On Conformal Deformations II

TL;DR

The paper advocates an index-based description of the conformal moduli space ${\cal M}_c$ in 4d ${\cal N}=1$ theories, identifying ${\cal M}_c$ locally with the holomorphic quotient of the space of supermarginals by the complexified global group $G_\mathbb{C}$ via D-terms. It shows that, at zero and weak coupling, this index matches the Leigh-Strassler construction, and it provides a practical method to compute ${\cal M}_c$ by constructing the q-matrix of charges, determining $G_0$, and performing the holomorphic quotient $R/G_\mathbb{C}$; in many examples it finds additional exactly marginal deformations beyond LS. Across a variety of theories including ${\cal N}=4$, ${\cal N}=2$ setups, SQCD at special $N_f/N_c$ ratios, scalar theories, and quiver gauge theories, the method either reproduces LS results or reveals a larger ${\cal M}_c$, with the Konishi anomaly ensuring the index remains invariant as couplings are turned on. The work fortifies a physically well-defined, scheme-independent framework for conformal moduli and highlights the pivotal role of the global symmetry and D-terms, with implications for AdS/CFT and the structure of exactly marginal deformations in 4d CFTs.

Abstract

The conformal index counts the number of exactly marginal deformations. In 4d the index is given by the number of chiral primary operators of dimension 3 moded out by the complexified global group, where the quotient is defined as usual by imposing a D-term. Here we show its consistency with the Leigh-Strassler method for weakly coupled theories, and we test it against known examples. In several examples this method discovers extra exactly marginal deformations beyond those of Leigh-Strassler. [This is an unpublished paper dated 3.3.03.]

Figures (1)

• Figure 1: Lowest order (1-loop) contribution to the two point function of a field $\Phi$: (a) gauge boson contribution (b) Fermion loop with a superpotential vertex. Note that the diagrams have opposite signs due to the fermion loop.