Charges, complex structures, and perturbations of instantons

TL;DR

The paper proves that ALF Hermitian, non-Kähler gravitational instantons are integrable and infinitesimally rigid. By exploiting the conformal relation $\uhat g_{ab}=mbe Omega^2 g_{ab}$ and the resulting conformally Kähler structure, it constructs a gauge-invariant linearized 2-form perturbation $\delta\uhat{oldsymbol{ om{}}}_{ab}$ whose periods on 2-cycles yield a linearized charge $\,elta Q[S]$. A Wu–LeBrun-type divergence identity shows $eltahat{oldsymbol{ om{}}}=0$, enabling a perturbative expansion around Hermitian-Einstein backgrounds and control over boundary terms for compact and ALF ends. The theory is illustrated via Ambi-Kähler and Chen-Teo examples, where the charges reproduce moduli and demonstrate that perturbations align with moduli directions, thereby clarifying the structure of the instanton moduli space and offering a Lorentzian-analogous conservation perspective in a Riemannian setting.

Abstract

The main result of this paper is that infinitesimal rigidity of the moduli space for Hermitian gravitational instantons holds. An important step in the proof is to show that, provided certain boundary conditions hold, a curve of Riemannian metrics passing through a Hermitian non-Kähler Einstein metric is conformally Kähler to second perturbative order. This uses ideas of Wu and LeBrun. Hermitian non-Kähler Einstein 4-manifolds have a quasi-locally conserved charge, which is shown to correspond to a parameter of the moduli space. This charge is evaluated for all explicitly known examples. It follows from the proof of our main theorem that infinitesimal Einstein deformations admit a closed 2-form that measures the perturbation in the moduli parameter.

Theorems & Definitions (25)

• Theorem 1.1
• Lemma 1.2
• Remark 1.3
• Proposition 3.1
• Remark 3.2
• proof
• Theorem 3.3
• proof
• Example 4.1: Kerr
• Example 4.2: Self-dual Taub-NUT