Abstract: I will discuss the small-weight degeneration of the Hitchin hyperkähler metric for strongly parabolic $\mathfrak{sl}_2(\mathbb C)$-Higgs bundles on the n-punctured sphere. Scaling the parabolic weights to $t\alpha$ and letting $t \to$ 0, we use the parabolic Deligne–Hitchin moduli space to relate twistor lines for the small-weight Higgs-bundle moduli spaces to twistor lines of the associated hyperpolygon space.
After natural identifications with bounded regions in the hyperpolygon space, the rescaled Hitchin metrics converge uniformly on compact subsets, in fact real analytically, to the hyperpolygon hyperkähler metric. This identifies the hyperpolygon space as the finite-dimensional model for the semiclassical degeneration. The talk is based on recent joint work with L. Heller and C. Meneses.