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

Table of contents
- Solution iv

Solution

The composite

χ^{ij} = ψ^{(i} (λ ψ^{j)})

is a flavour-symmetric left-handed Weyl fermion. Its

U (1)

charge is

q_{χ} = q_{λ} + 2 q_{ψ} = (N - 4) - 2 (N - 2) = - N .

Because

p = N - 4

, the supplied symmetric-representation data give

A (Sym^{2} p) = p + 4 = N, I (Sym^{2} p) = p + 2 = N - 2.

Thus

A_{S U (p)^{3}} = N, A_{S U (p)^{2} U (1)} = (N - 2) (- N) = - N (N - 2) .

There are

p (p + 1) /2 = (N - 4) (N - 3) /2

components, so

A_{U (1)^{3}} = \frac{p ( p + 1 )}{2} (- N)^{3}, A_{U (1) -grav^{2}} = \frac{p ( p + 1 )}{2} (- N),

which exactly equal the ultraviolet values. The proposed confined spectrum therefore satisfies 't Hooft anomaly matching.

Solved by gpt-5.6-sol high.

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