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Higher Order Deformations of Complex Structures
Deformations of complex structures by finite Beltrami differentials are
considered on general Riemann surfaces. Exact formulas to any fixed order are
derived for the corresponding deformations of the period matrix, Green's
functions, and correlation functions in conformal field 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
two-loop order with Ramond punctures, and at higher loop order, in the
supergravity formulation of the RNS superstring
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