# Operator Dimensions from Moduli

## TL;DR

This paper analyzes the operator spectrum of a 3D ${
m extbf{N}=2}$ superconformal field theory with moduli space, calculating anomalous dimensions of operators with large R-charge and revealing new structural insights about semi-short representations.

## Contribution

It introduces a method to compute anomalous dimensions of large R-charge operators and proves the existence of scalar semi-short states at all R-charges, linking superconformal symmetry with large-J expansion.

## Key findings

- The second-lowest scalar primary has dimension exactly J+1.
- The third-lowest scalar primary's dimension has a specific J-dependent correction.
- Positivity of a key coefficient is derived from unitarity and causality considerations.

## Abstract

We consider the operator spectrum of a three-dimensional ${\cal N} = 2$ superconformal field theory with moduli spaces of one complex dimension, such as the fixed point theory with three chiral superfields $X,Y,Z$ and a superpotential $W = XYZ$. By using the existence of an effective theory on each branch of moduli space, we calculate the anomalous dimensions of certain low-lying operators carrying large $R$-charge $J$. While the lowest primary operator is a BPS scalar primary, the second-lowest scalar primary is in a semi-short representation, with dimension exactly $J+1$, a fact that cannot be seen directly from the $XYZ$ Lagrangian. The third-lowest scalar primary lies in a long multiplet with dimension $J+2 - c_{-3} \, J^{-3} + O(J^{-4})$, where $c_{-3}$ is an unknown positive coefficient. The coefficient $c_{-3}$ is proportional to the leading superconformal interaction term in the effective theory on moduli space. The positivity of $c_{-3}$ does not follow from supersymmetry, but rather from unitarity of moduli scattering and the absence of superluminal signal propagation in the effective dynamics of the complex modulus. We also prove a general lemma, that scalar semi-short representations form a module over the chiral ring in a natural way, by ordinary multiplication of local operators. Combined with the existence of scalar semi-short states at large $J$, this proves the existence of scalar semi-short states at all values of $J$. Thus the combination of ${\cal N}=2$ superconformal symmetry with the large-$J$ expansion is more powerful than the sum of its parts.