Instantons on multi-Taub-NUT Spaces III: Down Transform, Completeness, and Isometry

TL;DR

The paper completes the circle between instanton data on the multi-Taub-NUT ALF space and bow data by formulating a Down transform that assigns a bow representation to an instanton via the index bundle of a Dirac-family. It proves that the Up and Down transforms are inverse and that each induces an isometry between the corresponding hyperkähler moduli spaces, thereby linking the instanton moduli to bow moduli. The analysis combines detailed operator-theoretic constructions with hyperkähler quotient geometry, establishing completeness, uniqueness, and a norm-compatible isometry for irreducible configurations. This work solidifies the bow transform as a robust, invertible, metric-preserving correspondence for ALF instantons on $ ext{TN}_k$ with generic asymptotic holonomy, enabling a quiver/bow description of the instanton moduli space.

Abstract

The index bundle of a family of Dirac operators associated to an instanton on a multi-Taub-NUT space forms a bow representation. We prove that the gauge equivalence classes of solutions of this bow representation are in one-to-one correspondence with the instantons. We also prove that this correspondence establishes an isometry of the bow and instanton moduli spaces.

Figures (1)

• Figure 1: The $A_k$ bow (presented here with $k=3$) is obtained by cutting the circle at points $p_\sigma$ and connecting the resulting intervals $J_\sigma=[p_{\sigma-1}+,p_\sigma-]$ by edges. The small representation of this bow has Hermitian line bundle $\underline{\mathpzc{e}}$ of rank one everywhere. The moduli space at level $i\nu$ of this representation is the multi-Taub-NUT space $\mathrm{TN}_k^\nu$Cherkis:2008ip.

Theorems & Definitions (66)

• Lemma 1
• Definition 2
• Proposition 3
• proof
• Proposition 4
• Proposition 5
• Proposition 6
• Proposition 7
• Lemma 8
• proof