A remark on nonlinear Dirac equations

Abstract

For an n n -dimensional spin manifold M M with a fixed spin structure and a spinor bundle Σ M \Sigma M , we prove an ϵ \epsilon -regularity theorem for weak solutions to the nonlinear Dirac equation \[ ∂ ̸ ψ = H j k l ⟨ ψ j , ψ k ⟩ ψ l , ot \partial \psi = H_{jkl}\langle \psi ^j, \psi ^k\rangle \psi ^l, \] of cubic nonlinearity. In particular, it implies that any weak solution is smooth when n = 2 n=2 , which answers a question raised by Chen, Jost, and Wang.