Hyperkähler Degenerations from Parabolic $$\text {SL}(2,\mathbb {C})$$-Higgs Bundle Moduli Spaces on the Punctured Sphere to Hyperpolygon Spaces

Abstract Complete hyperkähler 4-manifolds of finite energy are grouped into ALE, ALF, $$\hbox {ALG}^{(*)}$$ ALG ( ∗ ) , $$\hbox {ALH}^{(*)}$$ ALH ( ∗ ) , each of these being further classified according to the Dynkin type of their noncompact end. A family of ALG- $$D_4$$ D 4 spaces are modeled by certain moduli spaces of strongly parabolic $$\textrm{SL}(2,\mathbb {C})$$ SL ( 2 , C ) -Higgs bundles on the Riemann sphere with $$n=4$$ n = 4 punctures. Meanwhile, a family of ALE- $$D_4$$ D 4 spaces are modeled by certain Nakajima quiver varieties known as $$n=4$$ n = 4 hyperpolygon spaces. There is a map from hyperpolygon space to the moduli space of strong parabolic $$\textrm{SL}(2,\mathbb {C})$$ SL ( 2 , C ) -Higgs bundles that is a diffeomorphism onto its open and dense image. We show that under a fine-tuned degenerate limit, the pullback of a family of ALG- $$D_4$$ D 4 metrics parameterized by R converges pointwise to the ALE- $$D_4$$ D 4 metric as $$R \rightarrow 0$$ R → 0 . While the connection to gravitational instantons occurs in the $$n=4$$ n = 4 case, we prove our result for any finite n .