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?



Quiver varieties associated to D_4
It is [Kronhimer's result](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-29/issue-3/The-construction-of-ALE-spaces-as-hyper-K%C3%A4hler-quotients/10.4310/jdg/1214443066.full) 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.


