Supersymmetric σ-model with torsion in supergravity background and critical dimensions for string theories

The N=1 locally supersymmetric non-linear sigma -model with the generalised Wess-Zumino-Witten term in two spacetime dimensions is constructed by the Noether procedure. The consideration is generalised to the N=2 locally supersymmetric sigma -model with torsion, using superconformal twisted chiral multiplets in the framework of the N=2, d=2 superconformal tensor calculus. The two-loop superconformal anomalies for the above theories are calculated and the critical dimensions for various N-extended Fermi string theories in the same approximation are derived.