  We show that the cubic Dirac equation with zero mass is globally well-posed
for small data in the scale invariant space H^{\frac{n-1}{2}}(R^n) for n=2, 3.
The proof proceeds by using the Fierz identities to rewrite the equation in a
form where the null structure of the system is readily apparent. This null
structure is then exploited via bilinear estimates in spaces based on the null
frame spaces of Tataru. We hope that the spaces and estimates used here can be
applied to other nonlinear Dirac equations in the scale invariant setting. Our
work complements recent results of Bejenaru-Herr who proved a similar result
for n=3 in the massive case.
