$\mathcal{N}=2$ conformal gauge theories at large R-charge: the $SU(N)$ case

Abstract

Conformal theories with a global symmetry may be studied in the double scaling regime where the interaction strength is reduced while the global charge increases. Here, we study generic 4d $\mathcal{N}=2$ $SU(N)$ gauge theories with conformal matter content at large R-charge $Q_{\mathrm{R}}\to \infty$ with fixed 't Hooft-like coupling $\kappa =Q_{\mathrm{R}}{g}_{\mathrm{YM}}^2$. Our analysis concerns two distinct classes of natural scaling functions. The first is built in terms of chiral/anti-chiral two-point functions. The second involves one-point functions of chiral operators in presence of $\frac{1}{2}$-BPS Wilson-Maldacena loops. In the rank-1 $SU(2)$ case, the two-point sector has been recently shown to be captured by an auxiliary chiral random matrix model. We extend the analysis to $SU(N)$ theories and provide an algorithm that computes arbitrarily long perturbative expansions for all considered models, parametric in the rank. The leading and next-to-leading contributions are cross-checked by a three-loops computation in $\mathcal{N}=1$ superspace. This perturbative analysis identifies maximally non-planar Feynman diagrams as the relevant ones in the double scaling limit. In the Wilson-Maldacena sector, we obtain closed expressions for the scaling functions, valid for any rank and $\kappa$. As an application, we analyze quantitatively the large 't Hooft coupling limit $\kappa \gg 1$ where we identify all perturbative and non-perturbative contributions. The latter are associated with heavy electric BPS states and the precise correspondence with their mass spectrum is clarified.

52 pages, 13 pdf figures. v2: expanded bibliography

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