \psi_{\mu}^\alpha  {\mathcal F}_i +{\mathcal O}(\sigma^\mu \partial_\mu \phi_i)
\eeq
so the \Kahler function contains a term:
\beq
i K^i_j  \frac{1}{M_{\rm Pl}} {\bar \theta}\, {\overline {\wtilde \phi}}^j \theta^2
\psi_{\mu}  {\mathcal F}_i \sigma^\mu {\bar \theta}
~
\eeq
in analogy to the ordinary Higgs mechanism, that the \Red{gravitino  eats
the goldstino} if there is a nonvanishing ${\mathcal F}$ component 

in flat spacetime,  goldstino adds  right number of degrees of freedom to make a massive spin 3/2 particle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Gravitino Mass}
in flat spacetime
\beq
m_{3/2}^2 = {{{\mathcal F}^{* j} K^i_j {\mathcal F}_i }\over{3 M_{\rm Pl}^2}}~
\eeq
 use  (*) and
${\mathcal V}=0$ $\Rightarrow$
\beq
m_{3/2}^2 = e^{K/M_{\rm Pl}^2} \frac{|W|^2}{M_{\rm Pl}^4}~
\eeq
taking a canonical \Kahler function 
\beq
K= Z \Phi^{i \dagger} \Phi_i~
\eeq
and $M_{\rm Pl} \rightarrow \infty$ reproduces  usual global SUSY results

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Maximal Supergravity}
\label{Chap15:sugra:sec:higherdimensions}

massless supermultiplet with helicities $\le 2$
\newline SUSY charges
change the helicity by $\sfrac{1}{2}$ $\Rightarrow$ ${\mathcal N}\le8$
\newline arbitrary dimension cannot have more than $32=8 \times 4$ real SUSY charges
\newline maximal dimension: spinor in 11 dimensions has 32 components

supergravity theory must have  $e^a_\mu$ and  $\psi^\alpha_\mu$  massless gauge fields\newline  $D$ dimensions: ``little`` group $SO(D-2)$\newline graviton: symmetric tensor of $SO(D-2)$ has $(D-1)(D-2)/2 -1$ dof\newline 44 dof for $D=11$\newline gravitino is a vector-spinor and a vector has $D-2$ dof 

spinor   of $SO(D)$ has $d_S$ components, where
\beq
d_S=2^{(D-2)/2}\,\, \textrm{(for $D$ even)},\,\,\, d_S=2^{(D-1)/{2}}\,\, \textrm{(for $D$ odd)}~
\label{Chap11:spinordof}
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{11 dimensions}


Majorana spinor has $d_S=32$ real components,  16 dof 
on-shell
\newline  tracelessness condition $\Gamma^\mu \psi_\mu^\alpha=0$ leaves  $(D-3)d_S/2$ degrees of freedom for the vector-spinor\newline  gravitino has 128 real on-shell dof

\noindent gravitino - gaviton = 84 more fermionic dof  than bosonic
\newline difference made up
by three index antisymmetric tensor $A_{\mu\nu\rho}$\newline antisymmetric tensor with
$p$ indices (i.e. rank $p$) has 
\beq
\frac{1}{p!}\,(D-2)\ldots(D-p-1)
\label{Chap15:antisymmdof}
\eeq
dof on-shell, also called a $p$-form field
%It is traditional to write the coupling constant in from of $e R$ in the Lagrangian as $1/%(4 %\kappa^2$, so
%in $D=11$, $\kappa$ has mass dimension $-9/2$ however the theory has a symmetry which
%can rescale the fi
\beq \frac{(11-2)(11-3)(11-4)}{6}=3 \cdot 4 \cdot 7=84
\eeq


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{11 dimensions: BPS solitons}

