Two-loop superstrings in hyperelliptic language I: the main results

Following the new gauging fixing method of DʹHoker 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 Szegö 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 R4 term in type II superstring theories is vanishing at two loops. In this Letter, a summary of the main results is presented with detailed derivations to be provided in two subsequent publications.