Conformal invariants for the zero mode equation
Guofang Wang, Mingwei Zhang
TL;DR
The authors establish a sharp spin-geometry inequality linking the L^n norm of a gauge field A to the Yamabe invariant, and completely characterize when equality occurs, tying the result to Killing spinors and Sasaki-Einstein geometry. Their approach bypasses zeros of the spinor by a conformal reduction to scalar problems and introduces a family of conformal invariants that align with the Yamabe constant, enabling a precise equality analysis. They extend the framework to k-forms, derive analogous inequalities, and reveal new links between zero modes, twistor/killing spinors, and Sasaki-Einstein structures. A novel invariant Y_v is proposed to measure extremal zero-mode behavior while preserving conformal invariance, with conjectures guiding future exploration in odd dimensions.
Abstract
For non-trivial solutions to the zero mode equation on a closed spin manifold \[D \varphi=iA\cdot \varphi,\] we first provide a simple proof for the sharp inequality \eq{ \norm{A}_{L^n}^2 \ge \frac {n}{4(n-1)} Y(M,[g]), } where $Y(M,[g])$ is the Yamabe constant of $(M,g)$, which was obtained by Frank-Loss and Reuss. Then we classify completely the equality case by proving that equality holds if and only if $\varphi$ is a Killing spinor, and if and only if $(M,g)$ is a Sasaki-Einstein manifold with $A$ (up to scaling) as its Reeb field and $\varphi$ a vacuum up to a conformal transformation. More generalizations have been also studied.