Scaling dimensions of Coulomb branch operators of 4d N=2 superconformal field theories
Philip C. Argyres, Mario Martone
TL;DR
The paper shows that for 4d ${\mathcal N}=2$ SCFTs of rank $r$, the CB operator dimensions $\{∆_j\}$ lie in a finite rational set $\{\frac{n}{m}\mid {\varphi}(n)\le 2r,\; \gcd(m,n)=1,\; 0<m\le n\}$ by leveraging the topology of CB singularities, EM-duality monodromies in ${\rm Sp}_D(2r,\mathbb{Z})$, and positivity of the CB metric. It demonstrates that all $∆_j$ are commensurate and, more strongly, rational, with unit-norm eigenvalues of monodromies constrained by cyclotomic polynomials ${\mathscr C}_n$; the maximal dimension scales as $∆_{\max}\sim 2r e^{\gamma}\ln\ln r$, in agreement with Caorsi and Cecotti. The work connects CB geometry, unipotent monodromies, and number-theoretic constraints to yield precise spectral data, with implications for large-$r$ central charges $a,c$ and potential holographic duals. Overall, it provides a robust, model-independent framework to bound and compute CB operator dimensions in a broad class of ${\mathcal N}=2$ SCFTs.
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, $Δ$, of a Coulomb branch operator of a rank $r$ theory is allowed to take values in a finite set of rational numbers$Δ\in \big[\frac{n}{m}\big|n,m\in\mathbb N, 0<m\le n, gcd(n,m)=1,\ \varphi(n)\le2r\big]$ where $\varphi(n)$ is the Euler totient function. The maximal dimension grows superlinearly with rank as $Δ_\text{max} \sim r \ln\ln r$. This agrees with the recent result of Caorsi and Cecotti.