Carrollian $\mathbb{R}^\times$-bundles II: Sigma Models on Event Horizons

Carrollian field theories are usually understood as limits of relativistic theories. In this note, we use Carrollian $\mathbb{R}^\times$-bundles equipped with a principal connection to construct Carrollian sigma models intrinsically. The resulting theories are neither ``electric'' nor ``magnetic'' in the usual sense. As a physically suggestive example, we derive a Carrollian wave equation governing the dynamics of a scalar field on the event horizon of a Schwarzschild black hole.