Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 047, 14 pages      arXiv:1502.03673      https://doi.org/10.3842/SIGMA.2015.047
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

Higher Order Deformations of Complex Structures

Eric D'Hoker a and Duong H. Phong b
a) Department of Physics and Astronomy, University of California, Los Angeles 90024, USA
b) Department of Mathematics, Columbia University, New York, NY 10027, USA

Received March 27, 2015, in final form June 15, 2015; Published online June 23, 2015

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.

Key words: Beltrami differentials; deformations of covariant derivatives; stress tensor; conformal invariance.

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