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Global well-posedness and scattering for the cubic Dirac equation with smallinitial 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-Gordonequation in dimension $n=2$, which is captured by constructing an adaptedsystems of coordinate frames.

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