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 fully SO(2, 2) invariant tree-level three-point amplitude and the corresponding extension of the Plebanski action for self-dual gravity.