Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 307 / 3 / ii

Table of contents
- Solution ii

Solution

Classically the

p

identical fields

ψ^{i}

have

U (p) = S U (p) \times U (1)_{ψ}

, and

λ

has an independent

U (1)_{λ}

. One linear combination has an

S U (N)^{2} U (1)

anomaly. For the orthogonal combination choose

q_{λ} = p = N - 4, q_{ψ} = - (N - 2) .

Indeed,

I (antisym) q_{λ} + p I (\overline{□}) q_{ψ} = (N - 2) p - p (N - 2) = 0.

The continuous quantum global symmetry is therefore

S U (p) \times U (1),

up to discrete identifications. The anomalous orthogonal axial rotation does not survive as a continuous quantum symmetry.

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