Analyticity of the Free Energy of a Closed 3-Manifold

TL;DR

This paper proves that the free energy of any closed 3-manifold, related to the perturbative Chern-Simons invariant, is uniformly Gevrey-1, ensuring convergence properties and providing explicit formulas and asymptotic expansions.

Contribution

It establishes the Gevrey-1 nature of the free energy for all closed 3-manifolds and derives convergence results, with explicit calculations for Lens spaces and asymptotics for cubic graphs.

Findings

The free energy is uniformly Gevrey-1 for all closed 3-manifolds.

Genus g part of the free energy converges near zero, independent of g.

Explicit formula for free energy of Lens spaces and asymptotic expansion for cubic graphs.

Abstract

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 Painlev\'e differential equation, we obtain an asymptotic expansion for the number of cubic graphs to all orders, stengthening the results of Bender-Gao-Richmond.