δ h^{ab} = - 2 ω h^{ab}

δ_{ω} S = \frac{1}{2 π} \int d^{2} σ ∣ h ∣ ω h^{ab} T_{ab} .

The insertions are Weyl invariant by assumption, so varying the normalized path integral gives

δ_{ω} ⟨ ϕ_{1} \dots ϕ_{n} ⟩ = \frac{i}{2 π} \int d^{2} σ ∣ h ∣ ω ⟨ h^{ab} T_{ab} ϕ_{1} \dots ϕ_{n} ⟩_{conn} .

Consequently the stated condition

⟨ h^{ab} T_{ab} ϕ_{1} \dots ϕ_{n} ⟩ = 0

makes the correlation function Weyl invariant. Quantum failure of this condition is the worldsheet Weyl anomaly.

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