The quantum geometry of N = (2,2) non-linear σ-models

We consider a general N = (2,2) non-linear σ-model in (2,2) superspace. Depending on the details of the complex structures involved, an off-shell description can be given in terms of chiral, twisted chiral and semi-chiral superfields. Using superspace techniques, we derive the conditions the potential has to satisfy in order to be ultra-violet finite at one loop. We pay particular attention to the effects due to the presence of semi-chiral superfields. A complet description of N = (2,2) strings follows from this.