[Paper Review] Generic Ends of the Moduli Space of $SL(n,\mathbb{C})$-Higgs Bundles

Laura Fredrickson|arXiv (Cornell University)|Oct 3, 2018

Black Holes and Theoretical Physics17 references4 citations

TL;DR

This paper constructs explicit approximate solutions to Hitchin's equations for $SL(n,\mathbb{C})$-Higgs bundles along generic rays in the Hitchin moduli space, using gluing methods to build a family of approximate metrics $h_t^{\mathrm{app}}$ that differ from the true harmonic metrics $h_t$ by exponentially small errors $e^{-\delta t}$. The key contribution is a rigorous perturbation argument proving the existence and uniqueness of exact solutions close to these approximations, enabling finer asymptotic analysis of the moduli space geometry.

ABSTRACT

Given a generic ray of Higgs bundles $(\overline{\partial}_E, tφ)$, we describe the corresponding family of hermitian metrics $h_t$ solving Hitchin's equations via gluing methods. In the process, we construct a family of approximate solutions $h_t^{\mathrm{app}}$ which differ from the actual harmonic metrics $h_t$ by error terms of size $\mathrm{e}^{-δt}$. Such families of explicit approximate solutions have already proved useful for answering finer questions about the asymptotic geometry of the Hitchin moduli space.

Motivation & Objective

• To provide a finer analytic description of the asymptotic geometry of the $SU(n)$-Hitchin moduli space near its generic ends, beyond what algebraic geometry can achieve.
• To generalize prior results on $SU(2)$ to higher-rank $SL(n,\mathbb{C})$-Higgs bundles using constructive analytic techniques.
• To construct explicit approximate solutions $h_t^{\mathrm{app}}$ to Hitchin's equations that are exponentially close to the true harmonic metrics $h_t$, with error $\mathcal{O}(e^{-\delta t})$.
• To establish the existence and uniqueness of exact solutions via a contraction mapping argument on a small ball of perturbations, enabling detailed study of the moduli space structure.

Proposed method

• Uses gluing techniques to construct approximate solutions $h_t^{\mathrm{app}}$ to Hitchin's equations along generic rays in the moduli space, based on local models near degenerations.
• Employs a unitary formulation of Hitchin's equations, working with pairs $(\mathrm{d}_A, \Phi)$ and relating them to the holomorphic triple $(\overline{\partial}_E, \varphi, h)$ via gauge transformations.
• Applies a perturbation method: defines a nonlinear operator $\mathbf{T}_t$ on the space of hermitian sections $\gamma$, whose fixed point corresponds to the correction needed to turn $h_t^{\mathrm{app}}$ into the true harmonic metric $h_t$.
• Establishes uniform bounds on the linearized operator $L_t$ and nonlinear terms $Q_t$, showing $\|L_t^{-1}\|_{\mathcal{L}(L^2,H^2)} \leq \tilde{C}t^2$ and $\|Q_t(\gamma_1) - Q_t(\gamma_2)\|_{L^2} \leq \hat{C}\xi t^2 \|\gamma_1 - \gamma_2\|_{H^2}$ for small $\xi$.
• Applies a contraction mapping argument in weighted Sobolev spaces to prove existence and uniqueness of a fixed point $\gamma_t$ in a ball of radius $\rho_t = \mathcal{O}(e^{-(\delta + \epsilon)t})$ or $\mathcal{O}(t^{-4 - \epsilon})$, ensuring convergence of $h_t$ to a limit metric $h_\infty$.
• Uses local holomorphic and unitary gauges to handle the local structure of the Higgs bundle and spectral cover, with careful control over the size of disks $\mathbb{D}_p$ in the desingularization process.

Experimental results

Research questions

• RQ1 How can one construct explicit approximate solutions to Hitchin's equations for $SL(n,\mathbb{C})$-Higgs bundles along generic rays in the moduli space?
• RQ2 What is the rate of convergence between the approximate metrics $h_t^{\mathrm{app}}$ and the true harmonic metrics $h_t$?
• RQ3 Can a contraction mapping argument be applied to perturb the approximate solutions into exact solutions of Hitchin's equations?
• RQ4 How does the geometry of the spectral cover, particularly the coalescence of ramification points, affect the uniformity of the construction across the moduli space?
• RQ5 What is the asymptotic behavior of the harmonic metric $h_t$ as $t \to \infty$, and does it converge pointwise to a limiting metric $h_\infty$?

Key findings

• The approximate metrics $h_t^{\mathrm{app}}$ are constructed via gluing methods and differ from the true harmonic metrics $h_t$ by errors of size $\mathcal{O}(e^{-\delta t})$ for some $\delta > 0$, uniformly in the base curve $C$.
• The linearized operator $L_t$ associated with the perturbation equation satisfies $\|L_t^{-1}\|_{\mathcal{L}(L^2,H^2)} \leq \tilde{C}t^2$, showing that the inverse grows polynomially in $t$.
• The nonlinear term $Q_t$ satisfies $\|Q_t(\gamma_1) - Q_t(\gamma_2)\|_{L^2} \leq \hat{C}\xi t^2 \|\gamma_1 - \gamma_2\|_{H^2}$, ensuring the perturbation is sufficiently small for contraction mapping to apply.
• A fixed point $\gamma_t$ exists and is unique in a ball of radius $\rho_t = \mathcal{O}(e^{-(\delta + \epsilon)t})$ or $\mathcal{O}(t^{-4 - \epsilon})$, proving existence of a unique exact solution $h_t = h_t^{\mathrm{app}} \circ e^{-\gamma_t}$.
• The harmonic metrics $h_t$ converge pointwise to a limiting metric $h_\infty$ as $t \to \infty$, due to the uniform decay of $\gamma_t \to 0$.
• The construction is uniform within each stratum of the moduli space, but fails uniformly across strata when ramification points coalesce, as this corresponds to a collapsing cycle in the spectral cover.