The SUSY algebra of 11-D supergravity has two central charges \newline two Lorentz indices,  five Lorentz indices $\leftrightarrow$ BPS solitons\newline central charge acts as a \Red{topological charge},   spatial integral at  fixed $t$\newline
preserve  index structure,  solitons  extend in two and five spatial directions\newline called $p$-branes for  $p$ spatial directions \newline e.g. monopole is a 
$0$-brane, couples to a $1$-form gauge field $A_\mu$\newline $p$-brane
couples to a $(p+1)$-form gauge field\newline $2$-brane   couples to
 $3$-form gauge field $A_{\mu\nu\rho}$ \newline  \Blue{a $p$-form gauge field has a $(D-p-2)$-form dual gauge
field}
\newline   field strength $\leftrightarrow$ $A_{\mu\nu\rho}$ is a $4$-form: 
$F_{\mu\nu\rho\lambda}$\newline contract  with $\epsilon$ tensor gives  dual
$7$-form  $\leftrightarrow$  $6$-form dual gauge field\newline 
couples to the $5$-brane

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{10 dimensions}

compactify 1 dimension on a circle\newline decompose $D=11$ fields into massless $D=10$ fields (constant on circle) 
 \beq
 e_\mu^a \,(44) &\rightarrow&  e_\mu^a \,(35),B_\mu \,(8), \sigma \,(1) \nonumber \\
 A_{\mu\nu\rho} \,(84)&\rightarrow&  A_{\mu\nu\rho}\, (56),  A_{\mu\nu}\, (28)\\
 \psi_\mu^\alpha \,(128) &\rightarrow&  \psi_\mu^{+\alpha} \, (56), \psi_\mu^{-\alpha} \, (56), \lambda^{+\alpha} \,(8), \lambda^{-\alpha} \,(8)~\nonumber
 \label{Chap15:11decomp10}
 \eeq
32 supercharges of $D=11$ $\to$ 
two $D=10$ spinors\newline   spinors have opposite
chirality\newline  gravitino
splits into states of opposite chirality, labeled by $+$ and $-$ 
\newline this is \Blue{Type IIA} supergravity
\newline  two other supergravity theories in $D=10$\newline 
\Blue{Type I}: single spinor of supercharges \newline 
\Blue{Type IIB}: supercharges are two spinors with the same chirality

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Low-Energy Effective Theories}

 \noindent Type IIA $\leftrightarrow$ Type IIA  string theory \newline Type IIB  $\leftrightarrow$ Type IIB string theory\newline  Type I  with $E_8\times E_8$ or $SO(32)$ $\leftrightarrow$ heterotic string theory\newline  $D=11$ supergravity
$\leftrightarrow$ M-theory 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{4D helicities}

massless vector  $\to$  massless 4D vector and $D-4$ massless scalars\newline 
$\leftrightarrow$ two components with helicity $1$ and $-1$ and $D-4$ helicity $0$
states\newline $\Leftrightarrow$ $D-4$ lowering operators\newline e.g.  5D, the little group is
$SO(3)$, one lowering operator $\sigma^-=\frac{1}{2}(\sigma^1-i\sigma^2)$

