Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 307 / 1

For the stated two-dimensional N=(2,2) supersymmetry conventions, define the supersymmetric covariant derivativesThe terms in which a Grassmann derivative hits the explicit Grassmann coordinate cancel the spacetime-derivative terms, while derivatives involving different signs act on independent coordinates. Therefore

The twisted chiral superfield constraints $D_{-}U={\bar{D}}_{+}U=0$ are solved by the twisted chiral coordinatesThe superfield depends only on $({\tilde{y}}^{\pm };{\theta }^{+},{\bar{\theta }}^{-})$ and has the finite Grassmann variable expansionwhere every component on the right is evaluated at $\tilde{y}$; numerical $\sqrt{2}$ factors may be absorbed into the component definitions.

For one chiral superfield $\Phi$ and one twisted chiral superfield $U$, the most general local two-derivative supersymmetric action isHere $K$ is real, $W$ is a holomorphic superpotential, and $\tilde{W}$ is a holomorphic twisted superpotential. A full superspace integral, a chiral F-term, and a twisted F-term each vary by a spacetime or Berezin total derivative, so all three are supersymmetric.

Choose the Vector R-symmetry and Axial R-symmetry conventionswith conjugate coordinates transforming oppositely, and assign compatible charges to $\Phi$ and $U$. The D-term is invariant when $K$ is neutral, up to a generalized Kähler transformation. The chiral measure has vector R-charge $-2$ and axial charge zero, whereas the twisted chiral measure has axial R-charge $-2$ and vector charge zero. Hence classical invariance requirestogether with neutrality of $K$. Equivalently, $W$ and $\tilde{W}$ must be quasi-homogeneous with these charges; absent suitable charge assignments, the corresponding superpotential breaks that R-symmetry.