论文标题
抛物线寄生虫$ sl的模量空间的渐近几何（2，\ mathbb {c}）$ - higgs捆绑
Asymptotic Geometry of the Moduli Space of Parabolic $SL(2,\mathbb{C})$-Higgs Bundles
论文作者
论文摘要
给定一个通用稳定的强烈抛物线$ sl（2，\ mathbb {c}）$ - higgs捆绑$（\ mathcal {e}，φ）$，我们描述了Harmonic Metrics $ h_t $的家族，用于Higgs buggs bundles bun of Higgs bundles $（\ nathcal inter $ trical os $ tlaly $ tly $ t \ gg，构建近似解决方案的家族$ h_t^{\ mathrm {app}} $。然后，我们通过将其与更简单的“ Semi-Flat”HyperKählerMetric进行比较，描述了$ \ Mathcal {M} $上的天然Hyperkähler指标。我们证明$ g_ {l^2} - g _ {\ mathrm {sf}} = o（\ mathrm {e}^{ - γt}）沿通用射线，证明了Gaiotto-Moore-Moore-Moore-Neitzke的推测。 我们的结果扩展到弱抛物线$ SL（2，\ Mathbb {C}）$ - Higgs捆绑包。 在四函数球体的情况下，我们更明确地描述了模量空间和度量。在这种情况下，我们证明了Hyperkähler指标是ALG，并表明指数衰减的速率是猜想的最佳选择，$γ= 4L $，其中$ l $是基本曲线最短地理曲线的最短长度，以单数平面$ | | \ | | \ \ | \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ c. flose | $ | $。
Given a generic stable strongly parabolic $SL(2,\mathbb{C})$-Higgs bundle $(\mathcal{E}, φ)$, we describe the family of harmonic metrics $h_t$ for the ray of Higgs bundles $(\mathcal{E}, t φ)$ for $t\gg0$ by perturbing from an explicitly constructed family of approximate solutions $h_t^{\mathrm{app}}$. We then describe the natural hyperKähler metric on $\mathcal{M}$ by comparing it to a simpler "semi-flat" hyperKähler metric. We prove that $g_{L^2} - g_{\mathrm{sf}} = O(\mathrm{e}^{-γt})$ along a generic ray, proving a version of Gaiotto-Moore-Neitzke's conjecture. Our results extend to weakly parabolic $SL(2,\mathbb{C})$-Higgs bundles as well. In the case of the four-puncture sphere, we describe the moduli space and metric more explicitly. In this case, we prove that the hyperkähler metric is ALG and show that the rate of exponential decay is the conjectured optimal one, $γ=4L$, where $L$ is the length of the shortest geodesic on the base curve measured in the singular flat metric $|\mathrm{det}\, φ|$.
