On Conformal Deformations

TL;DR

The paper addresses the problem of characterizing the conformal moduli space $\\mathcal{M}_c$ of supersymmetric conformal field theories, focusing on its dimension and local geometry. It introduces the conformal index as the index of the SUSY variation operator, identifying the number of deformations with the difference between zero modes and obstructions, so that $\\dim\\mathcal{M}_c = \\mathrm{Index}[\\delta_{susy}]$. A key result is that for theories with at least 8 super-conformal charges the D-term of the global symmetry provides the obstruction, enforcing a holomorphic quotient of the marginal parameter space. As a concrete instance, the membrane theory in 3d yields a local moduli space $\\mathbf{35}/SL(4,\\mathbb{C})$, illustrating the framework's power and its connection to AdS/CFT and Leigh-Strassler-type analyses. The approach offers a route to study the local geometry and global structure of $\\mathcal{M}_c$ and to relate field-theory deformations to gravity-dual moduli.

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.