A matrix-model approach to integrated correlators in a $\mathcal{N}=2$ SYM theory

In a $\mathcal{N}=2$ superconformal gauge theory with matter hypermultiplets transforming in the symmetric and anti-symmetric representations of SU($N$), we study the integrated correlators of two Coulomb-branch operators and two moment-map operators using localization. In the corresponding matrix model we identify the operator associated with the integrated insertions of moment-map operators and provide for it an exact expression valid for all values of the coupling constant in the planar limit. This allows us to study the integrated correlators at strong-coupling where we show that they behave as the 2-point functions of the Coulomb-branch operators, up to an overall constant dependent only on the conformal dimensions of the latter. The strong-coupling relation between integrated correlators and 2-point functions turns out to be the same as in $\mathcal{N}=4$ SYM at large $N$, despite the reduced amount of supersymmetry in our theory.