formulae.tex

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\begin{document}
\begin{center}
        {\Large\bf Notations}
\end{center}

\begin{align*}
%% Phi
  \Phi  & ~~=~~  \phi  + \sqrt{2}\lambda\theta + F\theta^2  \\ \\
%% V  
  V  &  ~~=~~      - \overline{\theta\sigma_\mu}\theta A_\mu + 
		 i \theta^2 \overline{\theta\lambda} +
 		 i \overline{\theta^2}\theta\lambda -
          	 \frac{i}{2} \theta^2\overline{\theta}^2 D \\ \\
%% W		 
  W^\alpha & ~~=~~  -\lambda^\alpha + \theta^\alpha D  +
              \frac{1}{2}
	\left[\sigma_\mu\overline{\sigma}_\nu\theta\right]^\alpha F_{\mu\nu} -
	\theta^2 i \slashed{\mathcal{D}}^{\alpha\dot{\alpha}}
	\overline{\lambda}_{\dot{\alpha}} \\ \\
%% sigma-matrices	
  \sigma_\mu^{\alpha\dot{\alpha}} & ~~=~~ 
		\lgr 1,\; -i \tau^a \rgr~, ~~~~~~~
  \overline{\sigma}_{\mu\dot{\alpha}\alpha} ~~=~~ 
		\lgr 1,\; \phantom{+}i \tau^a \rgr \\ \\
%% Covariant derivatives -- D^\alpha
   D^\alpha &  ~~=~~  \partial^\alpha - 
            i \slashed{\partial}^{\alpha\dot{\alpha}}
	    \overline{\theta}_{\dot{\alpha}}  
\quad
   \epsilon_{\alpha\beta} ~~=~~ 
    \epsilon_{\dot{\alpha}\dot{\beta}} ~~=~~
    \lgr \begin{array}{rr}
         \!\!  0 & ~~1 \\ \\
         \!\! -1 & ~~0 
    \end{array} \rgr
\quad
	    \psi_\alpha ~~=~~ \epsilon_{\alpha\beta}\psi^\beta 
\quad
	    \overline{\psi}_{\dot{\alpha}} ~~=~~ 
			  \epsilon_{\dot{\alpha}\dot{\beta}}
			  \overline{\psi}^{\dot{\beta}}\\ 
%% Covariant derivatives -- D_{\dot \alpha}
   \overline{D}_{\dot{\alpha}} & ~~=~~ \overline{\partial}_{\dot{\alpha}} -
            i \ov{\slashed{\partial}}{}_{\dot{\alpha}\alpha}\theta^\alpha 
\quad
   \epsilon^{\alpha\beta} ~~=~~ 
    \epsilon^{\dot{\alpha}\dot{\beta}} ~~=~~
    \lgr\begin{array}{rr}
         ~0 & -1 \\ \\
         ~1 &  0
    \end{array} \rgr
\quad
	    \psi^\alpha ~~=~~ \epsilon^{\alpha\beta}\psi_\beta 
\quad
	    \overline{\psi}^{\dot{\alpha}} ~~=~~ 
			  \epsilon^{\dot{\alpha}\dot{\beta}}
			  \overline{\psi}_{\dot{\beta}}\\ \\
 \\
%%  Spinor-index contractions
  W^2 & ~~=~~  W_\alpha W^\alpha~,~~~~~~~ 
    \overline{W}^2 ~~=~~ \overline{W^{\dot{\alpha}} W_{\dot{\alpha}}} \\ \\
%%  Conjugate Field-strength
  \overline{W}_{\dot{\alpha}} & ~~=~~ 
        - \overline{\lambda}_{\dot{\alpha}} +
	  \overline{\theta}_{\dot{\alpha}} D -
	  \frac{1}{2} \left [ 
	      \overline{\sigma}_\mu\sigma_\nu \overline{\theta}
	              \right ]_{\dot{\alpha}} F_{\mu\nu} -
	  \overline{\theta}^2 (i \ov{\slashed{\mc{D}}}\lambda)_{\dot{\alpha}} \\ \\
%%  Chiral fields
  x_L & \equiv  ~~~y ~~~ 
	\equiv ~~~ x_\mu + i \overline{\theta\sigma_\mu}\theta~,~~~~~
   \Phi ~~=~~ \Phi(y)~,~~~~~~\overline{\Phi} ~~=~~ \overline{\Phi}(x_R) \\ \\
%%  Gauge derivative
  \nabla_\mu & ~~=~~  \partial_\mu - i A_\mu^a \frac{\tau^2}{2}~,~~~ 
           \text{for SU(2)} \\ \\
%%  Vector superfield in "y" variable
  V(y) & ~~=~~  - \overline{\theta\sigma_\mu}\theta A_\mu(y) + 
	    i \theta^2 \overline{\theta\lambda}(y) +
            i \overline{\theta}^2 \theta\lambda(y) -
            \frac{i}{2}\theta^2\overline{\theta}^2
	    \left [ D(y) - \partial_\mu A^\mu(y) \right ] \\ \\
%%  Definition of the field-strength
  W^\alpha & ~~=~~ - \frac{i}{4} \overline{DD} D^\alpha V 
  \qquad
  \lgr \text{has to be renormalized for $ e^{-2V} $ in the chiral representation} \rgr
	\\ \\
%%  Shift of the variable to the chiral variable (or vice versa)
  \text{Shift:} &\quad x_L ~~\to~~ x  \qquad {\rm is} \qquad e^{-i \ov{\theta\slashed{\partial}}\theta}
\end{align*}


