N = 1 , 2 d - induced Polyakov supergravity on a super Riemann surface

An effective action describing N=1, 2d-induced supergravity in the chi-ral gauge is obtained on the supertorus and on a compact SRS without boundary of arbitrary genus. This action integrates the superconformal Ward identity for the superdiffeomorphism anomaly; it yields the last step in the extension to more complicated topologies of the action initially obtained by A.M. Polyakov on the ordinary complex plane. Constructing the Weierstrass ζ function as the Cauchy kernel on the supertorus allows for solving the super Beltrami equations and thereby for computing the stress-energy tensor and 2-and 3-point Green functions. Using this result one checks the Polyakov conjecture which states that the Polyakov action resums the renormalized perturbative series. The stress-energy tensor is also constructed on a SRS. Obtaining these results was made possible by analysing new geometrical notions such as single-valued superdifferentials via the super Riemann-Roch theorem, multi-valued superdifferentials, a non-Berezinian method for integrating Grassmann variables, super Stokes theorem, Cauchy kernels and covariant derivatives.