On uniqueness for hyperbolic half-wave maps in dimension $d \geq 3$

Half-wave maps appear in the physics literature as the continuum limit of Calogero-Moser spin systems. We obtain a uniqueness result for the Half-Wave Maps equation in dimension $d \ge 3$ in the natural energy class with $\mathbb{H}^2$ target. In the proof, we differentiate in time to arrive at a wave-type equation and isometrically embed $\mathbb{H}^2$ into some $\mathbb{R}^m$ using the Nash embedding theorem. Relying on geometric properties of $\mathbb{H}^2$, combined with fractional Leibniz rules and commutator estimates, we then use a Gr\"{o}nwall inequality argument to obtain uniqueness.