Derivation of the half-wave maps equation from Calogero–Moser spin systems

We prove that the energy-critical half-wave maps equation\[$\partial_t \mathbf {S} = \mathbf {S} \times |\nabla |\mathbf {S}, \quad (\mathit{t}, \mathit{x}) \in \mathbb R \times \mathbb T$\]arises as an effective equation in the continuum limit of completely integrable Calogero–Moser classical spin systems with inverse square $1/r^2$ interactions on the circle. We study both the convergence to global-in-time weak solutions in the energy class as well as short-time strong solutions of higher regularity. The proofs are based on Fourier methods and suitable discrete analogues of fractional Leibniz rules and Kato–Ponce–Vega commutator estimates.