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

Abstract

Using supersymmetric localization, we study the sector of chiral primary operators $(\mathrm{Tr}{\varphi }^2{)}^n$ with large $R$-charge $4n$ in $\mathcal{N}=2$ four-dimensional superconformal theories in the weak coupling regime $g\to 0$, where $\lambda \equiv g^{2}n$ 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 $\alpha =\frac{1}{2}\dim (\mathfrak{g})$ 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 $\mathcal{N}=2$ $\mathfrak{su}(N)$ gauge theory with $2N$ fundamental hypermultiplets.

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