Analyticity of the Free Energy of a Closed 3-Manifold

Details Analyticity of the Free Energy of a Closed 3-Manifold Analyticity of the Free Energy of a Closed 3-Manifold

2008-09-15

arxiv.org/abs/0809.2572v5

SIGMA 4 (2008), 080, 20 pages

Mathematics

Geometric Topology

This is a contribution to the Special Issue on Deformation Quantization, published in SIGMA (Symmetry, Integrability and Geome

Scientific paper

10.3842/SIGMA.2008.080

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.

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