Аннотация:
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.
Образец цитирования:
Stavros Garoufalidis, Thang T. Q. Lê, Marcos Mariño, “Analyticity of the Free Energy of a Closed 3-Manifold”, SIGMA, 4 (2008), 080, 20 pp.
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\by Stavros Garoufalidis, Thang T.~Q.~L\^e, Marcos Mari\~no
\paper Analyticity of the Free Energy of a~Closed 3-Manifold
\jour SIGMA
\yr 2008
\vol 4
\papernumber 080
\totalpages 20
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Эта публикация цитируется в следующих 16 статьяx:
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Garoufalidis S., Gu J., Marino M., “The Resurgent Structure of Quantum Knot Invariants”, Commun. Math. Phys., 386:1 (2021), 469–493
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Borot G., Eynard B., Weisse A., “Root systems, spectral curves, and analysis of a Chern–Simons matrix model for Seifert fibered spaces”, Sel. Math.-New Ser., 23:2 (2017), 915–1025
Couso-Santamaria R., Schiappa R., Vaz R., “On Asymptotics and Resurgent Structures of Enumerative Gromov-Witten Invariants”, Commun. Number Theory Phys., 11:4 (2017), 707–790
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Aniceto I., Russo J.G., Schiappa R., “Resurgent Analysis of Localizable Observables in Supersymmetric Gauge Theories”, J. High Energy Phys., 2015, no. 3, 172
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Garoufalidis S., Its A., Kapaev A., Marino M., “Asymptotics of the Instantons of Painlevé I”, Int Math Res Not, 2012, no. 3, 561–606
Aniceto I., Schiappa R., Vonk M., “The Resurgence of Instantons in String Theory”, Commun. Number Theory Phys., 6:2 (2012), 339–496
Takata T., “On the SO(N) and Sp(N) free energy of a rational homology three-sphere”, Internat. J. Math., 22:4 (2011), 465–482
Bender E.A., Gao Zh., “Asymptotic enumeration of labelled graphs by genus”, Electron. J. Combin., 18:1 (2011), P13
Gao Zhicheng, “A formula for the bivariate map asymptotics constants in terms of the univariate map asymptotics constants”, Electron. J. Combin., 17:1 (2010), R155, 14 pp.
Garoufalidis S., Mariño M., “Universality and asymptotics of graph counting problems in non-orientable surfaces”, J. Combin. Theory Ser. A, 117:6 (2010), 715–740
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