= Perturbative gauge-coupling independence of the Wilsonian superpotential
{title2=$\partial_\tau W_{\mathrm{pert}}=0$}
The holomorphic coupling is $\tau=\vartheta/(2\pi)+4\pi i/g^2$. Perturbative coefficients are independent of the topological angle $\vartheta$. A <holomorphic function> invariant under continuous shifts of $\operatorname{Re}\tau$ is constant by the <Cauchy-Riemann equations>, hence a perturbative local <superpotential> coefficient cannot depend on $\tau$. Terms in $e^{2\pi i\tau}$ can evade this argument nonperturbatively. This does not forbid renormalization of the separate gauge kinetic <F-term>, or <wave-function renormalization> from the <Kähler potential>.
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