Asymptotic geometry of the moduli space of parabolic SL(2,C)$\operatorname{SL}(2,\mathbb {C})$‐Higgs bundles

Abstract

Given a generic stable strongly parabolic SL ( 2 , C ) $\operatorname{SL}(2,\mathbb {C})$ -Higgs bundle ( E , φ ) $({\mathcal {E}}, \varphi )$ , we describe the family of harmonic metrics h t $h_t$ for the ray of Higgs bundles ( E , t φ ) $({\mathcal {E}}, t \varphi )$ for t ≫ 0 $t\gg 0$ by perturbing from an explicitly constructed family of approximate solutions h t app $h_t^{\mathrm{app}}$ . We then describe the natural hyperkähler metric on M $\mathcal {M}$ by comparing it to a simpler ‘semiflat’ hyperkähler metric. We prove that g L 2 − g sf = O ( e − γ t ) $g_{L^2} \,{-}\, g_{\mathrm{sf}} \,{=}\, O({\mathrm{e}}^{-\gamma t})$ along a generic ray, proving a version of Gaiotto–Moore–Neitzke's conjecture. Our results extend to weakly parabolic SL ( 2 , C ) $\operatorname{SL}(2,\mathbb {C})$ -Higgs bundles as well. A centerpiece of this paper is our explicit description of the moduli space and its L 2 $L^2$ metric for the case of the four-punctured sphere. We prove that the hyperkähler manifold in this case is a gravitational instanton of type ALG and that its rate of exponential decay to the semiflat metric is the conjectured optimal one, γ = 4 L $\gamma =4L$ , where L $L$ is the length of the shortest geodesic on the base curve measured in the singular flat metric | det φ | $|\det \varphi |$ .