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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2026

Paper 305

The composite $\chi^{ij}=\psi^{(i}(\lambda\psi^{j)})$ is a flavour-symmetric left-handed Weyl fermion. Its $U(1)$ charge is
$$
q_\chi=q_\lambda+2q_\psi=(N-4)-2(N-2)=-N.
$$
Because $p=N-4$, the supplied symmetric-representation data give
$$
A(\operatorname{Sym}^2\mathbf p)=p+4=N,
\qquad
I(\operatorname{Sym}^2\mathbf p)=p+2=N-2.
$$
Thus
$$
\mathcal A_{SU(p)^3}=N,
\qquad
\mathcal A_{SU(p)^2U(1)}=(N-2)(-N)=-N(N-2).
$$
There are $p(p+1)/2=(N-4)(N-3)/2$ components, so
$$
\mathcal A_{U(1)^3}=\frac{p(p+1)}2(-N)^3,
\qquad
\mathcal A_{U(1)\text{-grav}^2}=\frac{p(p+1)}2(-N),
$$
which exactly equal the ultraviolet values. The proposed confined spectrum therefore satisfies <'t Hooft anomaly matching>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 305 / 3 / iv / Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 305 / 3 / iv / Solution

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Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 305 / 3 / iv / Solution

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