The cubic Dirac equation: Small initial data in H^(frac12)(mathbb{R}²)

Global well-posedness and scattering for the cubic Dirac equation with small initial data in the critical space $H^{\frac12}(\mathbb{R}^2)$ is established. The proof is based on a sharp endpoint Strichartz estimate for the Klein-Gordon equation in dimension $n=2$, which is captured by constructing an adapted systems of coordinate frames.