\vspace{24pt} traceless symmetric tensor field, $e^a_\mu$, $\Leftrightarrow$ symmetric product of two vectors:
\beq
\begin{array}{ccc}
 {\rm helicity} & \,\,{\rm degeneracy}\,\,D=11\,\,\,\,\,& \,\,\,\,{\rm degeneracy}\,\,D=10\\
 2 & 1& 1 \\
 1 & 7=1\cdot(11-4) & 6=1\cdot(10-4) \\
 0 & 28=7\cdot 8/2-1+1& 21=6\cdot 7/2-1+1 \\
 -1& 7=1\cdot(11-4) & 6=1\cdot(10-4) \\
 -2 & 1 & 1
\end{array}~
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{4D helicities: 2-form}
2-form field $\Leftrightarrow$
antisymmetric product of two vectors:
\beq
\begin{array}{ccc}
 {\rm helicity} & \,\,{\rm degeneracy}\,\,D=11\,\,\,\,\,&\,\, \,\,{\rm degeneracy}\,\,D=10\\
 1 & 7 & 6 \\
 0 & 7(7-1)/2+1=22& 6(6-1)/2+1=16 \\
 -1& 7 & 6 
\end{array}~
\eeq
where the $7(7-1)/2$ comes from antisymmetrizing the helicity 0 components, and the $+1$ corresponds to  combining the helicity 1 and $-1$ components of the two vectors
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{4D helicities: 3-form}
\beq
\begin{array}{ccc}
 {\rm helicity} & \,\,{\rm degeneracy}\,\,D=11\,\,\,\,\,& \,\,\,\,{\rm degeneracy}\,\,D=10\\
 1 & 21& 15 \\
 0 & 35+7=42& 20+6=26 \\
-1& 21& 15 
\end{array}~
\eeq
$35$ comes from antisymmetrizing three helicity 0 components, and the $+7$ corresponds to the combining  helicity 1 and $-1$ components and one helicity 0 component of the three vectors
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{4D helicities: gravitino}
$D=11$ spinor has 8 helicity $\sfrac{1}{2}$ components and 8 helicity $-\sfrac{1}{2}$ components, while for $D=10$ these components correspond to two opposite chirality spinors, we can reconstruct the gravitino by combining a vector and a spinor (remembering the
tracelessness condition) 
\beq
\begin{array}{rcc}
 {\rm helicity} & \,\,{\rm degeneracy}\,\,D=11\,\,\,\,\,& \,\,\,\,{\rm degeneracy}\,\,D=10\\
 \sfrac{3}{2} & 8\vphantom{\frac{1}{2}} & 8 \\
\sfrac{1}{2} & 56=8\cdot 7& 48=8\cdot 6 \\
-\sfrac{1}{2}& 56=8\cdot 7\vphantom{\frac{1}{2}}& 48=8\cdot 6 \\
-\sfrac{3}{2}& 8\vphantom{\frac{1}{2}}& 8
\end{array}~
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$D=11$ Supermultiplet}
starting with a helicity $-2$ state and raising the
helicity repeatedly by acting with 8 SUSY generators   (and remembering to antisymmetrize)
\beq
\begin{array}{rcc|ccc}
{\rm 11D}\,\, {\rm sugra.}\,\, {\rm state} & \,{\rm helicity}\, & \,{\rm degeneracy}\,& \,e^a_\mu\, & \,A_{\mu\nu\rho}\, & \,\psi^\alpha_\mu\\ \hline
\vphantom{ {{\overline Q}^8}^8}  {\overline Q}^8 |\Omega_{-2}\rangle & \hphantom{-}2 & 1 &1 && \\
 {\overline Q}^7 |\Omega_{-2}\rangle & \hphantom{-}\sfrac{3}{2} & 8 &&& 8\\
  {\overline Q}^6 |\Omega_{-2}\rangle & \hphantom{-}1& 28 &7 & 21 &\\
 {\overline Q}^5 |\Omega_{-2}\rangle & \hphantom{-}\sfrac{1}{2} &56 &&&56 \\
 {\overline Q}^4 |\Omega_{-2}\rangle & \hphantom{-}0 & 70&28&42 & \\
 {\overline Q}^3 |\Omega_{-2}\rangle & -\sfrac{1}{2} &56 &&&56\\
 {\overline Q}^2 |\Omega_{-2}\rangle & -1 & 28 &7&21& \\
 {\overline Q}\hphantom{2}  |\Omega_{-2}\rangle & -\sfrac{3}{2} &8 &&&8\\
|\Omega_{-2}\rangle & -2 & 1 &1&&
\end{array}~
\label{Chap15:gravitonmult:11}
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$D=10$ Type IIA Supermultiplet}
\beq
\begin{array}{rcc|ccccccc}
{\rm IIA}\,\, {\rm state} & \,{\rm helicity}\, & \,{\rm degen.}\,& \,e^a_\mu\, & \,A_{\mu\nu\rho}\,& 
\,A_{\mu\nu}\, & \,B_{\mu}\,&  \,\sigma\,&\,\psi^{\pm\alpha}_\mu\, & \,\lambda^{\pm\alpha} \\
\hline
\vphantom{ {{\overline Q}^8}^8}   {\overline Q}^8 |\Omega_{-2}\rangle & \hphantom{-}2 & 1 &1 &&&&&& \\
 {\overline Q}^7 |\Omega_{-2}\rangle & \hphantom{-}\sfrac{3}{2} & 8 &&&&&& 8&\\
  {\overline Q}^6 |\Omega_{-2}\rangle & \hphantom{-}1& 28 &6 & 15 &6&1&&&\\
 {\overline Q}^5 |\Omega_{-2}\rangle & \hphantom{-}\sfrac{1}{2} &56 &&&&&&48 &8 \\
 {\overline Q}^4 |\Omega_{-2}\rangle & \hphantom{-}0 & 70&21&26 &16&6&1&& \\
 {\overline Q}^3 |\Omega_{-2}\rangle & -\sfrac{1}{2} &56 &&&&&&48&8\\
 {\overline Q}^2 |\Omega_{-2}\rangle & -1 & 28 &6&15&6&1&&& \\
 {\overline Q}\hphantom{2}  |\Omega_{-2}\rangle & -\sfrac{3}{2} &8 &&&&&&8&\\
|\Omega_{-2}\rangle & -2 & 1 &1&&&&&&
\end{array}~
\label{Chap15:gravitonmult:IIA}
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{10D: BPS branes}
SUSY algebra of  Type IIA, central charges of rank 
 0, 1, 2, 4,  5, 6,  8\newline $\leftrightarrow$ p-branes\newline
gauge fields of rank 1, 2, and 3\newline dual gauge fields of rank 5, 6,  7\newline
 $1$-brane $\leftrightarrow$ fundamental string of
Type IIA string theory

