The effective action and geometry of closed N = 2 strings

N = 2 closed strings have been recently divided in [3] to two T-dual families denoted by α and β. In (2,2) signature both families have one scalar in the spectrum. The scalar in the β-string is known to be a deformation of the target space Kähler potential and the dynamics is that of self-dual gravity. In this paper we compute the effective action of the scalar in the α-string. The scalar is a deformation of a potential that determines the metric, torsion and dilaton. The scalar is free and the dynamics is that of a self-dual curvature with torsion. The result is in agreement with a σ-model computation of Hull.