On the $1/c$ expansion in $2d$ CFTs with degenerate operators

Abstract: We analytically determine the large central charge asymptotic expansion of the Virasoro conformal blocks entering in four-point functions with external degenerate operators on a sphere in $2d$ CFTs, and study its resurgence properties as a function of the conformal cross-ratio $z$. We focus on the cases of four heavy $(2,1)$ degenerate operators, and two $(2,1)$ heavy degenerate ones plus two arbitrary light operators. The $1/c$ asymptotic series is Borel summable for generic values of $z$, but it jumps when a Stokes line is crossed. Starting from the $1/c$ series of the identity block, we show how a resurgent analysis allows us to completely determine the other Virasoro block and in fact to reconstruct the full correlator. We also show that forbidden singularities, known to exist in correlators with two heavy and two light operators, appear with four heavy operators as well. In both cases, they are turning points emanating Stokes lines, artefacts of the asymptotic expansion, and we show how they are non perturbatively resolved. More general correlators and implications for gravitational theories in $\text{AdS}_{3}$ are briefly discussed. Our results are based on new asymptotic expansions for large parameters $(a,b,c)$ of certain hypergeometric functions $\,_2 F_1(a,b,c;z)$ which can be useful in general.