Perspectives on the Asymptotic Geometry of the Hitchin Moduli Space

Abstract

We survey some recent developments in the asymptotic geometry of the Hitchin moduli space, starting with an introduction to the Hitchin moduli space and hyperkähler geometry.Note: This is the model solution featured in [14,18,27]. The base curve is CP 1 with an irregular singularity at ∞ [15]. Nonabelian Hodge correspondenceThe Hitchin moduli space M is hyperkähler. As a consequence, it has a CP 1 -worth of complex structures, labeled by parameter ζ ∈ CP 1 . Two avatars of the Hitchin moduli space are• the Higgs bundle moduli space (ζ = 0), and• the moduli space of flat GL(n, C)-connections ζ ∈ C × .1 In a local holomorphic coordinate z and a local holomorphic frame for (E, ∂E), if ϕ = Φdz, then ϕ * h = h −1 Φ * hdz.

All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces

“…This behaviour is broadly classed as "ALG" 2 (see e.g. [25]) and entails a much slower volume growth. (In complex dimension 2, the volume growth for ALG is quadratic and is explicitly verified for the n = 4 parabolic Higgs moduli space in [26].)…”

Higgs Bundles—Recent Applications

“…Almost a decade ago Gaiotto-Moore-Neitzke gave a conjectural description of the hyperkähler metric on M G C , which surprisingly suggests that much of the asymptotic geometry of the moduli space can be derived from the abelian spectral data described before. Recent progress has been made by Mazzeo-Swoboda-Weiss-Witt, Dumas-Neitzke and Fredrickson but the global picture remains open (for a survey of the area, see [Fre19]).…”

“…Understanding the geometric and analytic properties of the harmonic maps arising from Hitchin's equations (1)-( 2) is of significant importance. For instance, one may ask how do those metrics behave at the boundaries of the moduli space, or how do the energy densities of the corresponding harmonic maps at different points of the Hitchin fibration determine each other (the reader may be interested in the reviews [Li19] and [Fre19], and references therein). From Hitchin's work, the moduli space of Higgs bundles has a natural C * -action λ • (E, Φ) = (E, λΦ), whose fixed point sets allow one to study different aspects of the topology and the geometry of the space, as done in [Hit87b] (see also [Ray18,Col19]).…”

An Introduction to Higgs Bundles via Harmonic Maps

“…Remark 8.6. There is also another approach to obtain the decoupling phenomenon in Theorem 8.5 for the Hitchin equation in [20,41] for generic Higgs bundles, which will be addressed in the survey paper [21] of L. Fredrickson. (1) For cyclic Higgs bundles in the Hitchin component, Theorems 8.4 and 8.5 were first proven in Loftin [38] for n = 3 and in Collier-Li [9] for n > 3.…”