A Short Primer on the Half-Wave Maps Equation

We review the current state of results about the half-wave maps equation on the domain ${\mathbb{R}}^d$ with target ${\mathbb{S}}^2$. In particular, we focus on the energy-critical case $d=1$, where we discuss the classification of traveling solitary waves and a Lax pair structure together with its implications (e.,g.~invariance of rational solutions and infinitely many conservation laws on a scale of homogeneous Besov spaces). Furthermore, we also comment on the one-dimensional space-periodic case. Finally, we list some open problem for future research.