I'm learning some conformal field theory.
I'm trying to use the formula $$ \partial_{\mu} \epsilon_\nu+\partial_\nu \epsilon_\mu=\frac{2}{d}(\partial \epsilon) \eta_{\mu \nu} $$ to derive the equation $$ \bigg( \eta_{\mu \nu} \square +(d-2)\partial_{\mu}\partial_{\nu} \bigg)(\partial \epsilon)=0 $$ where $$ \square=\eta^{\mu \nu} \partial_{\mu} \partial_{\nu} $$ And the dimension $ d $ can be any positive integer (e.g. $ d=p+q $ for $ \mathbb{R}^{p+q} $).
What is the best way to do this? And why do we expect the first equation to imply the second?
The reference I'm following is a bit physics-y so $ \epsilon $ is supposed to be an infinitesimal perturbation $$ x^\mu \mapsto x^\mu +\epsilon^\mu(x) $$ affecting the metric by $$ \eta_{\mu \nu}'= \eta_{\mu \nu } + (\partial_\mu \epsilon_\nu+\partial_{\nu}\epsilon_\mu )+\mathcal{O}(\epsilon^2) $$ which, since the perturbation is supposed to conformal, should imply that the factor $$ \partial_\mu \epsilon_\nu+\partial_{\nu}\epsilon_\mu $$ is proportional to the metric $ \eta_{\mu \nu} $. $ \partial \epsilon $ is the total divergence.
It's just algebra. I believe you are faked out by the double-duty use of the index ν, which I'll avoid.
You started from the relation $$ \partial_{\mu} \epsilon_\nu+\partial_\nu \epsilon_\mu= K \eta_{\mu \nu}.\tag{0} $$ Operate on it by $\eta^{\mu\nu}$ to determine $2\partial \epsilon= dK$, hence $$ \partial_{\mu} \epsilon_\nu+\partial_\nu \epsilon_\mu=\frac{2}{d}(\partial \epsilon) \eta_{\mu \nu} ~.\tag{1} $$ Next, operate on this by $\partial^\nu$ to obtain $$ \partial_{\mu} \partial\epsilon+ \square \epsilon_\mu=\frac{2}{d} \partial_\mu (\partial \epsilon) ~.\tag{2} $$
Now, operate on this by $\partial_\kappa$, symmetrize (μ,κ), and utilize (1) in the last step, inside the d'Alembertian, multiplying everything by d, $$ \partial_{\kappa}\partial_{\mu} \partial\epsilon+ \square \partial_{(\kappa}\epsilon_{\mu )}-\frac{2}{d} \partial_\mu \partial_{\kappa}(\partial \epsilon)=0 \qquad \leadsto \\ \bigg( \eta_{\mu \kappa} \square +(d-2)\partial_{\mu}\partial_{\kappa} \bigg)(\partial \epsilon)=0~,\tag{3} $$ your target expression. You had to use (1) to supplant the symmetric gradient tensor by the divergence multiplying the metric. One always does.