Restoring Reality for the Self-Dual N=2 String

TL;DR

This paper explores how world-sheet instantons and field configurations restore full Lorentz symmetry in the critical N=(2,2) string, leading to an SO(2,2) invariant formulation of self-dual gravity.

Contribution

It demonstrates the restoration of full Lorentz symmetry in the N=(2,2) string through instanton effects and constructs an SO(2,2) invariant amplitude and action for self-dual gravity.

Findings

Restoration of SO(2,2) symmetry via instantons and field configurations.

Explicit SO(2,2) invariant three-point amplitude.

Extension of Plebanski action for self-dual gravity.

Abstract

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.