Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 080, 20 pages      arXiv:0809.2572      https://doi.org/10.3842/SIGMA.2008.080
Contribution to the Special Issue on Deformation Quantization

Analyticity of the Free Energy of a Closed 3-Manifold

Stavros Garoufalidis a, Thang T.Q. Lê a and Marcos Mariño b
a) School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA
b) Section de Mathématiques, Université de Genève, CH-1211 Genève 4, Switzerland

Received September 15, 2008, in final form November 06, 2008; Published online November 15, 2008

Abstract
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.

Key words: 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; Painlevé I; WKB; asymptotic expansions; transseries; Riemann-Hilbert problem.

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References

  1. Aoki T., Kawai T., Koike T., Takei Y., On the exact WKB analysis of operators admitting infinitely many phases, Adv. Math. 181 (2004), 165-189.
  2. Bar-Natan D., On the Vassiliev knot invariants, Topology 34 (1995), 423-472.
  3. Bar-Natan D., Garoufalidis S., On the Melvin-Morton-Rozansky conjecture, Invent. Math. 125 (1996), 103-133.
  4. Bar-Natan D., Lawrence R., A rational surgery formula for the LMO invariant, Israel J. Math. 140 (2004), 29-60, math.GT/0007045.
  5. Bar-Natan D., Garoufalidis S., Rozansky L., Thurston D., The Aarhus integral of rational homology 3-spheres. I. A highly non trivial flat connection on S3, Selecta Math. (N.S.) 8 (2002), 315-339, q-alg/9706004.
    Bar-Natan D., Garoufalidis S., Rozansky L., Thurston D., The Aarhus integral of rational homology 3-spheres. II. Invariance and universality, Selecta Math. (N.S.) 8 (2002), 341-371, math.QA/9801049.
    Bar-Natan D., Garoufalidis S., Rozansky L., Thurston D., The Aarhus integral of rational homology 3-spheres. III. Relation with the Le-Murakami-Ohtsuki invariant, Selecta Math. (N.S.) 10 (2004), 305-324, math.QA/9808013.
  6. Bender E.A., Canfield E.R., The asymptotic number of rooted maps on a surface, J. Combin. Theory Ser. A 43 (1986), 244-257.
  7. Bender E.A., Gao Z., Richmond L.B., The map asymptotics constant tg, Electron. J. Combin. 15 (2008), no. 1, paper 51, 8 pages.
  8. Benna M.K., Benvenuti S., Klebanov I.R., Scardicchio A., A test of the AdS/CFT correspondence using high-spin operators, hep-th/0611135.
  9. Beisert N., Eden B., Staudacher M., Transcendentality and crossing, J. Stat. Mech. 2007 (2007), P01021, 30 pages, hep-th/0610251.
  10. Bessis D., Itzykson C., Zuber J.B., Quantum field theory techniques in graphical enumeration, Adv. in Appl. Math. 1 (1980), 109-157.
  11. Boutroux P., Recherches sur les transcendantes de M. Painlevé et l'étude asymptotique des équations différentielles du second ordre, Ann. Sci. École Norm. Sup. (3) 30 (1913), 255-375.
  12. Brézin E., Itzykson C., Parisi G., Zuber J.B., Planar diagrams, Comm. Math. Phys. 59 (1978), 35-51.
  13. Cohen H., Lewin L., Zagier D., A sixteenth-order polylogarithm ladder, Experiment. Math. 1 (1992), 25-34.
  14. Costin O., Garoufalidis S., Resurgence of the fractional polylogarithms, Math. Res. Lett., to appear, math.CA/0701743.
  15. Costin O., Kruskal M., Optimal uniform estimates and rigorous asymptotics beyond all orders for a class of ordinary differential equations, Proc. Roy. Soc. London Ser. A 452 (1996), no. 1948, 1057-1085, math.CA/0608412.
  16. Di Francesco P., Ginsparg P., Zinn-Justin J., 2D gravity and random matrices, Phys. Rep. 254 (1995), no. 1-2, 133 pages, hep-th/9306153.
  17. Eynard B., Zinn-Justin J., Large order behaviour of 2D gravity coupled to D < 1 matter, Phys. Lett. B 302 (1993), 396-402, hep-th/9301004.
  18. Fokas A.S., Its A.R., Kitaev A.V., The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys. 147 (1992), 395-430.
  19. Fokas A.S., Its A.R., Kapaev A., Novokshenov V.Yu., Painlevé transcendents. The Riemann-Hilbert approach, Mathematical Surveys and Monographs, Vol. 128, American Mathematical Society, Providence, RI, 2006.
  20. Garoufalidis S., Lê T.T.Q., Gevrey series in quantum topology, J. Reine Angew. Math. 618 (2008), 169-195, math.GT/0609618.
  21. Garoufalidis S., Lê T.T.Q., Asymptotics of the colored Jones function of a knot, math.GT/0508100.
  22. Garoufalidis S., Difference and differential equations for the colored Jones function, J. Knot Theory Ramifications 17 (2008), 495-510, math.GT/0306229.
