Asymptotic Geometry of the Hitchin Metric

Abstract

We study the asymptotics of the natural $L^2$ metric on the Hitchin moduli space with group $G=\mathrm{SU}(2)$. Our main result, which addresses a detailed conjectural picture made by Gaiotto, Neitzke and Moore \cite{gmn13}, is that on the regular part of the Hitchin system, this metric is well-approximated by the semiflat metric from \cite{gmn13}. We prove that the asymptotic rate of convergence for gauged tangent vectors to the moduli space has a precise polynomial expansion, and hence that the the difference between the two sets of metric coefficients in a certain natural coordinate system also has polynomial decay. Very recent work by Dumas and Neitzke indicates that the convergence rate for the metric is exponential, at least in certain directions.

45 pages, published version

References in corpus (2)

• Special Geometry of Euclidean Supersymmetry III: the local r-map, instantons and black holes
• Proper Affine Hyperspheres which fiber over Projective Special Kaehler Manifolds

Cited by in corpus (14)

• Singular Geometry and Higgs Bundles in String Theory
• Exponential Decay for the Asymptotic Geometry of the Hitchin Metric
• Asymptotic Geometry of the Moduli Space of Parabolic $SL(2,\mathbb{C})$-Higgs Bundles
• Perspectives on the Asymptotic Geometry of the Hitchin Moduli Space
• Generic Ends of the Moduli Space of $SL(n,\mathbb{C})$-Higgs Bundles
• The Ooguri-Vafa Space as a Moduli Space of Framed Wild Harmonic Bundles
• Higgs bundles, harmonic maps, and pleated surfaces
• Seiberg-Witten differentials on the Hitchin base
• Nilpotent Higgs bundles and families of flat connections
• The Algebraic and Analytic Compactifications of the Hitchin Moduli Space
• Adiabatic limits of anti-self-dual connections on collapsed K3 surfaces
• $\mathfrak{sl}(2)$-type singular fibres of the symplectic and odd orthogonal Hitchin system
• Spectral Networks and Non-abelianization
• Interpreting the Ooguri-Vafa symplectic form à la Atiyah-Bott