Non-planar corrections in orbifold/orientifold $\mathcal N=2$ superconformal theories from localization

We study non-planar corrections in two special $\mathcal N=2$ superconformal $SU(N)$ gauge theories that are planar-equivalent to $\mathcal N=4$ SYM theory: two-nodes quiver model with equal couplings and $\mathcal N=2$ vector multiplet coupled to two hypermultiplets in rank-2 symmetric and antisymmetric representations. We focus on two observables in these theories that admit representation in terms of localization matrix model: free energy on 4-sphere and the expectation value of half-BPS circular Wilson loop. We extend the methods developed in arXiv:2207.11475 to derive a systematical expansion of non-planar corrections to these observables at strong 't Hooft coupling constant $\lambda$. We show that the leading non planar corrections are given by a power series in $\lambda^{3/2}/N^2$ with rational coefficients. Sending $N$ and the coupling constant $\lambda$ to infinity with $\lambda^{3/2}/N^2$ kept fixed corresponds to the familiar double scaling limit in matrix models. We find that in this limit the observables in the two models are related in a remarkably simple way: the free energies differ by the factor of $2$, whereas the Wilson loop expectation values coincide. Surprisingly, these relations hold only at strong coupling, they are not valid in the weak coupling regime. We also discuss a dual string theory interpretation of the leading corrections to the free energy in the double scaling limit suggesting their relation to curvature corrections in type IIB string effective action.