\pagebreak
\renewcommand{\arraystretch}{2.5}

\begin{center}
        {\Large\bf Useful formulae}
\end{center}

\begin{eqnarray*}
\begin{array}{ll}
%%
  \theta^\alpha \theta^\beta ~~=~~ 
	\frac{1}{2}\theta^2 \epsilon^{\alpha\beta} 
  & \theta^1 \theta^2 ~~=~~ - \frac{1}{2} \theta^2 \\
%%
   \theta_\alpha \theta_\beta ~~=~~ 
	-\frac{1}{2}\theta^2 \epsilon_{\alpha\beta} 
  & \theta_1 \theta_2 ~~=~~ 
	-\frac{1}{2}\theta^2 \\
%%
   \overline{\theta^{\dot{\alpha}}\theta}{}^{\dot{\beta}} ~~=~~ 
	-\frac{1}{2}\overline{\theta}^2 \epsilon^{\dot{\alpha}\dot{\beta}}
  & \overline{\theta^{\dot{1}}\theta}{}^{\dot{2}} ~~=~~ 
	\frac{1}{2}\overline{\theta}^2 \\
%%
   \overline{\theta_{\dot{\alpha}}\theta}{}_{\dot{\beta}} ~~=~~
	\frac{1}{2}\overline{\theta}^2 \epsilon_{\dot{\alpha}\dot{\beta}}
  & \overline{\theta_{\dot{1}}\theta}{}_{\dot{2}} ~~=~~
	\frac{1}{2}\overline{\theta}^2 \\
%%
   \theta\lambda\, \theta\psi ~~=~~ - \frac{1}{2}\, \theta^2 \lambda\psi 
  &
   \ov{\theta\lambda}\, \ov{\theta\psi} ~~=~~ -\frac{1}{2}\, \ov{\theta}{}^2 \ov{\lambda\psi} \\
%%
   \sigma_\mu\,\ov{\sigma}{}_\nu\, \sigma_\rho ~~=~~ 
	\eta_{\mu\nu}\, \sigma_\rho ~+~ \eta_{\nu\rho}\,\sigma_\mu ~-~ \eta_{\mu\rho}\,\sigma_\nu
	~-~ \epsilon_{\mu\nu\rho\lambda}\,\sigma_\lambda 
&
   \multirow{2}{*}{
$	\sigma_\mu^{\beta\dot\beta}\, \ov{\sigma}{}_{\dot\alpha\alpha}^\mu ~~=~~
				2 \delta_\alpha^{\ \beta} \delta_{\dot\alpha}^{\ \dot\beta}  $
	} \\
%%
   \ov{\sigma}{}_\mu\,\sigma_\nu\,\ov{\sigma}{}_\rho ~~=~~
	\eta_{\mu\nu}\, \ov{\sigma}{}_\rho ~+~ \eta_{\nu\rho}\, \ov{\sigma}{}_\mu 
								   ~-~ \eta_{\mu\rho}\, \ov{\sigma}{}_\nu
	~+~ \epsilon_{\mu\nu\rho\lambda}\,\ov{\sigma}{}_\lambda \\
%%
   Tr \left ( \overline{\sigma}^\mu\sigma^\nu \right) ~~=~~
	2 \eta^{\mu\nu}
  & \overline{\theta\sigma}{}^\mu\theta 
    \overline{\theta\sigma}{}^\nu\theta ~~=~~
	\frac{1}{2} \theta^2\overline{\theta}^2 \eta^{\mu\nu} \\
%%
   \p_\alpha \theta_\beta ~~=~~ \epsilon_{\alpha\beta} 
  &
   \ov{\p^{\dot\alpha} \theta}{}^{\dot\beta} ~~=~~ \epsilon^{\dot\alpha\dot\beta} \\
%%
   \partial^\alpha \theta^2 ~~=~~ 2 \theta^\alpha
  & \overline{\partial}_{\dot{\alpha}} \overline{\theta}{}^2 ~~=~~
	2 \overline{\theta}_{\dot{\alpha}} \\
%%
   \partial^2 \theta^2 ~~=~~ -4
  &
   \ov{\p^2 \theta}{}^2 ~~=~~ -4 \\
%%
   \epsilon_{\alpha\gamma}\epsilon^{\gamma\beta} ~~=~~
	\delta_{\alpha}^{\beta}
  & \epsilon_{\dot{\alpha}\dot{\gamma}} 
    \epsilon^{\dot{\gamma}\dot{\beta}} ~~=~~ 
	\delta_{\dot{\alpha}}^{\dot{\beta}} \\
%% 
   \epsilon^{\alpha\gamma}\epsilon_{\gamma\beta} ~~=~~
	\delta^\alpha_\beta 
  & \epsilon^{\dot{\alpha}\dot{\gamma}}
    \epsilon_{\dot{\gamma}\dot{\beta}} ~~=~~
	\delta^{\dot{\alpha}}_{\dot{\beta}} \\
%%
  \sigma^{(\mu}\overline{\sigma}^{\nu)} ~~=~~ 
	2 \eta^{\mu\nu} 
  & \slashed{\p}\overline{\slashed{\p}} ~~=~~ \ov{\slashed{\p}}\slashed{\p} ~~=~~ \Box 
	\\
%%
  D_\alpha D^\alpha ~~=~~ \p^2 ~~-~~ 2\, i \p_\alpha\slashed{\p}^{\alpha\dot{\beta}}\ov{\theta}_{\dot{\beta}} 
	~~+~~ \Box \ov{\theta}^2
	\\
%%
  \overline{D^{\dot{\alpha}}D}{}_{\dot{\alpha}} ~~=~~ 
	\overline{\partial}^2  ~~+~~
	2 i \ov{\slashed{\partial}}{}_{\dot{\alpha}\alpha}
		\theta^\alpha \overline{\partial}^{\dot{\alpha}}
        ~~+~~ \theta^2 \Box
	\\
%%
   \left\{ D^{\alpha}\, \ov{D}{}^{\dot{\alpha}} \right\}  ~~=~~ 2i\,\slashed{\p}^{\alpha\dot{\alpha}}
\end{array}
\end{eqnarray*}

