Higher genus superstring amplitudes and the measure on the moduli space

Recent derivations of completely gauge-invariant two-loop superstring amplitudes renewed the interest in the possible form of higher genus contributions. In this respect, two main points are discussed: a. At genus greater than 3, the Schottky problem actually represents a formidable obstruction for the explicit expression of the integration measure on the moduli space in terms of Riemann period matrices. As a first step toward such a construction, a modular invariant measure for all genera is derived, corresponding to the restriction to the Schottky locus of the Siegel metric on the upper half-space. b. A class of natural generalizations of the 2 loop D'Hoker and Phong formula is discussed for the higher genus contributions to the 4-gravitons amplitude in type II theories. Such expressions also satisfy the non-trivial requirements of modular invariance and gauge slice independence. © 2007 WILEY-VCH Verlag GmbH & Co. KGaA.