  23. Garoufalidis S., Chern-Simons theory, analytic continuation and arithmetic, Acta Math. Vietnam., to appear, arXiv:0711.1716.
  24. Garoufalidis S., Habegger N., The Alexander polynomial and finite type 3-manifold invariants, Math. Ann. 316 (2000), 485-497, q-alg/9708002.
  25. Garoufalidis S., Mariño M., On Chern-Simons matrix models, math.GT/0601390.
  26. Garoufalidis S., Mariño M., in preparation.
  27. Garoufalidis S., Mariño M., in preparation.
  28. Garoufalidis S., Rozansky L., The loop of the Kontsevich integral, the null-move and S-equivalence, Topology 43 (2004), 1183-1210, math.GT/0003187.
  29. Gao Z.C., A pattern for the asymptotic number of rooted maps on surfaces, J. Combin. Theory Ser. A 64 (1993), 246-264.
  30. Gao Z.C., The number of rooted triangular maps on a surface, J. Combin. Theory Ser. B 52 (1991), 236-249.
  31. Gopakumar R., Vafa C., M-theory and topological strings-I, hep-th/9809187.
  32. Goulden I.P., Jackson D.M., The KP hierarchy, branched covers, and triangulations, Adv. Math., to appear, arXiv:0803.3980.
  33. Habegger N., Masbaum G., The Kontsevich integral and Milnor's invariants, Topology 39 (2000), 1253-1289.
  34. Habegger N., Thompson G., The universal perturbative quantum 3-manifold invariant, Rozansky-Witten invariants and the generalized Casson invariants, Acta Math. Vietnam., to appear, math.GT/9911049.
  35. 't Hooft G., On the convergence of planar diagram expansions, Comm. Math. Phys. 86 (1982), 449-464.
  36. Joshi N., Kruskal M.D., A direct proof that solutions of the six Painlevé equations have no movable singularities except poles, Stud. Appl. Math. 93 (1994), 187-207.
  37. Kapaev A.A., Asymptotic behavior of the solutions of the Painlevé equation of the first kind, Differentsial'nye Uravneniya 24 (1988), 1684-1695 (English transl.: Differential Equations 24 (1988), 1107-1115).
  38. Kapaev A.A., Quasi-linear stokes phenomenon for the Painlevé first equation, J. Phys. A: Math. Gen. 37 (2004), 11149-11167, nlin.SI/0404026.
  39. Kupergberg G., Thurston D.P., Perturbative 3-manifold invariants by cut-and-paste topology, math.GT/9912167.
  40. Kuriya T., On the LMO conjecture, arXiv:0803.1732.
  41. Lê T.T.Q., Murakami J., Ohtsuki T., On a universal perturbative invariant of 3-manifolds, Topology 37 (1998), 539-574.
  42. Lê T.T.Q., On perturbative PSU(n) invariants of rational homology 3-spheres, Topology 39 (2000), 813-849, math.GT/9802032.
  43. Lê T.T.Q., Quantum invariants of 3-manifolds: integrality, splitting, and perturbative expansion, Topology Appl. 127 (2003), 125-152, math.QA/0004099.
  44. Le Guillou J.C., Zinn-Justin J. (Editors), Large order behavior of perturbation theory, North-Holland, Amsterdam, 1990.
  45. Mariño M., Chern-Simons theory, matrix integrals and perturbative three-manifold invariants, Comm. Math. Phys. 253 (2005), 25-49, hep-th/0207096.
  46. Mariño M., Chern-Simons theory and topological strings, Rev. Modern Phys. 77 (2005), 675-720, hep-th/0406005.
  47. Mariño M., Schiappa R., Weiss M., Nonperturbative effects and the large-order behavior of matrix models and topological strings, Commun. Number Th. Phys. 2 (2008), 349-419, arXiv:0711.1954.
  48. Oesterlé J., Polylogarithmes, Séminaire Bourbaki, Vol. 1992/93, Astérisque no. 216 (1993), Exp. no. 762, 49-67.
  49. Ohtsuki T., A polynomial invariant of rational homology 3-spheres, Invent. Math. 123 (1996), 241-257.
  50. Ohtsuki T., Quantum invariants. A study of knots, 3-manifolds, and their sets, Series on Knots and Everything, Vol. 29, World Scientific Publishing Co., Inc., River Edge, NJ, 2002.
  51. Rozansky L., The universal R-matrix, Burau representation and the Melvin-Morton expansion of the colored Jones polynomial, Adv. Math. 134 (1998), 1-31.
  52. Takata T., On quantum PSU(n)-invariants for lens spaces, J. Knot Theory Ramifications 5 (1996), 885-901.
  53. Takei Y., On the connection formula for the first Painlevé equation - from the viewpoint of the exact WKB analysis, Surikaisekikenkyusho Kokyuroku 931 (1995), 70-99.
  54. Turaev V., The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), 527-553.
  55. Witten E., Two-dimensional gravity and intersection theory on moduli space, in Surveys in Differential Geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, 243-310.

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