 
\pagebreak
\renewcommand{\arraystretch}{1.0}

\begin{center}
        {\Large\bf Chiral Superfields}
\end{center} 

\begin{align*}
%%
	\Phi(y) & ~~=~~ \phi(y) ~~+~~ \sqrt{2}\, \psi\,\theta ~~+~~ F\, \theta^2~,   &
				y^\mu & ~~=~~ x^\mu ~~+~~ i\, \ov{\theta\sigma}{}^\mu\theta  \\
%%
	\ov{\Phi}(\ov{y}) & ~~=~~ \ov{\phi}(\ov y) ~~+~~ \sqrt{2}\, \ov{\psi\, \theta} ~~+~~ \ov{F\, \theta}{}^2~, &
				\ov{y}{}^\mu & ~~=~~ x^\mu ~~-~~ i\, \ov{\theta\sigma}{}^\mu\theta
\end{align*}

\begin{align*}
%%
	\Phi(y) & ~~=~~ \phi(x) ~+~ \sqrt{2}\,\theta\psi ~+~ \theta^2 F 
			~+~ i\,\ov{\theta\slashed{\p}}\,\theta\,\phi 
			~-~ i\,\frac{\sqrt{2}}{2}\, \theta^2\, \ov{\theta\slashed{\p}}\,\psi 
			~-~ \frac{1}{4}\,\theta^4\,\Box \phi
	\\
%%
	\ov{\Phi}(\ov y) & ~~=~~ 
			\ov\phi(x) ~+~ \sqrt{2}\,\ov{\theta\psi} ~+~ \ov\theta{}^2 \ov F 
			~+~ i\,\theta\slashed{\p}\,\ov\theta\,\ov\phi
			~-~ i\,\frac{\sqrt{2}}{2}\, \ov\theta{}^2\, \theta\slashed{\p}\,\ov{\psi}
			~-~ \frac{1}{4}\,\theta^4\,\Box \ov\phi
\end{align*}

\begin{align*}
%%
	\int d^4\theta & ~~=~~ \frac{1}{16}\, D^2\, \ov D{}^2 \Big|
	& 
	\int d^4\theta & ~~=~~ \frac{1}{16}\, \ov D{}^2\, D^2 \Big|
	\\
%%
	\int d^2\theta & ~~=~~ -\,\frac{1}{4}\, D^2 \Big|
	&
	\int d^2\ov\theta & ~~=~~ -\,\frac{1}{4}\, \ov D{}^2 \Big|
	\\ \\
%%
	D^\alpha \Phi \Big| & ~~=~~ \sqrt{2} \psi^\alpha     
	&
	\ov{D_{\dot{\alpha}}\Phi} \Big| & ~~=~~ \sqrt{2} \ov{\psi}{}_{\dot{\alpha}} \\
%%
	D^2 \Phi \Big| & ~~=~~ -4\, F  
	&
	\ov{D^2 \Phi} \Big| & ~~=~~ -4\, \ov{F} \\
%%
	\ov{D}{}^{\dot{\alpha}} D^{\alpha} \Phi & ~~=~~ 2i\, \slashed{\p}^{\alpha\dot{\alpha}} \Phi
	&
	D^{\alpha} \ov{D}{}^{\dot{\alpha}} \ov{\Phi} & ~~=~~ 2i\, \slashed{\p}^{\alpha\dot{\alpha}} \ov{\Phi} \\
%%
	\ov{D}{}_{\dot{\alpha}} D^2 \Phi & ~~=~~ 4i\, \ov{\slashed{\p}}_{\dot{\alpha}\beta} D^\beta \Phi
	&
	D^\alpha \ov{D}{}^2 \ov{\Phi} & ~~=~~ 4i\, \slashed{\p}^{\alpha\dot{\beta}} \ov{D_{\dot{\beta}} \Phi}
	\\
%%
	\ov{D}{}_{\dot{\alpha}} D^2 \Phi \Big| & ~~=~~ 4\sqrt{2}i\, \ov{\slashed{\p}}{}_{\dot{\alpha}\beta} \psi^\beta
	&
	D^\alpha \ov{D}{}^2 \ov{\Phi} \Big| & ~~=~~ 4\sqrt{2}i\, \slashed{\p}^{\alpha\dot{\beta}} \ov{\psi}{}_{\dot{\beta}}
	\\
%%
	\ov{D}{}^2 D^2 \Phi & ~~=~~ -16\, \Box \Phi
	&
	D^2\ov{D}{}^2 \ov{\Phi} & ~~=~~ - 16\, \Box \ov{\Phi}
\end{align*}

% Logarithm of an (anti) chiral field
\begin{align*}
%%
	\ln\, \Phi(y)    & ~~=~~    \ln\, \phi  ~~+~~  \sqrt{2}\, \theta\, \psi / \phi  ~~+~~
				    \theta^2 \lgr F / \phi  ~+~  \frac{1}{2}\, \Big( \psi / \phi \Big)^2 \rgr 
	\\
%%
%	\ln\, \ov \Phi (\ov y)    & ~~=~~    \ln\, \ov\phi  ~~+~~  \sqrt{2}\, \ov{\theta\, \psi / \phi}  ~~+~~
%					     \ov\theta{}^2 \lgr \ov{F / \phi} ~+~ \frac{1}{2}\, \Big( \ov{\psi / \phi} \Big)^2 \rgr
%	\\
%%
	\ln\, \ov \Phi (\ov y)    & ~~=~~    \ln\, \ov\phi  ~~+~~  \sqrt{2}\, \ov{\theta\, \psi} / {\ov \phi}  ~~+~~
					     \ov\theta{}^2 \lgr \ov F / \ov \phi ~+~ \frac{1}{2}\, \Big( \ov \psi / \ov \phi \Big)^2 \rgr
\end{align*}


