Hyperkähler Degenerations from Parabolic SL(2,C)-Higgs Bundle Moduli Spaces on the Punctured Sphere to Hyperpolygon Spaces

Complete hyperkähler 4-manifolds of finite energy are grouped into ALE, ALF, ALG(*) ALG ( ∗ ) , ALH(*) ALH ( ∗ ) , each of these being further classified according to the Dynkin type of their noncompact end. A family of ALG- D₄ D 4 spaces are modeled by certain moduli spaces of strongly parabolic SL(2,C) SL ( 2 , C ) -Higgs bundles on the Riemann sphere with $$n=4$$ n = 4 punctures. Meanwhile, a family of ALE- D₄ 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 SL(2,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- D₄ D 4 metrics parameterized by R converges pointwise to the ALE- D₄ D 4 metric as R → 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 .