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Analyticity of the Free Energy of a Closed 3-Manifold
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2008, том 4
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Analyticity of the Free Energy of a Closed 3-Manifold
Інший ID:
2000 Mathematics Subject Classification: 57N10; 57M25
Посилання:
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.
Підтримка:
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.
Дата:
2008
Переглядів:
583
Завантажень:
243
Анотація:
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.
