Analyticity of the Free Energy of a Closed 3-Manifold
Stavros Garoufalidis, Thang T. Q. Le, Marcos Marino
备注：
This is a contribution to the Special Issue on Deformation Quantization, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/
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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Graduate School of Mathematical Sciences, the University of Tokyo（东京大学大学院数理科学研究科）；RIKEN iTHEMS（理化学研究所 iTHEMS）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Department of Mathematics, University of Southern California（南加州大学数学系）；Department of Mathematics, State University of New York, Stony Brook（纽约州立大学石溪分校数学系）
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