\pagebreak
\begin{center}
{\Large\bf \boldmath{$ \mc{N}=1 $} Supersymmetry Transformations}
\end{center} 

\begin{align*}
%%
   Q^\alpha & ~~=~~  \partial^\alpha +
            i \slashed{\partial}^{\alpha\dot{\alpha}}
	    \overline{\theta}_{\dot{\alpha}}   \\
%%
   \overline{Q}_{\dot{\alpha}} & ~~=~~ \overline{\partial}_{\dot{\alpha}} +
            i \ov{\slashed{\partial}}{}_{\dot{\alpha}\alpha}\theta^\alpha 
\end{align*}

\begin{align*}
%%
	\delta\phi & ~~=~~ \sqrt{2}\, \epsilon\, \psi     &  
		\delta \ov{\phi} & ~~=~~ \sqrt{2}\, \ov{\epsilon\, \psi}\\
%%
	\delta\psi_\alpha & ~~=~~ i\sqrt{2}\, 
			\ov{\epsilon^{\dot{\alpha}} \slashed{\p}}{}_{\dot{\alpha}\alpha} \phi 
			~~+~~ \sqrt{2}\, \epsilon_\alpha F 
&
		\delta \ov{\psi}{}^{\dot{\alpha}} & ~~=~~ 
			i\sqrt{2}\, \epsilon_\alpha \slashed{\p}^{\alpha\dot{\alpha}} \ov{\phi} 
			~~+~~ \sqrt{2}\, \ov{\epsilon^{\dot{\alpha}}\, F} \\
%%
	\delta F & ~~=~~ -i \sqrt{2}\, \ov{\epsilon\, \slashed{\p}} \psi  &
		\delta F & ~~=~~ -i \sqrt{2}\, \epsilon\slashed{\p} \ov{\psi}
\end{align*}


\pagebreak
\section*{\centering Formulae for Two Dimensions}
\vspace{3mm}

% Left and Right Derivatives
\begin{align*}
%%
	\p_R \!\!\! &  ~~~=~ ~\, \p_0 ~+~ i\, \p_3 
	&
	\p_L \!\!\! &  ~~~=~ ~\, \p_0 ~-~ i\, \p_3 
	\\[1mm]
%%
	\slashed\p & ~~~=~ \lgr \begin{matrix}  \p_L  &       \\
						      & \p_R 
			   \end{matrix} \rgr^{\alpha\dot\alpha}
	&
	\ov{\slashed\p} & ~~~=~ \lgr \begin{matrix}  \p_R  &       \\
                                                           & \p_L				
				\end{matrix} \rgr_{\dot\alpha\alpha}
\end{align*}

% Reduction of indices from 4-d
\begin{align*}
%%
	\xi_1 & ~~\equiv~~ \xi_R  
	&
	\ov\xi{}_1 & ~~\equiv~~ \ov\xi{}_R
	&
	\epsilon\,\xi & ~~=~~ -\, \epsilon_{[R}\, \xi_{L]}
	&
	\xi^1 & ~~=~~ -\, \xi_L
	\\[2mm]
%%
	\xi_2 & ~~\equiv~~ \xi_L
	&
	\ov\xi{}_2 & ~~\equiv~~ \ov\xi{}_L
	&
	\ov{\epsilon\, \xi} & ~~=~~ \phantom{-\,\ } 
				\ov{\epsilon_{[R}\, \xi}{}_{L]}
	&
	\xi^2 & ~~=~~ \phantom{-\,\ }
		      \xi_R
\end{align*}

% Squares of spinors and of the right- and left-handed derivatives
\begin{align*}
%%
	\theta^2    & ~~=~~    -\, 2\, \theta_R\, \theta_L 
	&
	D^2         & ~~=~~    -\, 2\, D_R\, D_L
	\\[2mm]
%%
	\ov\theta{}^2    & ~~=~~    \phantom{+\,\ } 2\, \ov{\theta_R\, \theta}{}_L
	&
	\ov D{}^2              & ~~=~~    \phantom{+\,\ } 2\, \ov{D_R\, D}{}_L
\end{align*}

% The right- and left-handed derivatives
\begin{align*}
%%
	D_R    & ~~\equiv~~    D^1
	&
	\ov D{}_R    & ~~\equiv~ \phantom{+\,\ } \ov D{}_{\dot 2} 
	~~
	~~=~~ -\, \ov{D}{}^{\dot 1}
	\\[2mm]
%%
	D_L    & ~~\equiv~~    D^2
	&
	\ov D{}_L    & ~~\equiv~    -\, \ov D{}_{\dot 1}
	~~
	~~=~~ -\, \ov{D}{}^{\dot 2}
\end{align*}

% Anti-commutators of the D-derivatives
\begin{align*}
%%
	\left\{ \ov D{}_R\, D_L \right\}    & ~~=~~    0
	&
	\left\{ \ov D{}_R\, D_R \right\}    & ~~=~~    -\, 2 i\, \p_L
	\\
%%
	\left\{ \ov D{}_L\, D_R \right\}    & ~~=~~    0
	&
	\left\{ \ov D{}_L\, D_L \right\}    & ~~=~~    -\, 2 i\, \p_R
\end{align*}

