In https://arxiv.org/abs/1402.6334 on page 16 in their Eq. (5.21), they go from the equation $$S_G+S_\chi\approx\int d^2x\sqrt{-g}XR+\int dt\sqrt{-\gamma}XK$$ to the equation $$=\int d^2x \sqrt{-g}\partial^\mu X(g_{\mu\rho}\partial_\sigma-g_{\sigma\rho}\partial_\mu)g^{\sigma\rho},$$ where $X$ is a field, and the boundary is at fixed spatial coordinate $z$. I understand that this involves partial integration to go from the integral over just $t$ to that over $d^2x$, but I was confused about the exact steps, and which expressions to use for $K$ and $R$ to start. Does anyone understand what the steps here would be?
