SUGRA.tex

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\section{Supergravity}
\subsection{Minimal SUGRA Lagrangian}
Minimal SUGRA Lagrangian is constructed from supergravity multiplet
$(\bein a\mu,\psi^\alpha_\mu, B_\mu, F_\phi)$.
\begin{equation}
 \Lag = -\frac{M^2}{2}eR+e\epsilon^{\mu\nu\rho\sigma}\bar\psi_\mu\bSnd\DD_\rho\psi_\sigma
\end{equation}
where
\begin{align}
 \DD_\mu\psi_\nu
  &:=\Pm\psi_\nu+\frac12\omega\T_\mu^a^b\sigma_{ab}\psi_\nu\qquad
  \left[\omega\T_\mu^a^b : {\text{``spin接続''}}\right]\\
 e&:=\det \bein a\mu\\
 M&:=1/\sqrt{8\pi G} \quad \text{(Reduced Planck mass)}\\
 R&:={\bein a\mu}{\bein b\nu}R\T_\mu_\nu^a^b\\
 R\T_\mu_\nu^a^b
  &:=\Pm\omega\T_\nu^a^b-\Pn\omega\T_\mu^a^b
     -\omega\T_\mu^a^c\omega\T_\nu_c^b+\omega\T_\nu^a^c\omega\T_\mu_c^b.
\end{align}
\subsection{General SUGRA Lagrangian}
The components of general SUGRA Lagrangian is
\begin{equation}
 \Phi_i = (\phi_i,\chi^\alpha_i,F_i),\qquad
 V^{(a)}=(A^{(a)}_\mu,\lambda^{\alpha(a)},D^{(a)}),\qquad
 G=(\bein \mu a,\psi_\mu^\alpha,B_\mu,F_\phi),
\end{equation}
and described with following functions:
\begin{itemize}
 \item K\"ahler potential $K(\Phi,\Phi^*)$
\begin{itemize}
 \item Real function of chiral multiplets.
 \item In global SUSY, $\int\dd^4\theta K$ yields kinetic terms of the
       chiral multiplet.
 \item ``Minimal K\"ahler'' is (if no gauge interaction)
       $K=\Phi\Phi^\dagger$, which is
\begin{equation}
 \int\dd^4\theta\ \Phi\Phi^*=\Pm \phi^*\Pm\phi+\ii\bar\chi\bSm\Pm
  \chi + F^*F.
\end{equation}
\end{itemize}
 \item Super Potential $W(\Phi)$
 \item Gauge kinetic term $f_{(a)(b)}(\Phi)$
\begin{itemize}
 \item Some function which satisfies $f_{(a)(b)} = f_{(b)(a)}$.
 \item $(a), (b), ...$ are indices for adjoint representation of gauge
       group.
 \item Minimal one is $f_{(a)(b)} \propto \delta_{(a)(b)}.$
\end{itemize}
\end{itemize}

