Polyakov conjecture on the supertorus

We prove the Polyakov conjecture on the supertorus \((ST_2)\): we dermine an iterative solution at any order of the superconformal Ward identity and we show that this solution is resumed by the Wess-Zumino-Polyakov (WZP) action that describes the \((1,0)\) 2D-supergravity. The resolution of the superBeltrami equation for the Wess-Zumino (WZ) field is done by using on the one hand the Cauchy kernel defined on \(ST_2\) and on the other hand, the formalism developed to get the general solution on the supercomplex plane. Hence, we determine the n-points Green functions from the (WZP) action expressed in terms of the (WZ) field.