Restoring reality for the self-dual N = 2 string

It is known that the critical N = (2,2) string describes 2 + 2 dimensional self-dual gravity in a non-covariant form, since it requires the choice of a complex structure in the target, which leaves only U(1,1) Lorentz symmetry. We briefly review picture-changing and spectral flow and show that the world-sheet Maxwell instantons individually break the Lorentz group further to SU(1,1). However, their contributions conspire to restore full SO(2,2) global symmetry if dilaton and axion fields are assembled in a null anti-self-dual two-form, denying them the status of Lorentz scalars. We present the fullySO (2,2) invariant tree-level three-point amplitude and the corresponding extension of the Plebanski action for self-dual gravity.