On conformal deformations

For a conformal theory it is natural to seek the conformal moduli space, Mc to which it belongs, generated by the exactly marginal deformations. By now we should have the tools to determine Mc in the presence of enough supersymmetry. Here it is shown that its dimension is determined in terms of a certain index. Moreover, the D-term of the global group is an obstruction for deformation, in presence of a certain amount of preserved supersymmetry. As an example we find that the deformations of the membrane (3d) field theory, under certain conditions, are in 35/SL(4,C). Other properties including the local geometry of Mc are discussed. (author)