Asymptotic limits in the Hitchin moduli space
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Given a Higgs bundle ([Higgs bundle symbol]), Hitchin's equations are equations for a Hermitian metric on the underlying vector bundle. Hitchin's equations are a coupled system of non-linear PDEs, and as such are difficult to solve. Despite this, there are some situations in which it is possible to say something more concrete about the Hermitian metric solving Hitchin's equations. This is the unifying theme of this three-part dissertation. In the first chapter, we look at the ends of the Hitchin moduli space on a compact Riemann surface. We construct good approximate solutions of Hitchin's equations near the ends, taking advantage of the asymptotic abelianization of Hitchin's equations. In the second chapter, we consider solutions of Hitchin's equations on CP¹ which are fixed by a circle action. The circle action manifests in a radial symmetry which reduces Hitchin's equations from a coupled system of PDEs to a coupled system of ODEs. In the main result of the second part, we relate fixed points of the circle action to W-algebra representations. We prove that for each representation in the (K,K + N)-minimal model of W [subscript K], the effective central charge is equal to a number which can be computed from a solution of Hitchin's equations fixed by a certain circle action. The Hitchin moduli space has two interesting subspaces: the Hitchin section and the space of opers. In the third part, we relate two different families of flat connections corresponding to these two subspaces. To relate these families, we study how a certain harmonic metric blows up. This harmonic metric is related to the uniformizing metric on the underlying complex curve.