We exhibit a close relation between vortex configurations on the 2-sphere and magnetic zero-modes of the Dirac operator on $$\mathbb {R}^3$$R3 which obey an additional nonlinear equation. We show that both are best understood in terms of the geometry induced on the 3-sphere via pull-back of the round geometry with bundle maps of the Hopf fibration. We use this viewpoint to deduce a manifestly smooth formula for square-integrable magnetic zero-modes in terms of two homogeneous polynomials in two complex variables.

中文翻译：

---

磁零模式、涡流和 Cartan 几何

我们展示了 $$\mathbb {R}^3$$R3 上的狄拉克算子的 2 球体上的涡旋配置和磁零模式之间的密切关系，其遵循附加的非线性方程。我们表明，通过圆形几何形状的回拉以及 Hopf 纤维化的束图，在 3 球体上诱导的几何形状方面可以最好地理解两者。我们使用这个观点根据两个复变量中的两个齐次多项式推导出可平方积分磁零模式的明显平滑公式。