Abstract
AbstractRecent 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.
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