\vspace{24pt}
Type IIA supergravity $\leftrightarrow$ compactified
11D supergravity
\newline
2-brane of 11D  $\leftrightarrow$  2-brane of Type IIA when $\perp$ circle
\newline 2-brane of 11D  $\leftrightarrow$ 1-brane when wraps circle\newline  5-brane of 11D supergravity $\leftrightarrow$  5-brane and  4-brane of 
Type IIA 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Brane Tensions}
  11D supergravity has one coupling constant, $\kappa$,  11D Newton's constant
  \beq {\mathcal L}= \frac{1}{2\kappa^2} \, e R \eeq

11D Planck mass by $\kappa=M_{\rm Pl}^{-9/2}$\newline tension (energy per unit volume) of  branes
 given  powers of $M_{\rm Pl}$\newline energy per unit area of the 2-brane is $T_2 = M_{\rm Pl}^3$\newline 5-brane we have $T_5 = M_{\rm Pl}^6$\newline  2-brane and 5-brane
of Type IIA  have same tensions as 11D theory\newline  1-brane and 4-brane  have $T_1=R_{10}M_{\rm Pl}^3$ and  $T_4=R_{10}M_{\rm Pl}^6$\newline  1-brane is the fundamental string
of Type IIA string theory \newline $\Rightarrow$ \Blue{identify  tension with  string tension or string mass squared}:
\beq
T_1=R_{10}M_{\rm Pl}^3\equiv \frac{1}{4 \pi \alpha^\prime}\equiv m_s^2~ \eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{String Coupling} express the
 tensions in terms of $m_s$ and  Type IIA string coupling
\beq
g_s= (R_{10}M_{\rm Pl})^{3/2}~
\label{Chap15:IIAstringcouping}
\eeq
\beq
T_2 = \frac{m_s^3}{g_s}~,\,\,\, T_4 = \frac{m_s^5}{g_s}~,\,\,\, T_5 = \frac{m_s^6}{g_s^2}~
\label{Chap15:branetensions}
\eeq
 branes are nonperturbative BPS solitons not surprising to see inverse powers of
the coupling \newline  $1/g_s$ dependence  of the 2-brane and 4-brane 
significant \newline universal feature of what are now called \Red{D-branes}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$D=10$ Type IIB Supermultiplet}

