Lagrangian for $d=1$ $\mathcal{N}=4$ SUSY model on a $n$-complex dimensional Kahlerian target space is given as (see p.213, eqn. (10.251) in the Mirror Symmetry book (pdf)) $$\begin{equation} L= g_{i\bar{j}} \dot{\phi}^i \dot{\bar{\phi}}^{\bar{j}} + i g_{i\bar{j}} \bar{\psi}^{\bar{j}} D_t \psi^i + ig_{i\bar{j}} \bar{\psi}^i D_t \psi^{\bar{j}} + R_{i\bar{j}k\bar{l}} \bar{\psi}^i \psi^k \psi^{\bar{j}} \bar{\psi}^{\bar{l}} \end{equation},\tag{10.251}$$ which suggests that we have $n$ complex bosons : $\phi^i$, $\bar{\phi}^{\bar{i}}$, but $2n$ complex fermions : $\psi^i$, $\bar{\psi}^i$ , $\bar{\psi}^i$, $\bar{\psi}^{\bar{i}}$. I wonder:
- Why do we have fermions with both holomorphic and anti-holomorphic indices, i.e. what is the relation between ${\psi}^i$ and $\psi^{\bar{i}}$ terms? Are they two completely independent fermions, or do they depend on each other (as the notation suggests)?
- Does not this violate supersymmetry because we have than $2n$ fermionic dofs but only $n$ bosonic dofs? Do we have some additional auxiliary field accounting for this?
This is not an issue in the case of a Riemannian manifold where we only have $\psi^i$ and $\bar{\psi}^i$ (e.g : 2,3).
