 Perturbative gauge-coupling independence of the Wilsonian superpotential

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 $\mathrm{Re}\tau$ 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.