SUSY algebra
has central charges of  rank 1, 3, 5,  7\newline expect the corresponding 
$p$-branes to couple to gauge fields of rank 2 and 4,
$A_{\mu\nu}$ and $B_{\mu\nu\rho\lambda}$, and their duals\newline $e^a_\mu$, $\psi^\alpha_\mu$, $\lambda^\alpha$  have same dof as the Type IIA, difference being that $\psi_\mu^\alpha$ and $\lambda^\alpha$ have opposite chirality in the IIB theory\newline 
it turns out that $A_{\mu\nu}$ is complex, twice as many dof = 56\newline
so far the fermions have 37 more dof than the $e^a_\mu$ and $A_{\mu\nu}$
combined, while an unconstrained $4$-form field has 70 dof \newline $5$-form field strength corresponding to $B_{\mu\nu\rho\lambda}$ constrained to be self-dual, reduces dof to 35\newline  need a complex scalar, $a$,  to balance out the multiplet:
\beq
\begin{array}{c}
e_\mu^a \,(35), a \,(2), A_{\mu\nu}\, (56),  B_{\mu\nu\rho\lambda}\, (35)\\
\psi_\mu^\alpha \, (112), \lambda^\alpha \,(16)
\end{array}~
 \eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$D=10$ Type IIB Supermultiplet}

 two SUSY spinor charges of the Type IIB theory have  same chirality\newline 
transform as vector under an $SO(2)$ group, i.e. $R$-charges $\pm 1$\newline single Clifford vacuum
state with helicity $-2$ must have $SO(2)$ charge 0\newline gravitino splits into two parts
with charges $\pm 1$\newline to antisymmetrize  SUSY charges: \newline antisymmetrize  $SO(2)$, symmetrize  remaining four spinor indices\newline or \newline symmetrize  $SO(2)$,  antisymmetrize the remaining four spinor indices

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$D=10$ Type IIB Supermultiplet}

\beq
\begin{array}{rcc|cccccc}
{\rm IIB}\,\, {\rm state} & \,{\rm helicity}\, & \,{\rm degeneracy}\,& \,e^a_\mu\, & \,B_{\mu\nu\rho\lambda}\,& \,A_{\mu\nu}\,&  \,a \,&\,\psi^\alpha_\mu\, & \,\lambda^\alpha \\ \hline
\vphantom{ {{\overline Q}^8}^8}   {\overline Q}^8 |\Omega_{-2}\rangle & \hphantom{-}2 & 1 &1 &&&&& \\
 {\overline Q}^7 |\Omega_{-2}\rangle & \hphantom{-}\sfrac{3}{2} & 8 &&&&& 8&\\
  {\overline Q}^6 |\Omega_{-2}\rangle & \hphantom{-}1& 28 &6 & 10 &12&&&\\
 {\overline Q}^5 |\Omega_{-2}\rangle & \hphantom{-}\sfrac{1}{2} &56 &&&&&48 &8 \\
 {\overline Q}^4 |\Omega_{-2}\rangle & \hphantom{-}0 & 70&21 &15&32&2&& \\
 {\overline Q}^3 |\Omega_{-2}\rangle & -\sfrac{1}{2} &56 &&&&&48&8\\
 {\overline Q}^2 |\Omega_{-2}\rangle & -1 & 28 &6&10&12&&& \\
 {\overline Q}\hphantom{2}  |\Omega_{-2}\rangle & -\sfrac{3}{2} &8 &&&&&8&\\
|\Omega_{-2}\rangle & -2 & 1 &1&&&&&
\end{array}~
\label{Chap15:gravitonmult:IIB}
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$D=10$ Type IIB Supermultiplet}
symmetric combination of $4\times 4$ is $10$, antisymmetric under  $SO(2)$ 
\newline $\Rightarrow$ $B_{\mu\nu\rho\lambda}$ has  $SO(2)$ charge 0\newline antisymmetric combination of $4\times 4$ is $6$ and  graviton
has  $SO(2)$ charge 0\newline $\Rightarrow$  two 6's corresponding to $A_{\mu\nu}$ must have charges $\pm 2$.  \newline $\lambda^\alpha$ has $SO(2)$ charge $\pm 3$ \newline 
scalar $a$ has $SO(2)$ charge $\pm 4$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$D=10$ Type I Supermultiplet} 
parity  in Type IIB: 
$4$-form, half of  $2$-form,   half of scalar are odd
\newline  truncate by keeping only the even fields.
\newline Majorana condition
on the fermions reduces dof  by one half

