Higher order deformations of complex structures

Abstract

Deformations of complex structures by f inite Beltrami dif ferentials are considered on general Riemann surfaces. Exact formulas to any f ixed order are derived for the corresponding deformations of the period matrix, Green\u2019s functions, and correlation functions in conformal f ield theories with vanishing total central charge. The stress tensor is shown to give a simple representation of these deformations valid to all orders. Such deformation formulas naturally enter into the evaluation of superstring amplitudes at twoloop order with Ramond punctures, and at higher loop order, in the supergravity formulation of the RNS superstring.