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The Kähler potential $K(\Phi ,{\Phi }^{†})$ is a real function integrated over all four fermionic coordinates, $\int d^4\theta K$. Its complex Hessian gives the scalar Kähler metric and therefore the kinetic terms. The superpotential $W(\Phi )$ is holomorphic and is integrated over chiral superspace, $\int d^2\theta W+h.c.$; its derivatives determine Yukawa couplings and the F-term scalar potential.

For the canonical Kähler potential $K={\Phi }^{†}\Phi$, extracting the ${\theta }^2{\bar{\theta }}^2$ component of the supplied chiral-superfield component expansion and integrating by parts gives the bosonic actionin the mostly-plus convention. Extracting the ${\theta }^2$ component of a holomorphic function givesso its bosonic term is $F{W}^{\prime }(\varphi )$, with the Hermitian conjugate understood in a real action. For several fields, eliminating each algebraic auxiliary field by $F_i^{\ast }=-W_i$ produces $V_F={\sum }_i|W_i{|}^2$.

There is a conflict in the printed question: with dynamical $X$ and the displayed superpotential, $F_X=\tilde{\varphi }\varphi =M$ forces $M=0$, so the $N=1$ theory has no vacuum branch parametrized by nonzero $M$. The natural intended calculation is the Kähler quotient of the $N=1$ D-flat SQED matter fields before imposing that extra F-term.

On this quotient, $|\varphi |=|\tilde{\varphi }|$ and the gauge-invariant coordinate is $M=\tilde{\varphi }\varphi$. Hence $|\varphi {|}^2=|\tilde{\varphi }{|}^2=|M|$, and the canonical Kähler potential descends toThe associated Kähler metric isIt is smooth for $M\ne 0$ and has a conical singularity at $M=0$, locally $\mathbb{C}/{\mathbb{Z}}_2$. Physically, the charged fields become massless and the U(1) gauge symmetry is restored there, so integrating out the vector multiplet to obtain a sigma-model metric ceases to be valid.