Let $Q=(I,\Omega)$ be the $D_4$ affine quiver. We choose as dimension vector $(2,1,1,1,1)$ (where $2$ is on the central vertex). As this dimension vector is indivisible, we can choose a generic $\theta \in \mathbb{Z}^I$. We have the usual moment map $$\mu: R(\overline{Q},v) \to \mathbb{C}^I $$ from representations of the doubled quiver to $\mathbb{C}^I$.
We have different quiver varieties associated to these data (I do not know whether the notation is standard): $$\mathfrak{M}_{0,\theta}(v)=\mu^{-1}(0)//_{\theta}G(v)$$ and $$\mathfrak{M}_{\theta,\theta}(v)=\mu^{-1}(\theta)/G(v) $$
A computation shows that these varieties are actually of dimension $2$: is there an explicit geometric presentation of such objects? If we choose $\theta$ in an appropriate way, we can think of $\mathfrak{M}_{0,\theta}(v),\mathfrak{M}_{\theta,\theta}(v)$ as certain moduli spaces of parabolic Higgs bundles/parabolic connections over trivial vector bundle over $\mathbb{P}^1$.
I know that in this case we have a concrete description of the full moduli space (the so-called Hausel toy model). Also the character variety side is well understood as affine Del Pezzo surfaces. What about quiver side?
Best Answer
It is Kronhimer's result that $\mathfrak M_{\zeta_{\mathbb R},\zeta_{\mathbb C}}(\mathbf v)$ is $\mathbb C^2/\Gamma$ ($\zeta_{\mathbb R}=\zeta_{\mathbb C}=0$), its deformation ($\zeta_{\mathbb R}=0$), and the minimal resolution of the deformation (in general). Here $\Gamma$ is the binary dihedral group of type D4.