Analyticity of the Free Energy of a Closed 3-Manifold

The free energy of a closed 3-manifold is a 2-parameter formal power series which encodes the perturbative Chern–Simons invariant (also known as the LMO invariant) of a closed 3-manifold with gauge group $U(N)$ for arbitrary $N$. We prove that the free energy of an arbitrary closed 3-manifold is uniformly Gevrey-$1$. As a corollary, it follows that the genus $g$ part of the free energy is convergent in a neighborhood of zero, independent of the genus. Our results follow from an estimate of the LMO invariant, in a particular gauge, and from recent results of Bender–Gao–Richmond on the asymptotics of the number of rooted maps for arbitrary genus. We illustrate our results with an explicit formula for the free energy of a Lens space. In addition, using the Painlevé differential equation, we obtain an asymptotic expansion for the number of cubic graphs to all orders, stengthening the results of Bender–Gao–Richmond.