\beq
\begin{array}{c}
e_\mu^a \,(35), \sigma \,(1), A_{\mu\nu}\, (28)\\
\psi_\mu^\alpha \, (56), \lambda^\alpha \,(8)
\end{array}~
 \eeq
construction of multiplet has a further complication:  only four SUSY raising operators, starting with  $|\Omega_{-2}\rangle$   yields a maximum helicity of 0\newline adding CPT conjugate 
$\to$ two helicity 0 components and four helicity $\sfrac{1}{2}$\newline graviton  requires 21  helicity 0 components \newline gravitino and spinor require 28 helicity $\sfrac{1}{2}$ components\newline need to add 6 copies of a multiplet based on $|\Omega_{-1}\rangle$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$D=10$ Type I Supermultiplet}

\beq
\begin{array}{lcc|ccccc}
{\rm Type}\,\, {\rm I}\,\, {\rm state} & \,{\rm helicity}\, & \,{\rm degen.}\,& \,e^a_\mu\, & \,A_{\mu\nu}\,&  \,\sigma \,&\,\psi^\alpha_\mu\, & \,\lambda^\alpha \\ \hline
\vphantom{ {{\overline Q}^8}^8}      {\overline Q}^4 |\Omega_{0}\rangle & \hphantom{-}2 & 1 &1 &&&& \\
 {\overline Q}^3 |\Omega_{0}\rangle & \hphantom{-}\sfrac{3}{2} & 4 &&&& 4&\\
  {\overline Q}^2 |\Omega_{0}\rangle +6 \times   {\overline Q}^4 |\Omega_{-1}\rangle& \hphantom{-}1& 12 &6  &6&&&\\
 {\overline Q}\hphantom{^2} |\Omega_{0}\rangle +6 \times   {\overline Q}^3 |\Omega_{-1}\rangle& \hphantom{-}\sfrac{1}{2} &28 &&&&24 &4 \\
 {\overline Q}^4 |\Omega_{-2}\rangle +6 \times   {\overline Q}^2 |\Omega_{-1}\rangle+ |\Omega_{0}\rangle& \hphantom{-}0 & 38 &21 &16&1&& \\
 {\overline Q}^3 |\Omega_{-2}\rangle +6 \times   {\overline Q}\hphantom{^2} |\Omega_{-1}\rangle& -\sfrac{1}{2} &28 &&&&24&4\\
 {\overline Q}^2 |\Omega_{-2}\rangle +6 \times   |\Omega_{-1}\rangle& -1 & 12 &6&6&&& \\
 {\overline Q}\hphantom{^2}  |\Omega_{-2}\rangle & -\sfrac{3}{2} &4 &&&& 4&\\
 \hphantom{{\overline Q}^2}|\Omega_{-2}\rangle & -2 & 1 &1 &&&&
\end{array}~.~\,
\label{Chap15:gravitonmult:I}
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$D=10$ Type I and Yang-Mills} 
 $D=10$, ${\mathcal N}=1$ (16 real supercharges)
Yang-Mills \newline contains a vector and spinor with eight dof each 

couple to Type I supergravity \newline anomaly cancellation  allows  for the gauge group
$E_8\times E_8$ or $SO(32)$\newline  two  low-energy effective theories for
the two  heterotic string theories

\vspace{24pt} $D=10$, ${\mathcal N}=1$ 
Yang-Mills is  low-energy effective theory for\newline  Type I string theory

