Two-Loop Superstrings in Hyperelliptic Language II: the Vanishing of the   Cosmological Constant and the Non-Renormalization Theorem
  The vanishing of the cosmological constant and the non-renormali-zation theorem are verified at two loops by explicit computation using the hyperelliptic language and the newly obtained chiral measure of D'Hoker and Phong. A set of identities is found which is used in the verification of the non-renormalization theorem and leads to a great simplification of the calculation of the four-particle amplitude at two loops. 
Zhu-Jun Zheng, Jun-Bao Wu, Chuan-Jie Zhu