# Asymptotic Geometry of the Hitchin Metric

@article{Mazzeo2017AsymptoticGO,
title={Asymptotic Geometry of the Hitchin Metric},
author={R. Mazzeo and J. Swoboda and Hartmut Weiss and F. Witt},
journal={Communications in Mathematical Physics},
year={2017},
volume={367},
pages={151-191}
}
• R. Mazzeo, +1 author F. Witt
• Published 2017
• Mathematics, Physics
• Communications in Mathematical Physics
AbstractWe study the asymptotics of the natural L2 metric on the Hitchin moduli space with group $${G = \mathrm{SU}(2)}$$G=SU(2). Our main result, which addresses a detailed conjectural picture made by Gaiotto et al. (Adv Math 234:239–403, 2013), is that on the regular part of the Hitchin system, this metric is well-approximated by the semiflat metric from Gaiotto et al. (2013). We prove that the asymptotic rate of convergence for gauged tangent vectors to the moduli space has a precise… Expand
16 Citations
Asymptotics of Hitchin’s Metric on the Hitchin Section
• Mathematics, Physics
• 2018
We consider Hitchin’s hyperkähler metric g on the moduli space $${\mathcal{M}}$$M of degree zero SL(2)-Higgs bundles over a compact Riemann surface. It has been conjectured that, when one goes toExpand
Exponential Decay for the Asymptotic Geometry of the Hitchin Metric
We consider Hitchin’s hyperkähler metric $$g_{L^2}$$ g L 2 on the SU ( n )-Hitchin moduli space over a compact Riemann surface. We prove that the difference between the metric $$g_{L^2}$$ g L 2 and aExpand
The Ooguri-Vafa space as a moduli space of framed wild harmonic bundles
The Ooguri-Vafa hyperkahler metric is expected to be part of the local model of the $L^2$-hyperkahler metric of the Hitchin moduli spaces, near the generic part of the discriminant locus (seeExpand
Hyperk\"ahler metrics on the moduli space of weakly parabolic Higgs bundles
We use the theory of Gaiotto, Moore and Neitzke to construct a set of Darboux coordinates on the moduli space M of weakly parabolic SL(2,C)-Higgs bundles. For generic Higgs bundles (E , RΦ) with R 0Expand
ASYMPTOTIC GEOMETRY OF THE MODULI SPACE OF PARABOLIC SL(2, C)-HIGGS BUNDLES
Given a generic stable strongly parabolic SL(2, C)-Higgs bundle (E , φ), we describe the family of harmonic metrics ht for the ray of Higgs bundles (E , tφ) for t 0 by perturbing from an explicitlyExpand
Generic Ends of the Moduli Space of $SL(n,\mathbb{C})$-Higgs Bundles
Given a generic ray of Higgs bundles $(\overline{\partial}_E, t\varphi)$, we describe the corresponding family of hermitian metrics $h_t$ solving Hitchin's equations via gluing methods. In theExpand
$L^2$-cohomology of quasi-fibered boundary metrics
• Mathematics, Physics
• 2021
We develop new techniques to compute the weighted L-cohomology of quasi-fibered boundary metrics (QFB-metrics). Combined with the decay of L-harmonic forms obtained in a companion paper, this allowsExpand
Opers and nonabelian Hodge: numerical studies
• Mathematics, Physics
• 2020
We present numerical experiments that test the predictions of a conjecture of Gaiotto-Moore-Neitzke and Gaiotto concerning the monodromy map for opers, the nonabelian Hodge correspondence, and theExpand
Asymptotic geometric properties of Higgs bundle moduli spaces
on a hermitian vector bundle (E,h) of degree 0 and rank 2 over a compact Riemann surface X of genus γ ≥ 2, modulo unitary gauge transformations. The equations (1) are a system of nonlinear PDEs for aExpand
Adiabatic limits of anti-self-dual connections on collapsed $K3$ surfaces
• Mathematics
• 2018
We prove a convergence result for a family of Yang-Mills connections over an elliptic $K3$ surface $M$ as the fibers collapse. In particular, assume $M$ is projective, admits a section, and hasExpand

#### References

SHOWING 1-10 OF 21 REFERENCES
Asymptotics of Hitchin’s Metric on the Hitchin Section
• Mathematics, Physics
• 2018
We consider Hitchin’s hyperkähler metric g on the moduli space $${\mathcal{M}}$$M of degree zero SL(2)-Higgs bundles over a compact Riemann surface. It has been conjectured that, when one goes toExpand
Exponential Decay for the Asymptotic Geometry of the Hitchin Metric
We consider Hitchin’s hyperkähler metric $$g_{L^2}$$ g L 2 on the SU ( n )-Hitchin moduli space over a compact Riemann surface. We prove that the difference between the metric $$g_{L^2}$$ g L 2 and aExpand
Wall-crossing, Hitchin Systems, and the WKB Approximation
• Physics, Mathematics
• 2009
We consider BPS states in a large class of d = 4,N = 2 eld theories, obtained by reducing six-dimensional (2; 0) superconformal eld theories on Riemann surfaces, with defect operators inserted atExpand
Limiting configurations for solutions of Hitchin's equation
• Mathematics
• 2012
We review recent work on the compactification of the moduli space of Hitchin's self-duality equation. We study the degeneration behavior near the ends of this moduli space in a set of genericExpand
Moduli space of semistable pairs on a curve
Let X be a smooth projective curve over an algebraically closed field k of any characteristic. A stable pair (E, <p) on X, as defined by Hitchin [2], is a vector bundle E on X together with aExpand
Special geometry of Euclidean supersymmetry III: the local r-map, instantons and black holes
• Physics
• 2009
We define and study projective special para-Kahler manifolds and show that they appear as target manifolds when reducing five-dimensional vector multiplets coupled to supergravity with respect toExpand
Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces
Let $(E,\overline{\partial}_E,\theta)$ be a stable Higgs bundle of degree $0$ on a compact connected Riemann surface. Once we fix the flat metric $h_{\det(E)}$ on the determinant of $E$, we have theExpand
Ends of the moduli space of Higgs bundles
• Mathematics
• 2016
We associate to each stable Higgs pair (A(0), Phi(0)) on a compact Riemann surface X a singular limiting configuration (A(infinity), Phi(infinity)), assuming that det Phi has only simple zeroes. WeExpand
On the density of strebel differentials
• Mathematics
• 1975
w 1. Statement of the Main Result Let X be a compact Riemann surface of genus g > 2 and denote by O the sheaf of germs of holomorphic differential 1-forms on X. Let q~H~ ~| be a nonzero holomorphicExpand
Notes on a New Construction of Hyperkahler Metrics
I briefly review a new construction of hyperkahler metrics on total spaces of complex integrable systems, which we described in joint work with Davide Gaiotto and Greg Moore. The key ingredient inExpand