Abstract
Extended super-Riemann surfaces are studied and their underlying superconformal properties are derived as solutions of an operator equation. Superconformal tensors and differential forms are discussed in detail and are shown to be classified by means of a triplet of integers or half-integers. The integration on super-Riemann surfaces is developed. Finally, we derive the solution of the N=4 anomaly compatible with the non-locality feature and discuss the necessary conditions for its vanishing. A heuristic geometrical interpretation of these conditions is also given.
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