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Analyticity of the Free Energy of a Closed 3Manifold 

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Published in  Symmetry, Integrability and Geometry: Methods and Applications. 2008, vol. 4, no. 080, p. 20 pp.  
Abstract  The free energy of a closed 3manifold is a 2parameter formal power series which encodes the perturbative Chern–Simons invariant (also known as the LMO invariant) of a closed 3manifold with gauge group U(N) for arbitrary N. We prove that the free energy of an arbitrary closed 3manifold is uniformly Gevrey1. 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.  
Keywords  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 — Painlev´e I — WKB — Asymptotic expansions — Transseries — Riemann–Hilbert problem  
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Citation (ISO format)  GAROUFALIDIS, Stavros, LÊ, Thang T.Q., MARINO BEIRAS, Marcos. Analyticity of the Free Energy of a Closed 3Manifold. In: Symmetry, Integrability and Geometry: Methods and Applications, 2008, vol. 4, n° 080, p. 20 pp. https://archiveouverte.unige.ch/unige:9745 