On Conformal Deformations

TL;DR

This paper investigates the structure and dimension of the conformal moduli space in supersymmetric conformal theories, identifying obstructions to deformations and illustrating with membrane field theory examples.

Contribution

It establishes a formula for the dimension of the conformal moduli space using an index and discusses the role of D-terms as obstructions in supersymmetric contexts.

Findings

The moduli space dimension is given by a specific index.

D-terms act as obstructions to conformal deformations.

Membrane field theory deformations are characterized by a specific quotient.

Abstract

For a conformal theory it is natural to seek the conformal moduli space, M_c to which it belongs, generated by the exactly marginal deformations. By now we should have the tools to determine M_c 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 M_c are discussed.