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Local asymptotics for Hitchin's equations and high energy harmonic maps

Nathaniel Sagman, Peter Smillie

TL;DR

The paper addresses high-energy limits of Hitchin's self-duality equations for Higgs bundles, extending local asymptotics from generically regular semisimple to arbitrary Higgs fields by exploiting a Jordan–Chevalley decomposition into semisimple and nilpotent parts. It proves uniform nilpotent-boundedness (Theorem B) and almost-orthogonality (Theorem C), leading to asymptotic decoupling (Theorem A) on the spectral cover, where the nilpotent part governs a decoupled limit. The Hitchin WKB problem is resolved in generality (Theorem D), providing Weyl-chamber-distance bounds for monodromy with exponential improvement in the semisimple case, by extending Mochizuki/Mochizuki-type techniques. The work also analyzes high-energy harmonic maps to symmetric spaces and buildings (Theorem E), showing convergence of pullback metrics to toral data on cameral covers and establishing omega-convergence to harmonic maps into affine buildings, with quantitative control on pullback connections. Collectively, the results connect Higgs-bundle asymptotics, spectral geometry via cameral covers, WKB-type monodromy, and the large-scale geometry of harmonic maps, offering a robust framework for studying Hitchin-type equations and their geometric applications, including the Hitchin WKB problem and high-energy maps to buildings.

Abstract

We find new estimates and a new asymptotic decoupling phenomenon for solutions to Hitchin's self-duality equations at high energy. These generalize previous results for generically regular semisimple Higgs bundles to arbitrary Higgs bundles. We apply our estimates to the Hitchin WKB problem and to high energy harmonic maps to symmetric spaces and buildings.

Local asymptotics for Hitchin's equations and high energy harmonic maps

TL;DR

The paper addresses high-energy limits of Hitchin's self-duality equations for Higgs bundles, extending local asymptotics from generically regular semisimple to arbitrary Higgs fields by exploiting a Jordan–Chevalley decomposition into semisimple and nilpotent parts. It proves uniform nilpotent-boundedness (Theorem B) and almost-orthogonality (Theorem C), leading to asymptotic decoupling (Theorem A) on the spectral cover, where the nilpotent part governs a decoupled limit. The Hitchin WKB problem is resolved in generality (Theorem D), providing Weyl-chamber-distance bounds for monodromy with exponential improvement in the semisimple case, by extending Mochizuki/Mochizuki-type techniques. The work also analyzes high-energy harmonic maps to symmetric spaces and buildings (Theorem E), showing convergence of pullback metrics to toral data on cameral covers and establishing omega-convergence to harmonic maps into affine buildings, with quantitative control on pullback connections. Collectively, the results connect Higgs-bundle asymptotics, spectral geometry via cameral covers, WKB-type monodromy, and the large-scale geometry of harmonic maps, offering a robust framework for studying Hitchin-type equations and their geometric applications, including the Hitchin WKB problem and high-energy maps to buildings.

Abstract

We find new estimates and a new asymptotic decoupling phenomenon for solutions to Hitchin's self-duality equations at high energy. These generalize previous results for generically regular semisimple Higgs bundles to arbitrary Higgs bundles. We apply our estimates to the Hitchin WKB problem and to high energy harmonic maps to symmetric spaces and buildings.

Paper Structure

This paper contains 43 sections, 35 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. Previous results on local asymptotics
  4. Asymptotics of solutions to Hitchin's equations
  5. The Hitchin WKB problem
  6. Application to harmonic maps
  7. Outline of paper
  8. Acknowledgements
  9. Linear algebraic preliminaries
  10. The Schur decomposition
  11. Jordan-Chevalley and Schur
  12. Centres of centralizers
  13. Almost orthogonality
  14. Higgs bundles
  15. Higgs bundles and Hitchin's equations
  16. ...and 28 more sections

Key Result

Theorem 1.4

Let $\phi \in S_n(d,A)$, let $E=\oplus_{i=1}^m E_i$ denote the generalized eigenspace decomposition for $\phi$, and let $h$ be a harmonic metric for $\phi$. On any subdisk $D(r)$, the splitting $E = \oplus_{i=1}^m E_i$ is $Ce^{-cd}$-almost orthogonal for $h$ for constants $C=C(n, r, A)$ and $c=c(n,r

Theorems & Definitions (94)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3: Scaling properties
  • Theorem 1.4: Lemma 1.3 in Mo
  • Corollary 1.5: Theorem 1.2 in Mo
  • Corollary 1.6: section 1.2.1 in Mo
  • Remark 1.7
  • Remark 1.8
  • Proposition 2.1: Page 729 in S and Lemma 3.5 in LM1
  • Proposition 2.2
  • ...and 84 more