Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 080, 20 pages      arXiv:0809.2572
Contribution to the Special Issue on Deformation Quantization

Analyticity of the Free Energy of a Closed 3-Manifold

Stavros Garoufalidis a, Thang T.Q. Lê a and Marcos Mariño b
a) School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA
b) Section de Mathématiques, Université de Genève, CH-1211 Genève 4, Switzerland

Received September 15, 2008, in final form November 06, 2008; Published online November 15, 2008

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.

Key words: Chern-Simons theory; perturbation theory; gauge theory; free energy; planar limit; Gevrey series; LMO invariant; weight systems; ribbon graphs; cubic graphs; lens spaces; trilogarithm; polylogarithm; Painlevé I; WKB; asymptotic expansions; transseries; Riemann-Hilbert problem.

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