Perspectives on the Asymptotic Geometry of the Hitchin Moduli Space

We survey some recent developments in the asymptotic geometry of the Hitchin moduli space, starting with an introduction to the Hitchin moduli space and hyperk\"ahler geometry.

The Hitchin moduli space Fixed data: • C , a compact Riemann surface (possibly with punctures D) • G = SU(n), G C = SL(n, C) • E → C , a complex vector bundle of rank n with Aut(E ) = SL(E ) Hitchin moduli space, M.
• M ζ=0 is G C -Higgs bundle moduli space The Higgs bundle moduli space Definition A Higgs bundle is a pair (∂ E , ϕ) consisting of a holomorphic structurē ∂ E on E and a "Higgs field" ϕ ∈ Ω 1,0 (C , End 0 E ) such that∂ E ϕ = 0.
(Locally,∂ E =∂ and ϕ = Pdz, where P is a tracefree n × n matrix with holomorphic entries.) Ex: The GL(1)-Higgs bundle moduli space is M = Jac(C ) For C = T 2 τ , M = T 2 τ × C. Fact #2: In its avatar as the Higgs bundle moduli space, M is an algebraic completely integrable system. The Hitchin moduli space (with parameter t > 0) / ∼ arises as the moduli space of certain N = 2, 4d SUSY theories (namely "theories of class S", S[g, C , D]) compactified on a circle S 1 t . Gaiotto-Moore-Neitzke: • The BPS spectrum Ω(γ; u) u ∈ B, γ ∈ H1(Σu; Z)σ of the N = 2 4d theory S[g, C , D] can be recovered from the geometry of the family Mt as t → ∞. Satisfies Kontsevich-Soibelman wall-crossing.
• GMN also give a recipe for constructing hyperkähler metrics from integrable system data and BPS indices Ω(γ; u) Note: If M admits a C × ζ -action (E, ϕ) → (E, ζϕ), then conjecture is about the asymptotic geometry of a single Hitchin moduli space, M.
• g L 2 Hitchin's L 2 hyperkähler metric-uses h • g sf semiflat metric-from integrable system structure (1) Describe important elements of general proof.
• We can gain insight into physics conjecture from geometric analysis.
• Trying to prove intricate conjectures of physics stretches limits of geometric analysis.
(2) Specialize to 4d Hitchin moduli spaces, since 4d noncompact hyperkähler spaces are well-studied. In particular, I'll describe progress for SU(2)-Hitchin moduli space on the four-punctured sphere. (Here, we get optimal rate of exponential decay.) The main difficulty is dealing with the contributions to the integral · g L 2 = C · · · from infinitesimal neighborhoods around Z .
Idea #1: Semiflat metric is an L 2 -metric The semiflat metric, from the integrable system structure, on where the metric variationν ∞ of h ∞ is independent of t and solves Define an non-hyperkähler L 2 -metric g app on M using variations of the metric h app t .

Idea #2: Approximate solutions
Our goal is to show that the following sum is O(e −εt ): Since h app t differs from h ∞ only on disks around p ∈ Z , the difference g app − g sf localizes (up to exponentially-decaying errors) to disks around p ∈ Z .

Idea #3: Holomorphic variations
When Mazzeo-Swoboda-Weiss-Witt proved that g L 2 − g sf was at least polynomially-decaying in t, all of their possible polynomial terms came from infinitesimal variations in which the branch points move.
Dumas-Neitzke used a family of biholomorphic maps on local disks (originally defined by Hubbard-Masur) to match the changing location of the branch points. This uses subtle geometry of Hitchin moduli space. E.g. for SU(2), conformal invariance.
Remarkably, this can be generalized off of the Hitchin section and from SU(2) to SU(n).

4d Hitchin moduli spaces
Noncompact hyperkähler four-manifolds X There are several known families: the 'classical' spaces of types ALE, ALF, ALG, ALH, as well as two more recently discovered types, now frequently called ALG * ALH * .

Theorem [Chen-Chen]
If (X , g ) is a noncompact complete connected hyperkähler manifold of real dimension 4 (i.e. if (X , g ) is a "gravitational instanton") whose Riemannian curvature tensor decays faster than 1/r 2 , i.e., |Riemg (q)| ≤ C dist(p, q) −2− as q → ∞, where p is a fixed point in X , then (M, g ) necessarily belongs to one of the families ALE, ALF, ALG and ALH.
Categories based on asymptotic volume growth: ALE/ALF/ALG/ALH ALE: [Vol ∼ r 4 ] Any X is asymptotic to some standard model X • Γ = C 2 /Γ where Γ is a finite subgroup of SU(2). Curvature decay condition ⇒ volume growth is r 4 , r 3 , r 2 or r 1 . Rigidity! More recently, Chen-Chen-Chen classified ALG and ALH spaces.
ALG: Any X (with faster than quadratic curvature decay) is asymptotic to some standard model X • τ,β fibered over C β of angle 2πβ with fiber T 2 τ . [Chen-Chen] What about these other types ALG * and ALH * ? •

Question
Where do 4d Hitchin moduli spaces fit in to classification of gravitational instantons?

Modularity Conjecture
Every 4d Hitchin moduli space is of type ALG or ALG * . Conversely, every ALG & ALG * hyperkähler metric with Vol ∼ r 2 can be realized as the hyperkähler metric on a Hitchin moduli space.
Here are the types of ALG metrics, and the conjectural associated families of 4d Hitchin moduli spaces. The Hitchin moduli spaces associated to the SU(2) theories with N f = 0, 1, 2, 3 are conjecturally of type ALG * with fiber I * 4−N f . (They are not ALG since the modulus, τ , of torus does not converge at ends.) LeBrun gave a framework to describe all Ricci-flat Kähler metrics of complex-dimension two with a holomorphic circle action in terms of two functions u, w .

Regular
Generalized Gibbons-Hawking Ansatz specialized to our case: Consider a hyperkähler metric on T 2 x,y × R + r × S 1 θ with holomorphic circle action. The hyperkähler metric is g L 2 = e u u r (dx 2 + dy 2 ) + u r dr 2 + u −1 r dθ 2 where u : T 2 x,y × R + r → R solves ∆ T 2 u + ∂ 2 r e u = 0.
The semiflat metric g sf corresponds to u sf = log r .

Goal
Show that u − u sf has conjectured rate of exponential decay.