Strong coupling expansion in $\mathcal{N}=2$ superconformal theories and the Bessel kernel

Abstract

We consider strong 't Hooft coupling expansion in special four-dimensional $\mathcal{N}=2$ superconformal models that are planar-equivalent to $\mathcal{N}=4$ super Yang-Mills theory. Various observables in these models that admit localization matrix model representation can be expressed at large $N$ in terms of a Fredholm determinant of a Bessel operator. The latter previously appeared in the study of level spacing distributions in matrix models and, more recently, in four-point correlation functions of infinitely heavy half-BPS operators in planar $\mathcal{N}=4$ SYM. We use this relation and a suitably generalized Szego-Akhiezer-Kac formula to derive the strong 't Hooft coupling expansion of the leading corrections to free energy, half-BPS circular Wilson loop, and certain correlators of chiral primaries operators in the $\mathcal{N}=2$ models. This substantially generalizes partial results in the literature and represents a challenge for dual string theory calculations in AdS/CFT context. We also demonstrate that the resulting strong-coupling expansions suffer from Borel singularities and require adding non-perturbative, exponentially suppressed corrections. As a byproduct of our analysis, we determine the non-perturbative correction to the above mentioned four-point correlator in planar $\mathcal{N}=4$ SYM.

65 pages. v2: misprints and eq.(4.20) corrected, note added; v3: minor clarifications