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.
The Arakelov metric is given by the canonical 1-1 form $\mu_{\Sigma}=\frac{i}{2h}\sum_{i=1}^h \omega_i\wedge \overline{\omega_i}$ for an ON basis $\omega_1,\dots,\omega_h$ of 1-forms of $\Sigma$. It defines a Greeen function $G_{\Sigma}\colon \Sigma^2\to \mathbb{R}_{\ge0}$, which is unique by
(i) $G^2$ is $C^\infty$ on $\Sigma^2$
(ii) $\partial_x \overline{\partial}_x \log G(x,y)^2=2\pi i (\mu_{\Sigma}(x)-\delta_y(x))$
(iii) $\int_{\Sigma} \log G(x,y)\mu_{\Sigma}(x)=0$
This function $G$ defines a metric on the bundle $\mathscr{O}_{\Sigma^2}(\Delta)$ of the diagonal divisor $\Delta\subseteq\Sigma^2$. Denote $h_{\Delta}=\mathrm{curv}(\mathscr{O}_{\Sigma^2}(\Delta))$ for the curvature of this metrized bundle. That means we have $\partial_{\Sigma^2}\overline{\partial}_{\Sigma^2} \log G(x,y)^2=2\pi i (h_{\Delta}(x,y)-\delta_{\Delta}(x,y))$. Then the Zhang-Kawazumi invariant can be given by $$\varphi(\Sigma)=\int_{\Sigma^2}(\log G) h_{\Delta}^2.$$
Let me remark that the Zhang-Kawazumi invariant can be related to the Faltings invariant for any compact Riemann surface of any positive genus $h$ by the equality $$\delta_{Fal}(\Sigma)-2\varphi(\Sigma)=-24\int_{\mathrm{Pic}_{h-1}(\Sigma)}\log\|\theta\|\frac{\nu^h}{h!}-8h\log 2\pi,$$ where $\|\theta\|$ is the norm of the Riemann theta function and $\nu$ is the Kähler form of the Hodge metric on $\mathrm{Pic}_{h-1}(\Sigma)$, see my paper http://link.springer.com/article/10.1007/s00222-016-0713-1, where you can also find other formulas for $\varphi(\Sigma)$ only in terms of integrals of $\|\theta\|$.