Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 304 / 4 / b

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- Solution b

Solution

The product of generators satisfies

T_{a} T_{b} = \frac{1}{2 N} δ_{ab} I + \frac{1}{2} (d_{ab}^{c} + i f_{ab}^{c}) T_{c},

and therefore

Tr (T_{a} T_{b} T_{c}) = \frac{1}{4} (d_{ab c} + i f_{ab c}) .

Because

ϕ^{a} ϕ^{b} ϕ^{c}

is symmetric, its contraction with the antisymmetric structure constant of a Lie algebra

f_{ab c}

vanishes. Hence

- \frac{1}{3 !} Tr (ϕ^{3}) = - \frac{1}{24} d_{ab c} ϕ^{a} ϕ^{b} ϕ^{c} .

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