A remark on nonlinear Dirac equations

For an $n$-dimensional spin manifold $M$ with a fixed spin structure and a spinor bundle $\Sigma M$, we prove an $\epsilon$-regularity theorem for weak solutions to the nonlinear Dirac equation

$$\slashed\partial\psi= H_{jkl}\langle \psi^j, \psi^k\rangle \psi^l,$$

of cubic nonlinearity. In particular, it implies that any weak solution is smooth when $n=2$, which answers a question raised by Chen, Jost, and Wang.