Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 307 / 2

Write the scalar components as ${\varphi }_i,{\tilde{\varphi }}_i,x$. The superpotential givesChoosing the sign of the Fayet–Iliopoulos term so that positive $\zeta$ favors the positively charged fields, the full Supersymmetric quantum electrodynamics potential isChanging the FI sign convention exchanges $\varphi$ with $\tilde{\varphi }$ in the descriptions below.

Let $m_i=0$ and first take $\zeta >0$. The mass terms and D-flatness require $x=0$ and ${\sum }_i|{\varphi }_i{|}^2-{\sum }_i|{\tilde{\varphi }}_i{|}^2=\zeta$, while F-flatness adds ${\sum }_i{\tilde{\varphi }}_i{\varphi }_i=0$. Dividing by the U(1) gauge action gives the Higgs branchwith complex dimension $2N-2$. For $\zeta <0$, the same result follows after exchanging $\varphi$ and $\tilde{\varphi }$.

Suppose $\zeta >0$ and all $m_i$ are distinct. A nonzero ${\varphi }_j$ forces $x=m_j$. Distinctness then makes every other charged field vanish, and ${\sum }_i{\tilde{\varphi }}_i{\varphi }_i=0$ forces ${\tilde{\varphi }}_j=0$. The D-flatness equation fixes $|{\varphi }_j{|}^2=\zeta$, while its phase is removed by the gauge group. There is therefore one isolated supersymmetric vacuum for each flavor,for a total of $N$ vacua. Negative $\zeta$ gives the analogous vacua with ${\tilde{\varphi }}_j$ nonzero.

For $\zeta =0$ and distinct masses, any nonzero charged field can occur only at one $x=m_j$. At such a point, D-flatness gives $|{\varphi }_j|=|{\tilde{\varphi }}_j|$, whereas F-flatness gives ${\tilde{\varphi }}_j{\varphi }_j=0$; together they force both fields to vanish. Thus only the Coulomb branch remains,of complex dimension one. The points $x=m_i$ are special because the corresponding charged multiplets become massless there.

When every $m_i$ and $\zeta$ vanishes, the Coulomb branch again has ${\varphi }_i={\tilde{\varphi }}_i=0$ and arbitrary $x$. A Higgs branch also occurs at $x=0$: it is the zero-level quotient ofby U(1), and has complex dimension $2N-2$ for $N>1$. The two branches meet at the origin. For $N=1$, the F- and D-flat equations force both charged fields to vanish, so the Higgs branch collapses to that intersection point.

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.