Analyticity of the Free Energy of a Closed 3-Manifold

Understand

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.