For a compact Riemann surface $\Sigma$ of genus $h\geq 1$, the Kawazumi-Zhang invariant is defined as,

$$\varphi(\Sigma) = \sum_{\ell >0}\frac{2}{\lambda_\ell} \sum_{m,n=1}^h \bigg\vert \int_\Sigma \phi_\ell \omega_m \wedge \bar \omega_n\bigg\vert^2$$

where we have $\Delta_\Sigma \phi_\ell = \lambda_\ell \phi_\ell$ and $\{\omega_1, \dots, \omega_n\}$ form an orthonormal basis of holomorphic forms on $\Sigma$ and it is stressed $\Delta_\Sigma$ is with respect to the Arakelov metric on $\Sigma$.

There are other equivalent ways of expressing the invariant, which may be more suitable for explicit computation. For hyperbolic Riemann surfaces of certain genus, it can also be directly related to the Faltings invariant. However, many rely on this notion of an Arakelov metric, and as a string theorist, I have not delved into Arakelov theory.

As such, I would greatly appreciate if someone could elucidate what the Arakelov metric is, perhaps explicitly for a particular manifold, given this seems to be the only thing from Arakelov theory I need to be able to compute $\varphi(\Sigma)$.

For those curious, the motivation is that the integration of $\varphi(\Sigma)$ over the moduli space of Riemann surfaces of genus $h= 2$ arises in the evaluation of an amplitude in type II string theory.