% define a slanted fraction-style derivative
\newcommand{\deriv}[2]
{
  \raisebox{1.5ex}{\scalebox{0.86}{\ensuremath{#1}}}\hspace{-0.27em}\raisebox{1ex}{\scalebox{1.2}{/}} 
	\hspace{-0.36em}\scalebox{0.86}{\ensuremath{#2}}
}

% \vartheta's are the partial derivatives
\begin{align*}
%%
	\vartheta_R    & ~~=~~    \deriv \p {\p \theta_R}
	&
	\vartheta_L    & ~~=~~    \deriv \p {\p \theta_L}
	\\
%%
	\ov\vartheta{}_R    & ~~=~~    \deriv \p {\p \ov\theta{}_R}
	&
	\ov\vartheta{}_L    & ~~=~~    \deriv \p {\p \ov\theta{}_L}
\end{align*}

% D-derivatives explicitly
\begin{align*}
%%
	D_R    & ~~=~~    \vartheta_R  ~~-~~  i\, \ov\theta{}_R\,\p_L
	&
	\ov D{}_R    & ~~=~~    \ov\vartheta{}_R  ~~-~~  i\, \theta_R\,\p_L
	\\[2mm]
%%
	D_L    & ~~=~~    \vartheta_L  ~~-~~  i\, \ov\theta{}_L\,\p_R
	&
	\ov D{}_L    & ~~=~~    \ov\vartheta{}_L  ~~-~~  i\, \theta_L\,\p_R
\end{align*}

% Fierz Identities
\begin{align*}
%%
	\sigma_\mu^{\beta\dot\beta}\, \ov\sigma_{\dot\alpha\alpha}^\mu & ~~=~~ 
					2\, \delta_\alpha^{\ \beta} \delta_{\dot\alpha}^{\ \dot\beta} \cdot
						\delta_{\alpha\dot\alpha}\,,
	&
	\mu ~=~ 0,\, 3 
	\\[2mm]
%%
	\sigma_k^{\beta\dot\beta}\, \ov\sigma_{\dot\alpha\alpha}^k & ~~=~~
					2\, \delta_\alpha^{\ \beta} \delta_{\dot\alpha}^{\ \dot\beta} \cdot
						\epsilon_{\alpha\dot\alpha}\,,
	&
	k ~=~ 1,\, 2 
	\\[1mm]
%%
	\theta_\alpha \ov\theta_{\dot\alpha} & ~~=~~ 
		\frac{1}{2}\, \theta\sigma_\mu\ov{\theta} \cdot \ov{\sigma}_{\dot\alpha\alpha}^\mu
\end{align*}


\pagebreak
\subsection*{\centering\it Chiral superfields}
% Chiral Superfields in 2-d
\begin{align*}
%%
	\Phi(y)    & ~~=~~    \phi(y)  ~~-~~  \sqrt{2}\, \theta_R \psi_L
                                       ~~+~~  \sqrt{2}\, \theta_L \psi_R  ~~-~~  2\, \theta_R\theta_L\, F
	\\[2mm]
%%
	\Phi(\ov y)    & ~~=~~    \ov\phi(\ov y)  ~~+~~  \sqrt{2}\, \ov{\theta_R \psi}{}_L
                                                  ~~-~~  \sqrt{2}\, \ov{\theta_L \psi}{}_R
                                                  ~~+~~  2\, \ov{\theta_R \theta}{}_L\, \ov{F}
\end{align*}

% Chiral variables "y"
\begin{align*}
%%
	y^0    & ~~=~~  x^0  ~+~ i \left( \ov\theta{}_R\theta_R  ~+~  \ov\theta{}_L\theta_L \right)
	&
	y^3    & ~~=~~  x^3  ~+~ \left( \ov\theta{}_R\theta_R  ~-~  \ov\theta{}_L\theta_L \right)
	\\[2mm]
%%
	\ov y{}^0    & ~~=~~   x^0  ~-~ i \left( \ov\theta{}_R\theta_R  ~+~  \ov\theta{}_L\theta_L \right)
	&
	\ov y{}^3    & ~~=~~   x^3  ~-~ \left( \ov\theta{}_R\theta_R  ~-~  \ov\theta{}_L\theta_L \right)
\end{align*}

\vspace{2mm}
\subsection*{\centering \it Twisted Chiral superfields}
\begin{align*}
%%
	\Sigma(\wt y)    & ~~=~~    \sigma(\wt y)  ~~-~~  \sqrt{2}\, \theta_R \ov\lambda{}_L
						   ~~+~~  \sqrt{2}\, \ov\theta{}_L \lambda_R
						   ~~-~~  i\, \sqrt{2}\, \theta_R \ov\theta{}_L\! \lgr iD ~+~ F_{03} \rgr
	\\[2mm]
%%
	\ov\Sigma(\ov{\wt y})    & ~~=~~    \ov\sigma(\ov{\wt y})  ~~-~~ \sqrt{2}\, \theta_L \ov\lambda{}_R
								   ~~+~~ \sqrt{2}\, \ov\theta{}_R \lambda_L
								   ~~-~~ i\, \sqrt{2}\, \theta_L \ov\theta{}_R\! \lgr iD ~-~ F_{03} \rgr
\end{align*}

% Twisted Chiral variables "\tilde y"
\begin{align*}
%%
	\wt y{}^\mu    & ~~=~~    x^\mu  ~+~  i\,\ov{\theta \wt\sigma}{}_\mu \theta
	&
	\wt \sigma{}_0    & ~~=~~\,    i\, \sigma_3
	&
	\wt \sigma{}_3    & ~~=~\,    -\, i\, \sigma_0
	\\[2mm]
%%
	\ov{\wt y}{}^\mu    & ~~=~~    x^\mu  ~+~  i\,\theta \wt\sigma{}_\mu \ov\theta
	&
	\ov{\wt \sigma}{}_0    & ~~=~~\,    i\, \ov \sigma{}_3
	&
	\ov{\wt \sigma}{}_3    & ~~=~\,    -\, i\, \ov \sigma{}_0
\end{align*}

