A limit for large $R$-charge correlators in $\mathcal{N}=2$ theories

TL;DR

This paper investigates the asymptotic behavior of large R-charge correlators in four-dimensional  superconformal theories using supersymmetric localization, revealing a universal scaling form and deriving new formulas for partition functions and correlators.

Contribution

It introduces a universal asymptotic formula for large R-charge correlators in  theories and provides new explicit formulas for partition functions and perturbative correlators.

Findings

Correlation functions scale as ^{2n} n^{rac{1}{2} ext{dim}(rak{g})} with a universal function F_\u221e(\u03bb).

Derived new formulas for partition functions in   gauge theories with 2N hypermultiplets.

Identified a simple asymptotic behavior in the weak coupling limit for large R-charge operators.

Abstract

Using supersymmetric localization, we study the sector of chiral primary operators $({\rm Tr} \, \phi^2 )^n$ with large $R$-charge $4n$ in $\mathcal{N}=2$ four-dimensional superconformal theories in the weak coupling regime $g\rightarrow 0$, where $\lambda\equiv g^2n$ is kept fixed as $n\to\infty $, $g$ representing the gauge theory coupling(s). In this limit, correlation functions $G_{2n}$ of these operators behave in a simple way, with an asymptotic behavior of the form $G_{2n}\approx F_{\infty}(\lambda) \left(\frac{\lambda}{2\pi e}\right)^{2n}\ n^\alpha $, modulo $O(1/n)$ corrections, with $\alpha=\frac{1}{2} \mathrm{dim}(\mathfrak{g})$ for a gauge algebra $\mathfrak{g}$ and a universal function $F_{\infty}(\lambda)$. As a by-product we find several new formulas both for the partition function as well as for perturbative correlators in ${\cal N}=2$ $\mathfrak{su}(N)$ gauge theory with $2N$ fundamental hypermultiplets.