Higher Order Deformations of Complex Structures

Understand

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.