Two-Loop Superstrings in Hyperelliptic Language I: the Main Results

Following the new gauging fixing method of D'Hoker and Phong, we study two-loop superstrings in hyperelliptic language. By using hyperelliptic representation of genus 2 Riemann surface we derive a set of identities involving the Szeg\"o kernel. These identities are used to prove the vanishing of the cosmological constant and the non-renormalization theorem point-wise in moduli space by doing the summation over all the 10 even spin structures. Modular invariance is maintained at every stage of the computation explicitly. The 4-particle amplitude is also computed and an explicit expression for the chiral integrand is obtained. We use this result to show that the perturbative correction to the $R^4$ term in type II superstring theories is vanishing at two loops. In this paper, a summary of the main results is presented with detailed derivations to be provided in two subsequent publications.