Higher Order Deformations of Complex Structures

TL;DR

This paper develops exact formulas for higher order deformations of complex structures on Riemann surfaces, with applications to superstring theory and conformal field theories, providing tools for precise amplitude calculations.

Contribution

It derives all-order deformation formulas for period matrices, Green's functions, and correlation functions, linking complex structure deformations to superstring amplitude computations.

Findings

Exact formulas for deformations of period matrices and Green's functions.

Representation of deformations via the stress tensor valid to all orders.

Application to superstring amplitudes at two-loop and higher orders.

Abstract

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.