% Twisted Chiral variables "\tilde y" in components
\begin{align*}
%%
	\wt y{}^0    & ~~=~~  x^0  ~+~ i \left( \ov\theta{}_R\theta_R  ~-~  \ov\theta{}_L\theta_L \right)
	&
	\wt y{}^3    & ~~=~~  x^3  ~+~ \left( \ov\theta{}_R\theta_R  ~+~  \ov\theta{}_L\theta_L \right)
	\\[2mm]
%%
	\ov{\wt y}{}^0    & ~~=~~   x^0  ~-~ i \left( \ov\theta{}_R\theta_R  ~-~  \ov\theta{}_L\theta_L \right)
	&
	\ov{\wt y}{}^3    & ~~=~~   x^3  ~-~ \left( \ov\theta{}_R\theta_R  ~+~  \ov\theta{}_L\theta_L \right)
\end{align*}

% Slashed derivatives contracted with twisted sigma matrices
\begin{align*}
%%
	\wt{\slashed\p} & ~~~=~ \lgr \begin{matrix}  \, \p_L  &       \\
						              & - \p_R 
				\end{matrix} \rgr^{\alpha\dot\alpha}
	&
	\ov{\slashed{\wt\p}} & ~~~=~ \lgr \begin{matrix}  - \p_R  &       \\
                                                                  & \p_L				
				     \end{matrix} \, \rgr_{\dot\alpha\alpha}
\end{align*}

% D derivatives expressed in ``\tilde y'' variables
\begin{align*}
%%
	D_R    & ~=~    \deriv \p {\p \theta_R}  ~-~  2 i \ov\theta{}_R\p_L \Big|_{\wt y}
		 ~=~    \deriv \p {\p \theta_R} \Big|_{\ov{\wt y}}
	&
	\ov D{}_R    & ~=~    \deriv \p {\p \ov\theta{}_R} \Big|_{\wt y}
		       ~=~    \deriv \p {\p \ov\theta{}_R}  ~-~  2 i \theta_R \p_L \Big|_{\ov {\wt y}}
	\\[3mm]
%%
	D_L    & ~=~    \deriv \p {\p \theta_L} \Big|_{\wt y}
		 ~=~    \deriv \p {\p \theta_L} ~-~ 2 i \ov\theta{}_L \p_R \Big|_{\ov {\wt y}}
	&
	\ov D{}_L    & ~=~    \deriv \p {\p \ov\theta{}_L} ~-~ 2 i \theta_L \p_R \Big|_{\wt y}
		       ~=~    \deriv \p {\p \ov\theta{}_L} \Big|_{\ov {\wt y}}
\end{align*}

% Integration and applying D derivatives
\begin{align*}
%%
	\int d^4\theta & ~~=~~ \frac{1}{4}\, \ov D{}_R\, D_R\, \ov D{}_L\, D_L \Big|
	&
	\int d^4\theta & ~~=~~ \frac{i}{2}\, \ov D{}_R\, D_L \cdot \frac{i}{2}\, \ov D{}_L D_R \Big|
	\\
%%
	\int d^2\wt\theta & ~~=~~ \frac{1}{2}\, \ov D{}_L\, D_R \Big|
	&
	\int d^2\ov{\wt\theta} & ~~=~~ \frac{1}{2}\, \ov D{}_R\, D_L \Big|
	\\ \\
%%
	D_R\, \sqrt{2}\, \Sigma \Big| & ~~=~~ -\, 2\, \ov\lambda{}_L
	&
	\ov D{}_R\, \sqrt{2}\, \ov\Sigma \Big| & ~~=~~ +\, 2\, \lambda_L
	\\
%%
	\ov D{}_L\, \sqrt{2}\, \Sigma \Big| & ~~=~~ +\, 2\, \lambda_R
	&
	D_L\, \sqrt{2} \ov\Sigma \Big| & ~~=~~ -\, 2\, \ov\lambda{}_R
	\\
%%
	\frac{i}{2}\, \ov D{}_L D_R\, \sqrt{2}\, \Sigma \Big| & ~~=~~ iD \,+\, F_{03}
	&
	\frac{i}{2}\, \ov D{}_R D_L\, \sqrt{2}\, \ov\Sigma \Big| & ~~=~~ iD \,-\, F_{03}
\end{align*}

% Logarithm of a twisted chiral field
\begin{align*}
%%
	\ln\, \Sigma(\wt y) & ~~=~~ \ln\,\sigma 
				~-~ \sqrt{2}\, \theta_R \ov\lambda{}_L / \sigma
				~+~ \sqrt{2}\, \ov\theta{}_L \lambda_R / \sigma
				~-~ i\, \theta_R \ov\theta{}_L \!\!
				    \lgr
					\frac{ \sqrt{2}\, \left( iD \,+\, F_{03} \right) }{ \sigma }
					\,-\,
					2\,i\, \frac{\ov\lambda{}_L}{\sigma} \frac{\lambda_R}{\sigma}
				    \rgr
	\\
%%
	\ln\, \ov\Sigma(\ov{\wt y}) & ~~=~~ \ln\,\ov\sigma 
				~-~ \sqrt{2}\, \theta_L\, \ov\lambda{}_R / {\ov \sigma}
				~+~ \sqrt{2}\, \ov\theta{}_R\, \lambda_L / {\ov \sigma}
				~-~ i\, \theta_L \ov\theta{}_R \!\!
				    \lgr
					\frac{ \sqrt{2}\, \left( iD \,-\, F_{03} \right) }{ \ov\sigma }
					\,-\,
					2\,i\, \frac{\ov \lambda{}_R}{\ov\sigma} \frac{\lambda_L}{\ov\sigma}
				    \rgr
\end{align*}