\def\epsmnrs{\epsilon^{\mu\nu\rho\sigma}}
\def\fR{{f\suprm{R}}}
\def\fI{{f\suprm{I}}}
\begin{align}
 \Lag
={}&
 -\frac12eR
 +eg_{ij^*}\DD_\mu\phi^i\DD^\mu\phi^{*j}
 -\frac12eg^2D_{(a)}D^{(a)}
\notag\\&
 +\ii eg_{ij^*}\bar\chi^j\bSm\DD_\mu\chi^i
 +e\epsmnrs\bar\psi_\mu\bSnd\DD_\rho\psi_\sigma
\notag\\&
 -\frac14e\fR_{(ab)}F^{(a)}_{\mu\nu}F^{\mu\nu(b)}
 +\frac18e\epsmnrs\fI_{(ab)}f^{(a)}_{\mu\nu}f^{(b)}_{\rho\sigma}
\notag\\&
 +\frac\ii2e\left[\lambda_{(a)}\Sm\DD_\mu\bar\lambda^{(a)}
                + \bar\lambda_{(a)}\bSm\DD_\mu\lambda^{(a)}\right]
 -\frac12\fI_{(ab)}\DD_\mu\left[e\lambda^{(a)}\Sm\bar\lambda^{(b)}\right]
\notag\\&
 +\sqrt2egg_{ij^*}X^{*j}_{(a)}\chi^i\lambda^{(a)}
 +\sqrt2egg_{ij^*}X^{i}_{(a)}\bar\chi^j\bar\lambda^{(a)}
\notag\\&
 -\frac\ii4\sqrt2 eg\partial_if_{(ab)}D^{(a)}\chi^i\lambda^{(b)}
 +\frac\ii4\sqrt2 eg\partial_{i*}f^*_{(ab)}D^{(a)}\bar\chi^i\bar\lambda^{(b)}
\notag\\&
 -\frac14\sqrt2e\partial_if_{(ab)}\chi^i\sigma^{\mu\nu}\lambda^{(a)}F^{(b)}_{\mu\nu}
 -\frac14\sqrt2e\partial_{i^*}f^*_{(ab)}\bar\chi^i\bar\sigma^{\mu\nu}\bar\lambda^{(a)}F^{(b)}_{\mu\nu}
\notag\\&
 +\frac12egD_{(a)}\psi_\mu\Sm\bar\lambda^{(a)}
 -\frac12egD_{(a)}\bar\psi_\mu\bSm\lambda^{(a)}
\notag\\&
 -\frac12\sqrt2eg_{ij^*}\DD_\nu\phi^{*j}\chi^i\Sm\bSn\psi_\mu
 -\frac12\sqrt2eg_{ij^*}\DD_\nu\phi^i\bar\chi^j\bSm\Sn\bar\psi_\mu
\notag\\&
-\frac\ii4e\left[\psi_\mu\sigma^{\nu\rho}\Sm\bar\lambda_{(a)}
               + \bar\psi_\mu\bar\sigma^{\nu\rho}\bSm\lambda_{(a)}\right]
           \left[F^{(a)}_{\nu\rho}+\hat F^{(a)}_{\nu\rho}\right]
\notag\\&
+\frac14eg_{ij^*}\left[\ii\epsmnrs\psi_\mu\sigma_\nu\bar\psi_\rho
                     + \psi_\mu\Ss\bar\psi^\mu\right]
                 \chi^i\sigma_\sigma\bar\chi^i
\notag\\&
-\frac18e\left[g_{ij^*}g_{kl^*}-2R_{ij^*kl^*}\right]\chi^i\chi^k\bar\chi^j\bar\chi^l
\notag\\&
+\frac1{16}e\left[2g_{ij^*}\fR_{(ab)}
                + {\fR^{(cd)}}^{-1}\partial_if_{(bc)}\partial_{j^*}f^*_{(ad)}\right]
            \bar\chi^j\bSm\chi^i\bar\lambda^{(a)}\bar\sigma_\mu\lambda^{(b)}
\notag\\&
+\frac18e\nabla_i\partial_jf_{(ab)}\chi^i\chi^j\lambda^{(a)}\lambda^{(b)}
+\frac18e\nabla_{i^*}\partial_{j^*}f^*_{(ab)}\bar\chi^i\bar\chi^j\bar\lambda^{(a)}\bar\lambda^{(b)}
\notag\\&
+\frac1{16}e{\fR^{(cd)}}^{-1}\partial_if_{(ac)}\partial_jf_{(bd)}
             \chi^i\lambda^{(a)}\chi^j\lambda^{(b)}
\notag\\&
+\frac1{16}e{\fR^{(cd)}}^{-1}\partial_{i^*}f^*_{(ac)}\partial_{j^*}f^*_{(bd)}
             \bar\chi^i\bar\lambda^{(a)}\bar\chi^j\bar\lambda^{(b)}
\notag\\&
-\frac1{16}eg^{ij^*}\partial_if_{(ab)}\partial_{j^*}f^*_{(cd)}
             \lambda^{(a)}\lambda^{(b)}\bar\lambda^{c}\bar\lambda^{(d)}
\notag\\&
+\frac3{16}e\lambda_{(a)}\Sm\bar\lambda^{(a)}\lambda_{(b)}\sigma_\mu\bar\lambda^{(b)}
\notag\\&
+\frac\ii4\sqrt2 e\partial_{i}f_{(ab)}\left[
                \chi^i\sigma^{\mu\nu}\lambda^{(a)}\psi_\mu\sigma_\nu\bar\lambda^{(b)}
              - \frac14\bar\psi_\mu\bSm\chi^i\lambda^{(a)}\lambda^{(b)}\right]
\notag\\&
+\frac\ii4\sqrt2 e\partial_{i^*}f^*_{(ab)}\left[
              \bar\chi^i\bar\sigma^{\mu\nu}\bar\lambda^{(a)}\bar\psi_\mu\bar\sigma_\nu\lambda^{(b)}
            - \frac14\psi_\mu\Sm\bar\chi^i\bar\lambda^{(a)}\bar\lambda^{(b)}\right]
\notag\\&
-e\ee^{K/2}\left[W^*\psi_\mu\sigma^{\mu\nu}\psi_\nu
               + W  \bar\psi_\mu\bar\sigma^{\mu\nu}\bar\psi_\nu\right]
\notag\\&
+\frac\ii2\sqrt2e\ee^{K/2}\left[D_iW\chi^i\Sm\bar\psi_\mu
                              + D_{i^*}W^*\bar\chi^i\bSm\psi_\mu\right]
\notag\\&
-\frac12e\ee^{K/2}\left[\DD_i D_jW\chi^i\chi^j+\DD_{i^*}D_{j^*}W^*\bar\chi^i\bar\chi^j\right]
\notag\\&
+\frac14e\ee^{K/2}g^{ij^*}\left[ D_{j^*}W^*\partial_if_{(ab)}\lambda^{(a)}\lambda^{(b)}
                               + D_iW\partial_{j^*}f^*_{(ab)}\bar\lambda^{(a)}\bar\lambda^{(b)}
                          \right]
\notag\\&
-e\ee^{K}\left[g^{ij^*}(D_iW)(D_{j^*}W^*)-3W^*W\right]
\end{align}

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