Exponential Decay for the Asymptotic Geometry of the Hitchin Metric

Abstract

We consider Hitchin's hyperkähler metric g L 2 on the SU(n)-Hitchin moduli space over a compact Riemann surface. We prove that the difference between the metric g L 2 and a simpler "semiflat" hyperkähler metric g sf is exponentially-decaying along generic rays in the Hitchin moduli space, as conjectured by Gaiotto-Moore-Neitzke. 1 arXiv:1810.01554v2 [math.DG] 29 May 2019 2 h 0z dz. (Note that because C is Kähler, the Hodge star on 1-forms depends only on the conformal class of the metric g C .)An infinitesimal h 0 -unitary gauge transformation γ gives rise to a deformation (Ȧ γ ,Φ γ ). The map is given by ρ : Ω 0 (su(E)) → Ω 0,1 (sl(E)) ⊕ Ω 1,0 (sl(E)) [DN19, Eq. 2.5], where ρ(γ) = (Ȧ 0,1 γ ,Φ γ ) = (−∂ A γ, [γ, Φ]) (1.6)

Asymptotic Geometry of the Hitchin Metric

“…This is accomplished with careful attention to the rate of exponential decay, but unfortunately they miss the conjectured sharp numerical value of this rate by a factor of 2. Their result has successfully been extended to the entire space M ′ , including non-horizontal directions and the region off of the Hitchin section, in the very recent preprint [Fr18] by Laura Fredrickson.…”

Twisted cyclic quiver varieties on curves

“…For cyclic Higgs bundles, this was improved to pointwise monotonicity in [10,11]. These observations are intertwined with the "Hitchin WKB problem" [16,25,27,21,26,14,15,12] in the nonabelian Hodge theory of X, asking about the behaviour of harmonic metrics, harmonic maps, and their corresponding flat connections as t goes to infinity in (E, tΦ). Again, this problem has been resolved for cyclic Higgs bundles in the Hitchin section [8].…”

Perspectives on the Asymptotic Geometry of the Hitchin Moduli Space

“…In these notes, we give an introduction to the hyperkähler geometry of the Hitchin moduli space, focusing on the geometry of the ends of the Hitchin moduli space. In the last section (Section 4), we briefly survey some recent developments in the description of the asymptotic geometry of M. We start with Gaiotto-Moore-Neitzke's conjectural description in [17,18] and survey recent progress in [12,13,14,27,28]. We take a meandering path through more classical geometric results to get there.…”

“…Mazzeo-Swoboda-Weiss-Witt [28] have shown that along a generic ray g M −g sf decays polynomially in t. Dumas-Neitzke [12] have shown that -restricted to the Hitchin section -g M − g sf decays exponentially in t like O e −2M t . The author has shown that along a ray in M SU(2) g M − g sf decays exponentially in t [13]. However, the constant of exponential decay is not sharp.…”

“…The proof in [13] can be seen as an extension of the method of Dumas-Neitzke [12] to all of M using the analysis and approximate solutions of Mazzeo-Swoboda-Weiss-Witt in [27,28].…”