Asymptotics of certain families of Higgs bundles in the Hitchin component

Abstract

Using Hitchin's parameterization of the Hitchin-Teichmüller component of the $SL(n,\mathbb{R})$ representation variety, we study the asymptotics of certain families of representations. In fact, for certain Higgs bundles in the $SL(n,\mathbb{R})$-Hitchin component, we study the asymptotics of the Hermitian metric solving the Higgs bundle equations. This analysis is used to estimate the asymptotics of the corresponding family of flat connections as we scale the differentials by a real parameter. We consider Higgs fields that have only one holomorphic differential $q_n$ of degree $n$ or $q_{n-1}$ of degree $n-1.$ We also study the asymptotics of the associated family of equivariant harmonic maps to the symmetric space $SL(n,\mathbb{R})/SO(n,\mathbb{R})$ and relate it to recent work of Katzarkov, Noll, Pandit and Simpson.

48 pages, v2: ~20 pages added to v1, substantial changes were made to the proof of parallel transport asymptotics

References in corpus (2)

• Ends of the moduli space of Higgs bundles
• G2 geometry and integrable systems

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• Hitchin harmonic maps are immersions
• The Extended Bogomolny Equations and Generalized Nahm Pole Boundary Condition
• Twisted cyclic quiver varieties on curves
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