Resurgent analysis associates, to a perturbative series, a rich structure of exponentially small corrections, or trans-series. It turns out that, in some cases, this structure encodes the counting of BPS states in an appropriate quantum field theory. I will review this connection in the context of the exact WKB method, and I will present a richer example: the resurgent structure of the perturbative Chern-Simons invariants of hyperbolic knots. In this example, singularities in the Borel plane appear in infinite towers, and the corresponding Stokes constants are integer numbers. This gives a new mechanism for obtaining integer invariants in these theories. We compute/conjecture these integers in terms of q-series, and we relate them toBPS counting and to the DGG index of the knot. If time permits, I will sketch how some of these structures appear in topological string theory.

No abstract available

Very often in physics and mathematics we have an asymptotic expansion in small parameter which is divergent because the coefficients have factorial growth an ∼ n! exp(O(n)). The property of resurgence means that the Borel transform ∑n anζn/n! admits an analytic continuation, which allows to reconstruct the exact value of the original divergent series.

I will review the theory of resurgence and resummation for the asymptotic expansions of exponential integrals. I will illustrate the theory by finite-dimensional examples (Airy function, Stirling formula for Γ-function), and infinite-dimensional ones (heat kernel, quantum Chern-Simons theory).