Curvature Decay and the Spectrum of the Non-Abelian Laplacian on $\mathbb{R}^3$
Michael Wilson
TL;DR
The paper addresses how the pointwise decay of the non-abelian curvature $F_A$ influences the essential spectrum of the covariant Laplacian on $\mathbb{R}^3$. It establishes a sharp threshold at $|F_A(x)| \lesssim (1+|x|)^{-3-\varepsilon}$ ensuring the operator is a relatively compact perturbation of the free Laplacian, preserving $\sigma_{ess}=[0,\infty)$, and constructs a borderline example at $|F_A(x)| \sim |x|^{-3}$ with $0$ in the essential spectrum. Key contributions include a curvature-adjusted Kato inequality, a gauge-fixing scheme, and explicit non-abelian hedgehog constructions that demonstrate sharpness. The results illuminate the stability of gauge-field fluctuations and have implications for semiclassical analyses and infrared aspects of gauge theories.
Abstract
I study the spectral behavior of the covariant Laplacian $Δ_A = d_A^* d_A$ associated with smooth $\mathrm{SU}(2)$ connections on $\mathbb{R}^3$. The main result establishes a sharp threshold for the pointwise decay of curvature governing the essential spectrum of $Δ_A$. Specifically, if the curvature satisfies the bound $|F_A(x)| \le C(1 + |x|)^{-3-\varepsilon}$ for some $\varepsilon > 0$, then $Δ_A$ is a relatively compact perturbation of the flat Laplacian and hence $σ_{\mathrm{ess}}(Δ_A) = [0,\infty)$. At the critical decay rate $|F_A(x)| \sim |x|^{-3}$, I construct a smooth connection for which $0 \in σ_{\mathrm{ess}}(Δ_A)$, showing that the threshold is sharp. Moreover, a genuinely non-Abelian example based on the hedgehog ansatz is given to demonstrate that the commutator term $A \wedge A$ contributes at the same order. This work identifies the exact decay rate separating stable preservation of the essential spectrum from the onset of delocalized modes in the non-Abelian setting, providing a counterpart to classical results on magnetic Schrödinger operators.