The Geometry of N = 2 Strings with Torsion

N = 2 string theories are formulated in space-times with 2 space and 2 time dimensions. If the world-sheet matter system consists of 2 chiral superfields, the space-time is Kahler and the dynamics are those of anti-self-dual gravity. If instead one chiral superfield and one twisted chiral superfield are used, the space-time is a hermitian manifold with torsion and a dilaton. The string spectrum consists of a scalar, which is a potential K determining the metric, torsion and dilaton. The dynamics imply that the curvature with torsion is anti-self-dual, and an action is found for the potential K. It is argued that any N = 2 sigma-model with twisted chiral multiplets in any dimension can be deformed to a conformally invariant theory if the lowest order contribution to the conformal anomaly vanishes.