On four dimensional N=3 superconformal theories

TL;DR

The paper investigates the existence and properties of four-dimensional N=3 superconformal theories that do not enhance to N=4. By embedding N=3 in N=2 and analyzing anomaly relations, it proves the universal relation $a=c$ and determines explicit anomaly coefficients, suggesting tight constraints on such theories. It then shows that pure N=3 SCFTs cannot have any $N=3$-preserving exactly marginal deformations or non-R global symmetries, and it delineates the structure of their potential moduli spaces as Coulomb-branch–like, governed by specific N=3 multiplets whose bottom components have dimensions $\\Delta = a \\ge 3$. The results, together with further discussion of Coulomb branches and short multiplets, indicate strong rigidity for pure N=3 theories and guide future explorations via bootstrap or moduli-space classification.

Abstract

In this note we study four dimensional theories with N=3 superconformal symmetry, that do not also have N=4 supersymmetry. No examples of such theories are known, but their existence is also not ruled out. We analyze several properties that such theories must have. We show that their conformal anomalies obey a=c. Using the N=3 superconformal algebra, we show that they do not have any exactly marginal deformations preserving N=3 supersymmetry, or global symmetries (except for their R-symmetries). Finally, we analyze the possible dimensions of chiral operators labeling their moduli space.