Resurgence of the Kontsevich-zagier Power Series

Abstract. Perturbative quantum field theory associates formal power series invariants to knotted objects, that is to knots or homology 3-spheres. These formal power series are known to be Gevrey, and are expected to be factorially divergent, and somehow linked to quantum invariants of knotted objects at complex roots of unity. The latter are the well-known Witten-Reshetikhin-Turaev invariants of 3-manifolds, and the Kashaev invariant of knots in 3-space. In the paper, we formulate a resurgence conjecture for the formal power series of knotted objects, which among other things explains the existence of asymptotic (and more generally, transseries) expansions of the quantum and perturbative invariants. The bulk of the paper considers a test case of the above conjecture, namely a complete proof of our conjecture for the simplest non-trivial knot: the trefoil 31. The formal power series of this knot is a power series introduced by Zagier-Kontsevich. We give an explicit formula for its Borel transform which proves resurgence in a manifest way. This is a new construction of resurgent functions that do not seem to satisfy any differential equations (linear or not) with polynomial coefficients. Our results extend without change to the case of torus knots Seifert fibered 3-manifolds, as we explain for the case of the Poincaré homology sphere. In a subsequent publication we will study resurgence for a class of geometrically interesting knotted 3-dimensional objects that include the simplest hyperbolic 41 knot.