  We prove global existence from $L^2$ initial data for a nonlinear Dirac
equation known as the Thirring model. Local existence in $H^s$ for $s>0$, and
global existence for $s>1/2$, has recently been proven by Selberg and Tesfahun
by using $X^{s, b}$ spaces together with a type of null form estimate. In
contrast, motivated by the recent work of Machihara, Nakanishi, and Tsugawa, 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 $s>0$.
