Geometric origin of the Kahler potential for SUSY $\sigma$-models

Question

I was reading Cecotti's book on Supersymmetric Field Theories, and there is a statement that confuses me. It is proven that if one considers a $\sigma$-model $\phi^i:~\Sigma\to\mathcal{M}$ with Lagrangian $$ \mathcal{L}=\frac{1}{2}g_{ij}\partial_\mu\phi^i\partial^\mu\phi^j+\frac{i}{2}h_{ij}\bar{\psi}^i\overline{\sigma}^\mu\partial_\mu\psi^j, $$ if we have $\mathcal{N}=1$, then $g=h$, and moreover, if one wants $\mathcal{N}>1$ supersymmetry, the target manifold must admit $\mathcal{N}-1$ parallel structures $I^a_{ij}$ satisfying the Clifford algebra $I^aI^b+I^bI^a=-2\delta^{ab}$, with $a=2,...,\mathcal{N}$ (a full explicit proof can also be found in Bagger, Supersymmetric Sigma Models, 1984).

In particular it means that for $\mathcal{N}=2,\, \mathcal{M}$ admits a complex structure $I^2=-1$, and a corollary is that $\mathcal{N}=2$ supersymmetric target manifolds must be Kähler.

This is confusing to me, because I would have thought that target manifolds of $\mathcal{N}=1$ must be Kähler. One of the first result in any SUSY course is that the most general SUSY kinetic term (for scalar multiplets) is $$ \mathcal{L}_\text{kin}=\frac{1}{2}K_{i\bar{\jmath}}\partial_\mu\phi^i\partial^\mu\bar{\phi}^\bar{\jmath}\qquad K_{i\bar{\jmath}}=\partial_i\overline{\partial}_\bar{\jmath}K $$ For some Kähler potential $K$.

Is the difference that in the second case, I assume already that I have complex scalars? In that case does that mean that despite the metric descending from a Kähler potential, the target space is not Kähler? If the target space of $\mathcal{N}=1$ is indeed Kähler, what is the complex structure?

Answer 1

It depends on the dimensionality of the space one works with. In 2d, the kinetic term that you presented appears after integration over 4 Grassmann coordinates in the expression $\int \mathrm{d}^2 \sigma \ \mathrm{d}^4 \theta K(\Phi, \overline{\Phi})$ which means $\cal{N}=2$. Perhaps it is the reason of your confusion?