= Loewner variation of the Dirichlet Green function
{c}
{title2=$\partial_tG_{D_t}(z,w)=-4\operatorname{Im}(1/f_t(z))\operatorname{Im}(1/f_t(w))$}
With <half-plane capacity> $2t$, $f_t=g_t-W_t$ and $G_{\mathbb H}(z,w)=\log|(z-\overline w)/(z-w)|$, differentiating $G_{D_t}(z,w)=G_{\mathbb H}(f_t(z),f_t(w))$ gives the displayed identity. The translation by the driver cancels in the Green function. This rank-one <covariance> loss balances the quadratic covariation of the harmonic mean in the <SLE4 coupling with a Gaussian free field>.
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