\pagebreak
\subsection*{\centering \it Twisted Chiral Superfield \boldmath{$S$}}

% Definition of S and \ov{S}
\begin{align*}
%
	S &    ~~=~~    \frac{i}{2}\, \ov D{}_R D_L\, \ln \sqrt{2}\, \ov\Sigma
	&
	\ov S &    ~~=~~    \frac{i}{2}\, \ov D{}_L D_R\, \ln \sqrt{2}\, \Sigma
\end{align*}

% Components of S and \ov{S}
\begin{align*}
	S \Big| &    ~~=~~    \frac{ \sqrt{2}\ov\sigma\, ( iD \,-\, F_{03} ) ~-~ 2\,i\, \ov\lambda{}_R \lambda_L }
						{ \big( \sqrt{2}\, \ov\sigma \big)^2 }
	\quad
	&
	\ov S \Big| &    ~~=~~    \frac{ \sqrt{2}\sigma\, ( iD \,+\, F_{03} ) ~-~ 2\,i\, \ov\lambda{}_L \lambda_R }
						{ \big( \sqrt{2}\, \sigma \big)^2 }
\end{align*}

\begin{align*}
%
	D_R\, S &    ~~=~~    \p_L\, \frac{ D_L\, \sqrt{2}\,\ov\Sigma}{\sqrt{2}\,\ov\Sigma}
	&
	D_R\, S \Big| &    ~~=~~    -\, 2\, \p_L\, \frac{\ov\lambda{}_R}{\sqrt{2}\ov\sigma}
	\\[2mm]
%
	\ov {D_R\, S} &    ~~=~~    -\, \p_L\, \frac{\ov D{}_L\, \sqrt{2}\,\Sigma}{\sqrt{2}\, \Sigma}
	&
	\ov {D_R\, S} \Big| &    ~~=~~    -\, 2\, \p_L\, \frac{\lambda_R}{\sqrt{2}\sigma}
	\\[2mm]
%
	\ov D{}_L S &    ~~=~~    -\, \p_R\, \frac{ \ov{D_R\, \sqrt{2}\, \Sigma} }{ \sqrt{2}\, \ov\Sigma }
	&
	\ov D{}_L S \Big| &    ~~=~~    -\, 2\, \p_R\, \frac{\lambda_L}{\sqrt{2}\ov\sigma}
	\\[2mm]
%
	D_L\, \ov S &    ~~=~~    \p_R\, \frac{D_R\, \sqrt{2}\, \Sigma}{\sqrt{2}\,\Sigma}
	&
	D_L\, \ov S \Big| &    ~~=~~    -\, 2\, \p_R\, \frac{\ov\lambda{}_L}{\sqrt{2}\sigma}
	\\[6mm]
%
	\frac{i}{2}\, \ov D{}_L D_R\, S &    ~~=~~    \Box\, \ln \sqrt{2}\, \ov\Sigma
	&
	\frac{i}{2}\, \ov D{}_L D_R\, S \Big| &    ~~=~~    \Box\, \ln \sqrt{2}\ov\sigma
	\\[2mm]
%
	\frac{i}{2}\, \ov D{}_R D_L \ov S &    ~~=~~    \Box\, \ln \sqrt{2}\Sigma
	&
	\frac{i}{2}\, \ov D{}_R D_L \ov S \Big| &    ~~=~~    \Box\, \ln \sqrt{2}\sigma
\end{align*}


\pagebreak
\subsection*{\centering \it Vector superfield}
% Vector superfield
\begin{align*}
%%
	V(x) & ~~=~~ -~~ A_L\, \ov\theta{}_R\theta_R  ~-~ A_R\, \ov\theta{}_L\theta_L  
                   ~~-~~ i\, \sqrt{2}\,\sigma\, \ov\theta{}_R\theta_L  
                   ~~-~~ i\, \sqrt{2}\, \ov{\sigma\, \theta}{}_L\theta_R
                   ~~-
	\\[2mm]
	& \phantom{~~=~~}
	             -~~ 2\,i\, \theta_R\theta_L\,\ov{\theta_{[R} \lambda}{}_{L]}  
                   ~~-~~  2\,i\, \ov{\theta_R\theta}{}_L\, \theta_{[R}\lambda_{L]}
	           ~~-~~ 2\,i\, \ov\theta{}_R\theta_R \ov\theta{}_L\theta_L\, D(x)
	\\ \\[2mm]
%%
% Vector superfield in ``\tilde y'' variables
	V(x) & ~~=~~ -~~ A_L\, \ov\theta{}_R\theta_R  ~-~ A_R\, \ov\theta{}_L\theta_L  
                   ~~-~~ i\, \sqrt{2}\,\sigma\, \ov\theta{}_R\theta_L  
                   ~~-~~ i\, \sqrt{2}\, \ov{\sigma\, \theta}{}_L\theta_R
                   ~~-
	\\[2mm]
	& \phantom{~~=~~}
	             -~~ 2\,i\, \theta_R\theta_L\,\ov{\theta_{[R} \lambda}{}_{L]}  
                   ~~-~~  2\,i\, \ov{\theta_R\theta}{}_L\, \theta_{[R}\lambda_{L]}
	           ~~-~~ 2\, \ov\theta{}_R\theta_R \ov\theta{}_L\theta_L\, \!\! \lgr iD \,\pm\, F_{03} \rgr_{\wt y,\, {\ov{\wt y}}}
\end{align*}

