Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 304 / 4 / b

Table of contents
- Solution b

Solution

The gauge-invariant Yang-Mills and matter part

L_{0}

is BRST invariant because a BRST variation is a gauge transformation with ghost-valued parameter. For the gauge-fixing fermion

Ψ = \overset{c}{ˉ}^{a} (G^{a} (A) - \frac{ξ}{2} B^{a}),

the graded Leibniz rule gives

Q Ψ = B^{a} G^{a} - \frac{ξ}{2} B^{a} B^{a} - \overset{c}{ˉ}^{a} \frac{δ G ^{a}}{δ A _{μ}^{b}} D_{μ}^{b c} c^{c},

up to the common sign convention used to define the ghost term. This is precisely the gauge-fixing, auxiliary-field, and ghost sector of the displayed Lagrangian. Therefore

L = L_{0} + Q Ψ, Q L = Q L_{0} + Q^{2} Ψ = 0.

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