Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension

We prove global existence from $L^2$ initial data for a nonlinear Dirac equation known as the Thirring model [12]. Local existence in $H^s$ for $s>0$, and global existence for $s>\frac{1}{2}$, has recently been proven by Selberg and Tesfahun in [9] where they used $X^{s, b}$ spaces together with a type of null form estimate. In contrast, motivated by the recent work of Machihara, Nakanishi, and Tsugawa, [7] we first prove local existence in $L^2$ by using null coordinates, where the time of existence depends on the profile of the initial data. To extend this to a global existence result we need to rule out concentration of $L^2$ norm, or charge, at a point. This is done by decomposing the solution into an approximately linear component and a component with improved integrability. We then prove global existence for all $s\geqslant 0$.