Semiclassical Limits of Strongly Parabolic Higgs Bundles and Hyperpolygon Spaces

We investigate the Hitchin hyperkähler metric on the moduli space of strongly parabolic $\mathfrak{s}\mathfrak{l}(2,\text{C})$-Higgs bundles on the $n$-punctured Riemann sphere and its degeneration obtained by scaling the parabolic weights $t\alpha$ as $t\to 0$. 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 hyperkähler metric on the hyperpolygon space ${\mathcal{X}}_{\alpha }$, which thus serves as the natural finite-dimensional model for the degeneration of the infinite-dimensional hyperkähler 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.