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Nahm Poles and 0-Instantons

Marco Usula

Abstract

We study self-dual 0-connections, or 0-instantons, on asymptotically hyperbolic 4-manifolds. These connections develop a uniform singularity along the conformal infinity, and are asymptotic, at each point of the boundary, to a "Nahm pole" model solution on $H^4$. Examples include the Levi-Civita spin connections on $S^+$ over spin Poincaré-Einstein 4-manifolds. Inspired by the Fefferman-Graham expansion for Poincaré-Einstein metrics, we study the asymptotic expansion of these 0-instantons. We prove that the expansion is log-smooth, and that the coefficient of the first log term - which we call the 0-instanton obstruction tensor - is a conformal invariant related to the Weyl curvature of the ambient conformal metric. We then show that this invariant vanishes if and only if the 0-instanton is smooth modulo gauge. Finally, we study the renormalized Yang-Mills energy: we prove that, if the metric is asymptotically Poincaré-Einstein to third order, then this energy is a well-defined conformal invariant, and equals the negative Chern-Simons invariant of the conformal infinity.

Nahm Poles and 0-Instantons

Abstract

We study self-dual 0-connections, or 0-instantons, on asymptotically hyperbolic 4-manifolds. These connections develop a uniform singularity along the conformal infinity, and are asymptotic, at each point of the boundary, to a "Nahm pole" model solution on . Examples include the Levi-Civita spin connections on over spin Poincaré-Einstein 4-manifolds. Inspired by the Fefferman-Graham expansion for Poincaré-Einstein metrics, we study the asymptotic expansion of these 0-instantons. We prove that the expansion is log-smooth, and that the coefficient of the first log term - which we call the 0-instanton obstruction tensor - is a conformal invariant related to the Weyl curvature of the ambient conformal metric. We then show that this invariant vanishes if and only if the 0-instanton is smooth modulo gauge. Finally, we study the renormalized Yang-Mills energy: we prove that, if the metric is asymptotically Poincaré-Einstein to third order, then this energy is a well-defined conformal invariant, and equals the negative Chern-Simons invariant of the conformal infinity.
Paper Structure (19 sections, 28 theorems, 210 equations)

This paper contains 19 sections, 28 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. Conformally compact geometry
  3. Definitions and main properties
  4. Poincaré--Einstein metrics
  5. $0$-connections
  6. The Nahm pole boundary condition
  7. Nahm poles and self-duality
  8. Nahm poles and spin structures on the boundary
  9. Comparison with Nahm poles for the KW equations
  10. Examples
  11. Asymptotic expansion of $0$-instantons
  12. A geodesic normal form for $0$-connections
  13. Asymptotic expansions of $0$-instantons
  14. The $0$-instanton obstruction tensor
  15. The boundary value of a $0$-instanton
  16. ...and 4 more sections

Key Result

Theorem 1

(Theorems thm:index-set-0-instantons, thm:coefficients-of-the-instanton-expansion) If $A$ is a polyhomogeneous self-dual $0$-instanton, then the expansion of the geodesic normal family $\alpha_{h_{0}}\left(x\right)$ is log-smooth; more precisely, it takes the form In this expansion $\alpha_{0}$ is an $\mathop{\mathrm{\mathrm{SU}}}\nolimits\left(2\right)$ connection on $\mathbb{S}$, the other coef

Theorems & Definitions (91)

  • Theorem
  • Theorem
  • Theorem
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 81 more