Analyticity of the Free Energy of a Closed 3-Manifold
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 uniform...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2008 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2008
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/148975 |
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| Cite this: | Analyticity of the Free Energy of a Closed 3-Manifold / S. Garoufalidis, Thang T.Q. Lê, M. Mariño // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 55 назв. — англ. |
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Garoufalidis, S. Thang T.Q. Lê Mariño, M. 2019-02-19T12:21:34Z 2019-02-19T12:21:34Z 2008 Analyticity of the Free Energy of a Closed 3-Manifold / S. Garoufalidis, Thang T.Q. Lê, M. Mariño // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 55 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 57N10; 57M25 https://nasplib.isofts.kiev.ua/handle/123456789/148975 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. This paper is a contribution to the Special Issue on Deformation Quantization. Much of the paper was conceived during conversations of the first and third authors in Geneva in the spring of 2008. S.G. wishes to thank M.M. and R. Kashaev for the wonderful hospitality, E. Witten who suggested that we look at the U(N) Chern–Simons theory for arbitrary N and A. Its for enlightening conversations on the Riemann–Hilbert problem. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Analyticity of the Free Energy of a Closed 3-Manifold Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Analyticity of the Free Energy of a Closed 3-Manifold |
| spellingShingle |
Analyticity of the Free Energy of a Closed 3-Manifold Garoufalidis, S. Thang T.Q. Lê Mariño, M. |
| title_short |
Analyticity of the Free Energy of a Closed 3-Manifold |
| title_full |
Analyticity of the Free Energy of a Closed 3-Manifold |
| title_fullStr |
Analyticity of the Free Energy of a Closed 3-Manifold |
| title_full_unstemmed |
Analyticity of the Free Energy of a Closed 3-Manifold |
| title_sort |
analyticity of the free energy of a closed 3-manifold |
| author |
Garoufalidis, S. Thang T.Q. Lê Mariño, M. |
| author_facet |
Garoufalidis, S. Thang T.Q. Lê Mariño, M. |
| publishDate |
2008 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
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.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/148975 |
| citation_txt |
Analyticity of the Free Energy of a Closed 3-Manifold / S. Garoufalidis, Thang T.Q. Lê, M. Mariño // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 55 назв. — англ. |
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AT garoufalidiss analyticityofthefreeenergyofaclosed3manifold AT thangtqle analyticityofthefreeenergyofaclosed3manifold AT marinom analyticityofthefreeenergyofaclosed3manifold |
| first_indexed |
2025-12-07T19:52:46Z |
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2025-12-07T19:52:46Z |
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1850880469035384832 |