A limit for large $R$-charge correlators in $\mathcal{N}=2$ theories
Using supersymmetric localization, we study the sector of chiral primary operators $({\rm Tr} \, ϕ^2 )^n$ with large $R$-charge $4n$ in $\mathcal{N}=2$ four-dimensional superconformal theories in the weak coupling regime $g\rightarrow 0$, where $λ\equiv g^2n$ 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 $G_{2n}\approx F_{\infty}(λ) \left(\fracλ{2πe}\right)^{2n}\ n^α$, modulo $O(1/n)$ corrections, with $α=\frac{1}{2} \mathrm{dim}(\mathfrak{g})$ for a gauge algebra $\mathfrak{g}$ and a universal function $F_{\infty}(λ)$. 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.