Quick notes on Supersymmetric Quantum Mechanics II

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Getting to sigma models 

Consider $\phi(x^i,t)$

      Let $\phi(t)$ be  $0 + 1D$ QFT

      Let $\mathcal{L} = \frac{1}{2} \frac{d \phi(t)}{dt} \frac{d \phi(t)}{dt}$


       $\phi: \mathcal{R} \rightarrow \mathcal{R}$

        $\phi$: base space $\rightarrow $ target space 


      Consider now $\mathcal{L} = \frac{1}{2} g_{ij} \frac{d \phi^i}{dt} \frac{d \phi^j}{dt}$

               $\phi: \mathcal{R} \rightarrow \mathcal{M}$

      $\mathcal{L} = \frac{1}{2} g_{ij} \frac{d \phi^i}{dt} \frac{d \phi^j}{dt}$

         Setting $g_{ij}$ to $\delta_{ij}$

         $\mathcal{L} = \frac{1}{2} \dot{\phi_1}^2 +  . . . . . \frac{1}{2} \dot{\phi_n}^2$

       $\phi$ is a scalar

     

      Let  Dirac spinors be $\psi^i$

             $\mathcal{L_f} = g_{ij}\frac{1}{2} {\psi^i}_\alpha  (\gamma^{\mu}  \partial_\mu)_{\alpha \beta} {\psi^j} _{\beta}$

     where $\alpha$ is the spinor index


        $\mathcal{L_f} = \frac{i}{2} \delta_{ij}  \psi^i_\alpha \partial_t \psi^j_\alpha$

        $\mathcal{L_f} = \frac{i}{2}   \psi^i_\alpha \dot{\psi^j_\alpha}$


$\mathcal{L_{susy}} = \mathcal{L_{b}}+ \mathcal{L_{f}}$

$\mathcal{L_{susy}} =  \frac{1}{2} \dot{\phi_1}^2 +  . . . . . \frac{1}{2} \dot{\phi_n}^2+ \frac{i}{2}   \psi^i_\alpha \dot{\psi^j_\alpha}$


  $\delta \phi^i = i \epsilon \psi$      and     $\delta \psi^i =  - \epsilon \dot{\phi}^i$

 

    1 epsilon so 1 supercharge. Finding Noether charge.


   $\epsilon Q =  \epsilon \sqrt{2} \psi^i \dot{\phi}_i$





         

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