Deformations of Superconformal Theories

TL;DR

This work addresses how to systematically classify supersymmetry-preserving deformations of unitary SCFTs in $3\le d\le 6$ without relying on Lagrangians. It hinges on the structure of superconformal multiplets, distinguishing long and short multiplets and identifying top components that yield allowed deformations, with the Racah-Speiser algorithm aiding level-by-level decompositions. A central finding is that many theories harbor deformations in multiplets containing conserved currents, potentially altering the SUSY algebra through central or non-central charges, while highly supersymmetric theories often lack relevant or marginal deformations. The results yield both general constraints and concrete tables across dimensions, and are used to derive implications for moduli-space effective actions via irrelevant deformations. This framework provides model-independent insights into the landscape of SCFT deformations and their impact on low-energy dynamics.

Abstract

We classify possible supersymmetry-preserving relevant, marginal, and irrelevant deformations of unitary superconformal theories in $d \geq 3$ dimensions. Our method only relies on symmetries and unitarity. Hence, the results are model independent and do not require a Lagrangian description. Two unifying themes emerge: first, many theories admit deformations that reside in multiplets together with conserved currents. Such deformations can lead to modifications of the supersymmetry algebra by central and non-central charges. Second, many theories with a sufficient amount of supersymmetry do not admit relevant or marginal deformations, and some admit neither. The classification is complicated by the fact that short superconformal multiplets display a rich variety of sporadic phenomena, including supersymmetric deformations that reside in the middle of a multiplet. We illustrate our results with examples in diverse dimensions. In particular, we explain how the classification of irrelevant supersymmetric deformations can be used to derive known and new constraints on moduli-space effective actions.