Semiclassical Limits of Strongly Parabolic Higgs Bundles and Hyperpolygon Spaces

We investigate the Hitchin hyperk\"ahler metric on the moduli space of strongly parabolic $\mathfrak{sl}(2,\C)$-Higgs bundles on the $n$-punctured Riemann sphere and its degeneration obtained by scaling the parabolic weights $t\alpha$ as $t\to0$. Using the parabolic Deligne--Hitchin moduli space, we show that twistor lines of hyperpolygon spaces arise as limiting initial data for twistor lines at small weights, and we construct the corresponding real-analytic families of $\lambda$-connections. On suitably shrinking regions of the moduli space, the rescaled Hitchin metric converges, in the semiclassical limit, to the hyperk\"ahler metric on the hyperpolygon space $\mathcal X_\alpha$, which thus serves as the natural finite-dimensional model for the degeneration of the infinite-dimensional hyperk\"ahler reduction. Moreover, higher-order corrections of the Hitchin metric in this semiclassical regime can be expressed explicitly in terms of iterated integrals of logarithmic differentials on the punctured sphere.