% Left and right gauge field and the definition of \sigma
\begin{equation*}
\begin{array}{lll}
%%
	A_R       ~~=~~    A_0  ~+~  i\,A_3
	&
	\qquad\quad
	\raisebox{-1.5ex}{
		\multirow{2}{*}{
		$ \sigma    ~~=~~    -\, \frac{\raisebox{0.5ex}{$\displaystyle A_1 \,+\, i A_2$}}
							{\raisebox{-1.4ex}{$ \displaystyle \sqrt 2$}} $
		}
	}
	&
	\qquad
	\raisebox{-0.8ex}{
		\multirow{2}{*}{
	 	   $ \p_{[R} A_{L]}    ~~=~~    -\, 2i F_{03} $
		}
	}
	\\[4mm]
%%
	A_L       ~~=~~    A_0  ~-~  i\,A_3
\end{array}
\end{equation*}

% Twisted Chiral superfield from V
\begin{align*}
%%
	\Sigma    & ~~=~~    \frac{i}{\sqrt 2}\, D_L\, \ov D{}_R\, V
	&
	\ov \Sigma    & ~~=~~    \frac{i}{\sqrt 2}\, D_R\, \ov D{}_L\, V
\end{align*}



\pagebreak
\section*{\centering Vortices}

\begin{equation*}
e^{i \alpha \nnbar} \cdot \hat{M} \cdot e^{- i \alpha \nnbar}    ~~=~~
\hat{M}  ~~+~~  i\, \sin \alpha \: \big[\, \nnbar,\, \hat{M} \,\big]
~~-~~  ( 1 ~-~ \cos \alpha )\, \big[\, \nnbar\, \big[\, \nnbar,\, \hat{M} \,\big] \,\big]
\end{equation*}


\vspace{2mm}
\subsection*{\centering Profiles}

\begin{align*}
%
	A_\pm^\text{U(1)}    & ~~=~~    \mp\, i\, \frac{x^\pm}{r^2}\, \frac{1}{N}\, f(r)
	\\[2mm]
%
	A_\pm^\text{SU(N)}    & ~~=~~    \mp\, i\, \frac{x^\pm}{r^2}\, f_N(r)\, \lgr \nnbar \,-\, 1/N \rgr
\end{align*}

\begin{align*}
%
	\varphi & ~~=~~
		\phi_2 ~+~ n\nbar\, \bigl( \phi_1 ~-~ \phi_2 \bigr)
	\\[2mm]
%
	& ~~=~~
		\frac{1}{N} \bigl( \phi_1 ~+~ (N-1)\phi_2 \bigr)
			~+~ \bigl( \phi_1 ~-~ \phi_2 \bigr) \lgr n\nbar ~-~ 1/N \rgr \,,
	\\[2mm]
%
	\p_r\, \big( \phi_1\, \phi_2^{N-1} \big)    & ~~=~~    \frac{f}{r} \cdot \phi_1\, \phi_2^{N-1}
	\\[2mm]
%
	\p_r\, \frac{\phi_1}{\phi_2}    & ~~=~~    \frac{f_N}{r}\, \frac{\phi_1}{\phi_2}
\end{align*}


\vspace{2mm}
\subsection*{\centering \boldmath{$ F $}-term Profiles}

\begin{align*}
%
	q^f    & ~~=~~    \lgr \phantom{+\,}\frac \phi {\sqrt 2}~~~	\phantom{+\,}\frac \phi {\sqrt 2} \phantom{\,+} \rgr
	&
	\ov q{}_f    & ~~=~~    \lgr \phantom{+\,}\frac{\phi^\dag}{\sqrt 2}~~~ \phantom{+\,}\frac{\phi^\dag}{\sqrt 2} \phantom{\,+} \rgr
	\\[2mm]
%
	q_f    & ~~=~~    \lgr \phantom{+\,}\frac \phi {\sqrt 2}~~~ -\,\frac \phi {\sqrt 2} \phantom{+} \rgr
	&
	\ov q{}^f    & ~~=~~    \lgr -\,\frac{\phi^\dag}{\sqrt 2}~~~ \phantom{+\,}\frac{\phi^\dag}{\sqrt 2} \phantom{\,+} \rgr
\end{align*}


\vspace{2mm}
\subsection*{\centering Gauge Field Normalization}

\begin{equation*}
	\hat{\mu}{}_1    ~~=~~    \sqrt{\frac{2}{N}}\, \mu_1
	\qquad\qquad
	\hat{A}{}^a    ~~=~~    \frac{1}{\sqrt{2 N}}\, A^a
	\qquad\qquad
	\hat{T}{}^a    ~~=~~    \sqrt{2N}\, T^a
\end{equation*}


\vspace{2mm}
\subsection*{\centering Slashed Vectors}

\begin{align*}
%
	\slashed{a}    & ~~=~~
				\lgr
					\begin{matrix}
						\phantom{-\, i\,}a_L	&	-\, i\, a_-	\\[3mm]
						-\, i\, a_+		&	\phantom{-\, i\,}a_R\phantom{+\,}
					\end{matrix}
				\rgr
	&
	\overline{\slashed{a}}    & ~~=~~
				\lgr
					\begin{matrix}
						\phantom{+\, i\,}a_R	&	\phantom{+\,}i\, a_-	\\[3mm]
						\phantom{+\,}i\, a_+	&	\phantom{+\, i\,}a_L\phantom{+\,}
					\end{matrix}
				\rgr
\end{align*}

\begin{align*}
%
	a_{R,L}    & ~~=~~    a_0  ~~\pm~~  i\, a_3	&
	a_\pm      & ~~=~~    a_1  ~~\pm~~  i\, a_2
\end{align*}


\end{document}