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$D=5$, ${\mathcal N=8}$, gauged supergravity}
\label{Chap15:sugra:sec:fivedimensions}

 \noindent consider Type IIB supergravity compactified
on  $S^5$\newline  integrate out nonzero modes on $S^5$ \newline  $SO(6) \sim SU(4)$ isometry $\to$  gauge symmetry in the effective theory
 \newline 5D little group is $SO(3)$\newline 
each massless field has one component for each helicity
\newline massless 5D  have the same dof as the corresponding massive 4D 

\vspace{24pt}\noindent graviton has helicities: 2,1,0,-1, and -2: five dof\newline  vector and 2-form have  three  helicity components 1,  0, and  -1
\newline  gravitino has  four helicity components:  3/2, 1/2,  -1/2, and -3/2
\newline  spinor has helicity components  1/2 and -1/2

\vspace{24pt}\noindent in addition to the $SU(4)$ gauge symmetry, there is  $SO(2)$ $R$-symmetry  from  Type IIB  theory
\newline SUSY generators transform as $(\fund,+1)+(\afund,-1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{5D Graviton Supermultiplet}

\beq
\begin{array}{rcc|cccccc}
{\rm 5D}\,\,{\rm  sugra.}\,\, {\rm state} & \,{\rm helicity}\, & \,{\rm degeneracy}\,& \,e^a_\mu\, & \,A_\mu \,& B_{\mu\nu} \,&  \,\phi \,&\,\psi^\alpha_\mu\, & \,\lambda^\alpha \\ \hline
\vphantom{ {{\overline Q}^8}^8}   {\overline Q}^8 |\Omega_{-2}\rangle & \hphantom{-}2 & 1 &1 &&&&& \\
 {\overline Q}^7 |\Omega_{-2}\rangle & \hphantom{-}\sfrac{3}{2} & 8 &&&&& 8&\\
  {\overline Q}^6 |\Omega_{-2}\rangle & \hphantom{-}1& 28 &1 & 15 &12&&&\\
 {\overline Q}^5 |\Omega_{-2}\rangle & \hphantom{-}\sfrac{1}{2} &56 &&&&&8 &48 \\
 {\overline Q}^4 |\Omega_{-2}\rangle & \hphantom{-}0 & 70&1 &15&12&42&& \\
 {\overline Q}^3 |\Omega_{-2}\rangle & -\sfrac{1}{2} &56 &&&&&8&48\\
 {\overline Q}^2 |\Omega_{-2}\rangle & -1 & 28 &1&15&12&&& \\
 {\overline Q}\hphantom{2}  |\Omega_{-2}\rangle & -\sfrac{3}{2} &8 &&&&&8&\\
|\Omega_{-2}\rangle & -2 & 1 &1&&&&&
\end{array}~
\label{Chap15:gravitonmult:5D}
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{5D Graviton Supermultiplet}
 representations of  $SU(4) \times SO(2)$
\beq
\begin{array}{c|cc}
{\rm graviton}& e^a_\mu & (1,0) \\
 {\rm vector} &A_\mu &\left(\Yoneoone,0\right)\\
\textrm{  2-form} &B_{\mu\nu} &\left(\asymm,2\right)+\left(\asymm,-2\right)\\\
{\rm scalars} &\phi &(1,\pm1), \left(\symm\vphantom{\overline{ \symm}}, 2\right)+ \left(\overline{ \symm}, - 2 \right)+\left(\Yfourbox,0\right)\\
 {\rm gravitino}& \psi^\alpha_\mu& (\fund,1)+(\afund,-1)\\
 {\rm ``gauginos``} & \lambda^\alpha& \left(\fund\vphantom{\afund},3\right),  + \left(\afund,-3\right)+\left( \Yadjoint ,1\right)+\left(\overline{\Yadjoint} ,-1\right)
\end{array}~
\label{Chap15:gravitonmult:5Dreps}
\eeq 



\end{document}

 
 
