Analyticity of the Free Energy of a Closed 3-Manifold

Stavros Garoufalidis; Thang T. Q. Le; Marcos Marino

doi:10.3842/sigma.2008.080

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arXiv · 0809.2572

Analyticity of the Free Energy of a Closed 3-Manifold

Stavros Garoufalidis·Thang T. Q. Le·Marcos Marino

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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 Painlevé 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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Stavros Garoufalidis, Thang T. Q. Le, Marcos Marino. 2008-11-15. Analyticity of the Free Energy of a Closed 3-Manifold. https://doi.org/10.3842/sigma.2008.080

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