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The Asymptotic Structure of Monopoles in R^3, Calorons, and ALF Instantons

Sergey A. Cherkis, Mark Stern

TL;DR

The paper proves that instantons on ALF spaces, including calorons and monopoles on $\mathbb{R}^3$, always admit a large-radius decomposition into sums of $U(1)$ constituents even without maximal symmetry breaking. It develops a unified framework based on spectral truncations of a Higgs-like operator $\nabla_\theta$, Hilbert bundles, and spectral $\zeta$-functions to control asymptotics, establishing quadratic curvature decay and precise blockwise decompositions labeled by $(\lambda,\beta)$ with $\beta\in\tfrac{1}{2}\mathbb{Z}$ and integrality constraints. The approach extends to $k$-centered Taub-NUT manifolds and calorons, using the same truncation and spectral-geometry toolkit to deduce the asymptotic structure. These results remove a major obstacle to studying the Nahm transform and moduli spaces with arbitrary asymptotic holonomy, enabling a more complete understanding of gauge-theory configurations on ALF geometries. In short, the work provides a robust mechanism to dissect ASD connections into canonical $U(1)$ building blocks at infinity, irrespective of symmetry-breaking assumptions, with broad implications for geometric analysis and mathematical physics.

Abstract

We study the asymptotic structure of instantons on multi-centered Taub-NUT manifolds, calorons, and monopoles on R^3. We show that, without any assumptions on symmetry breaking, these instantons and monopoles asymptotically decompose as a sum of U(1) instantons and monopoles, respectively.

The Asymptotic Structure of Monopoles in R^3, Calorons, and ALF Instantons

TL;DR

The paper proves that instantons on ALF spaces, including calorons and monopoles on , always admit a large-radius decomposition into sums of constituents even without maximal symmetry breaking. It develops a unified framework based on spectral truncations of a Higgs-like operator , Hilbert bundles, and spectral -functions to control asymptotics, establishing quadratic curvature decay and precise blockwise decompositions labeled by with and integrality constraints. The approach extends to -centered Taub-NUT manifolds and calorons, using the same truncation and spectral-geometry toolkit to deduce the asymptotic structure. These results remove a major obstacle to studying the Nahm transform and moduli spaces with arbitrary asymptotic holonomy, enabling a more complete understanding of gauge-theory configurations on ALF geometries. In short, the work provides a robust mechanism to dissect ASD connections into canonical building blocks at infinity, irrespective of symmetry-breaking assumptions, with broad implications for geometric analysis and mathematical physics.

Abstract

We study the asymptotic structure of instantons on multi-centered Taub-NUT manifolds, calorons, and monopoles on R^3. We show that, without any assumptions on symmetry breaking, these instantons and monopoles asymptotically decompose as a sum of U(1) instantons and monopoles, respectively.
Paper Structure (14 sections, 30 theorems, 196 equations)

This paper contains 14 sections, 30 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Monopoles in $\mathbb{R}^3$
  3. Analyzing $\Phi$
  4. Spectral truncation
  5. Estimating $T(R)$
  6. Angular Estimate
  7. Connection Asymptotics
  8. $k-$ centered Taub-NUT
  9. Hilbert bundles and Spectral Functions
  10. Geometry of Hilbert bundles
  11. Spectral functions
  12. Spectral Truncation
  13. Angular Instanton Estimates
  14. Connection Asymptotics

Key Result

Theorem 1.4

Let $K\subset \mathbb{R}^3$ be a compact set (possibly empty). Let $(E,d_A,\Phi)$ be a hermitian vector bundle over $\mathbb{R}^3\setminus K$, with connection $d_A$ and Higgs field $\Phi$, satisfying the monopole equations, with curvature $F_A\in L^2(\mathbb{R}^3\setminus K)$. Then $F_A$ decays quad and for $\beta_1\not = \beta_2$

Theorems & Definitions (51)

  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 2.2
  • Lemma 2.3: See for example Tian
  • Remark 2.6
  • Corollary 2.7
  • proof
  • ...and 41 more