Polyakov conjecture on the supertorus

TL;DR

This paper proves the Polyakov conjecture on the supertorus by deriving an iterative solution to the superconformal Ward identity, showing it is encapsulated by the Wess-Zumino-Polyakov action, and computing n-point Green functions.

Contribution

It provides a rigorous proof of the Polyakov conjecture on the supertorus and explicitly constructs solutions to the superconformal Ward identity using advanced supergeometry techniques.

Findings

Solution to the superBeltrami equation using the supertorus kernel

Expression of the WZP action in terms of the WZ field

Derivation of n-point Green functions from the WZP action

Abstract

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.