Scaling dimensions of Coulomb branch operators of 4d N=2 superconformal field theories

Abstract

Under reasonable assumptions about the complex structure of the set of singularities on the Coulomb branch of $\mathcal{N}=2$ superconformal field theories, we present a relatively simple and elementary argument showing that the scaling dimension, $\Delta$, of a Coulomb branch operator of a rank $r$ theory is allowed to take values in a finite set of rational numbers$\Delta \in \left[\frac{n}{m}\right|n,m\in \mathbb{N},0<m\le n,gcd(n,m)=1,\phi (n)\le 2r]$ where $\phi (n)$ is the Euler totient function. The maximal dimension grows superlinearly with rank as ${\Delta }_{\text{max}}\sim r\ln \ln r$. This agrees with the recent result of Caorsi and Cecotti.

6 pages, 1 figure

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