# Marginal Deformations of 3d N=4 Linear Quiver Theories

## TL;DR

This paper investigates superconformal deformations of 3d N=4 theories, identifying mixed-branch operators and their moduli space, with implications for holography and supergravity descriptions.

## Contribution

It provides a detailed analysis of marginal operators in 3d N=4 theories using superconformal index computations, revealing their structure and holographic interpretation.

## Key findings

- Mixed moduli are double-string operators in (Adjoint, Adjoint) representation.
- Superconformal index unambiguously extracts all marginal N=2 operators.
- Gauged supergravities can capture the entire moduli space including boundary condition moduli.

## Abstract

We study superconformal deformations of the $T_\rho^{\hat\rho}[SU(N)]$ theories of Gaiotto-Hanany-Witten, paying special attention to mixed-branch operators with both electrically- and magnetically-charged fields. We explain why all marginal ${\cal N}=2$ operators of an ${\cal N}=4$ CFT$_3$ can be extracted unambiguously from the superconformal index. Computing the index at the appropriate order we show that the mixed moduli in $T_\rho^{\hat\rho}[SU(N)]$ theories are double-string operators transforming in the (Adjoint, Adjoint) representation of the electric and magnetic flavour groups, up to some overcounting for quivers with abelian gauge nodes. We comment on the holographic interpretation of the results, arguing in particular that gauged supergravities can capture the entire moduli space if, in addition to the (classical) parameters of the background solution, one takes also into account the (quantization) moduli of boundary conditions.