The adjoint fermion is a flavour singlet, so
$$
\mathcal A[SU(N_f)_L^3]=2,
\qquad
\mathcal A[SU(N_f)_L^2U(1)_X]=-2.
$$
There are $4N_f$ fundamental Weyl components of charge $-1$ and three adjoint components of charge $q=N_f/2$. Hence
$$
\boxed{\mathcal A[U(1)_X^3]=-4N_f+\frac{3N_f^3}{8}},
\qquad
\boxed{\mathcal A[U(1)_X\text{-gravity}]=-4N_f+\frac{3N_f}{2}=-\frac{5N_f}{2}}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 305 / 3 / v / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-305/3/v/solution