An internal generator $T_a$ is a Lorentz scalar. The <Coleman–Mandula theorem> and the graded extension allow it to act nontrivially on supercharges only as an R-symmetry. Since R-symmetries are excluded,
$$
\boxed{[T_a,Q_\alpha]=[T_a,\bar Q_{\dot\alpha}]=0}.
$$

Translation invariance of a conserved global supercharge and the graded Jacobi identities give
$$
\boxed{[P_\mu,Q_\alpha]=[P_\mu,\bar Q_{\dot\alpha}]=0}.
$$
Lorentz covariance requires the supercharges to transform as Weyl spinors,
$$
\boxed{[M^{\mu\nu},Q_\alpha]=(\sigma^{\mu\nu})_\alpha{}^\beta Q_\beta},
$$
with the complex-conjugate dotted-spinor relation for $\bar Q_{\dot\alpha}$, up to the sign convention used for the action of generators.

The anticommutator $\{Q_\alpha,\bar Q_{\dot\beta}\}$ transforms as a Lorentz vector because $(1/2,0)\otimes(0,1/2)=(1/2,1/2)$. The only translation generator with that transformation law is $P_\mu$, and a normalization of $Q$ fixes
$$
\boxed{\{Q_\alpha,\bar Q_{\dot\beta}\}
=2\sigma^\mu_{\alpha\dot\beta}P_\mu}.
$$
An equal-chirality anticommutator could only contain an antisymmetric spinor contraction times a central charge, but the anticommutator is symmetric under exchange of the complete supercharges. For one supercharge and no central extension this forces
$$
\boxed{\{Q_\alpha,Q_\beta\}=0,
\qquad
\{\bar Q_{\dot\alpha},\bar Q_{\dot\beta}\}=0}.
$$
Together with the stated Poincare brackets, these are the four-dimensional $\mathcal N=1$ Super-Poincare relations.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 307 / 1 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-307/1/b/solution