Semiclassical Limits of Strongly Parabolic Higgs Bundles and Hyperpolygon Spaces

TL;DR

This paper studies the degeneration of the Hitchin hyperk"ahler metric on moduli spaces of strongly parabolic Higgs bundles, showing it converges to hyperpolygon space metrics and analyzing higher-order corrections.

Contribution

It demonstrates the semiclassical limit of the Hitchin metric converges to hyperpolygon space metrics and explicitly describes higher-order corrections using iterated integrals.

Findings

Rescaled Hitchin metric converges to hyperpolygon space metric in the semiclassical limit.

Twistor lines of hyperpolygon spaces arise as limits of twistor lines at small weights.

Higher-order corrections are expressed via iterated integrals of logarithmic differentials.

Abstract

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.