There is a conflict in the printed question: with dynamical $X$ and the displayed <superpotential>, $F_X=\widetilde\phi\phi=M$ 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, $|\phi|=|\widetilde\phi|$ and the gauge-invariant coordinate is $M=\widetilde\phi\phi$. Hence $|\phi|^2=|\widetilde\phi|^2=|M|$, and the canonical <Kähler potential> descends to
$$
\boxed{K(M,\overline M)=2\sqrt{M\overline M}=2|M|}.
$$
The associated <Kähler metric> is
$$
\boxed{g_{M\overline M}=\partial_M\partial_{\overline M}K
=\frac1{2|M|}}.
$$
It is smooth for $M\ne0$ 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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 307 / 2 / N = 1 quotient metric / Solution

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A Kähler quotient imposes a moment-map level condition and divides by the compact symmetry group. Under suitable stability assumptions it is equivalent to a quotient by the complexified group and inherits a Kähler metric away from singular orbits.


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 Incoming links: Kähler quotient