Knot state asymptotics I: AJ conjecture and Abelian representations

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• 作者：L. Charles ; J. Marché
• 刊名：Publications math篓娄matiques de l'IH篓娄S
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：121
• 期：1
• 页码：279-322
• 全文大小：1,080 KB
• 参考文献：[A05] J. E. Andersen , Deformation quantization and geometric quantization of Abelian moduli spaces, Commun. Math. Phys., 255 (2005), 727-45. View Article MATH
  [BNG96] D. Bar-Natan and S. Garoufalidis , On the Melvin-Morton-Rozansky conjecture, Invent. Math., 125 (1996), 103-33. View Article MATH MathSciNet
  [BHMV95] C. Blanchet , N. Habegger , G. Masbaum and P. Vogel , Topological quantum field theories derived from the Kauffman bracket, Topology, 34 (1995), 883-27. View Article MATH MathSciNet
  [C03a] L. Charles , Berezin-Toeplitz operators, a semi-classical approach, Commun. Math. Phys., 239 (2003), 1-8. View Article MATH MathSciNet
  [C03b] L. Charles , Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators, Commun. Partial Differ. Equ., 28 (2003), 1527-566. View Article MATH MathSciNet
  [C06] L. Charles , Symbolic calculus for Toeplitz operators with half-forms, J. Symplectic Geom., 4 (2006), 171-98. View Article MATH MathSciNet
  [C10a] L. Charles , On the quantization of polygon spaces, Asian J. Math., 14 (2010), 109-52. View Article MATH MathSciNet
  [C10b] L. Charles, Asymptotic properties of the quantum representations of the modular group, Trans. Am. Math. Soc. (2010), arXiv:-005.-452 .
  [C11] L. Charles, Torus knot state asymptotics, arXiv:-107.-692 , 2011.
  [CM11] L. Charles and J. Marché , Knot state asymptotics. II. Witten conjecture and irreducible representations, Publ. Math. (2015). doi:10.-007/?s10240-015-0069-x .
  [CCGLS94] D. Cooper , M. Culler , H. Gillet , D. Long and P. Shalen , Plane curves associated to character varieties of 3-manifolds, Invent. Math., 118 (1994), 47-4. View Article MATH MathSciNet
  [D74] J. J. Duistermaat , Oscillatory integrals, Lagrange immersions and unfolding of singularities, Commun. Pure Appl. Math., 27 (1974), 207-81. View Article MATH MathSciNet
  [FK91] C. D. Frohman and E. P. Klassen , Deforming representations of knot groups in SU2, Comment. Math. Helv., 66 (1991), 340-61. View Article MATH MathSciNet
  [G98] S. Garoufalidis , Applications of TQFT invariants in low dimensional topology, Topology, 37 (1998), 219-24. View Article MATH MathSciNet
  [G04] S. Garoufalidis , On the characteristic and deformation varieties of a knot, in Proceedings of the Casson Fest, Geometry and Topology Monographs, vol.?7, pp. 91-09, 2004.
  [GL05] S. Garoufalidis and T. T. Q. Le , The colored Jones function is q-holonomic, Geom. Topol., 9 (2005), 1253-293. View Article MATH MathSciNet
  [GL] S. Garoufalidis and T. T. Q. Le , Asymptotics of the colored Jones function of a knot, Geom. Topol., 15 (2011), 2135-180. View Article MATH MathSciNet
  [GS10] S. Garoufalidis and X. Sun , The non-commutative A-polynomial of twist knots, J. Knot Theory Ramif., 19 (2010), 1571-595. View Article MATH MathSciNet
  [GU10] R. Gelca and A. Uribe, From classical theta functions to topological quantum field theory, arXiv:-006.-252 .
  [GM10] P. Gilmer and G. Masbaum , Maslov index, Lagrangians, mapping class groups and TQFT, Forum Math., 25 (2013), 1067-106. MATH MathSciNet
  [Ha01] K. Habiro , On the quantum sl(2) invariants of knots and integral homology spheres, in Invariant of Knots and 3-Manifolds, Kyoto, 2001, Geometry and Topology Monographs, vol.?4, pp. 55-8, 2002.
  [Hi04] K. Hikami , Difference equation of the colored Jones polynomial for torus knot, Int. J. Math., 15 (2004), 959-65. View Article MATH MathSciNet
  [H?90] L. H?rmander , The Analysis of Linear Partial Differential Operators, I, Grundlehren der Mathematischen Wissenschaften, vol.?256, Springer, Berlin, 1990. View Article MATH
  [J92] L. C. Jeffrey , Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Commun. Math. Phys., 147 (1992), 563-04. View Article MATH MathSciNet
  [KM04] P. B. Kronheimer and T. S. Mrowka , Witten’s conjecture and property P, Geom. Topol., 8 (2004), 295-10. View Article MATH MathSciNet
  [Le06] T. Q. T. Le , The colored Jones polynomial and the A-polynomial of knots, Adv. Math., 207 (2006), 782-04. View Article MATH MathSciNet
  [Ma03] G. Masbaum , Skein-theoretical derivation of some formulas of Habiro, Algebr. Geom. Topol., 3 (2003), 537-56. View Article MATH MathSciNet
  [Mo95] H. C. Morton , The coloured Jones function and Alexander polynomial for torus knots, Math. Proc. Camb. Philos. Soc., 117 (1995), 129-35. View Article MATH MathSciNet
  [Mu83] D. Mumford , Tata Lectures on Theta, I, Progress in Mathematics, vol.?28, Birkh?user, Boston, 1983. MATH
  [RT91] N. Reshetikhin and V. G. Turaev , Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math., 103 (1991), 547-97. View Article MATH MathSciNet
  [R98] L. Rozansky , The universal R-matrix, Burau representation and the Melvin-Morton expansion of the colored Jones polynomial, Adv. Math., 134 (1998), 1-1. View Article MATH MathSciNet
  [S96] C. Sorger ,
• 作者单位：L. Charles (1)
  J. Marché (1)
  
  1. Institut de Mathématiques de Jussieu, UMR 7586, Université Pierre et Marie Curie-Paris 6, 75005, Paris, France
  
• 刊物主题：Mathematics, general; Algebra; Analysis; Geometry; Number Theory;
• 出版者：Springer Berlin Heidelberg
• ISSN：1618-1913

文摘

Consider the Chern-Simons topological quantum field theory with gauge group \(\operatorname {SU}_{2}\) and level k. Given a knot in the 3-sphere, this theory associates to the knot exterior an element in a vector space. We call this vector the knot state and study its asymptotic properties when the level is large. The latter vector space being isomorphic to the geometric quantization of the \(\operatorname {SU}_{2}\)-character variety of the peripheral torus, the knot state may be viewed as a section defined over this character variety. We first conjecture that the knot state concentrates in the large level limit to the character variety of the knot. This statement may be viewed as a real and smooth version of the AJ conjecture. Our second conjecture says that the knot state in the neighborhood of Abelian representations is a Lagrangian state. Using microlocal techniques, we prove these conjectures for the figure eight and torus knots. The proof is based on q-difference relations for the colored Jones polynomial. We also provide a new proof for the asymptotics of the Witten-Reshetikhin-Turaev invariant of the lens spaces and a derivation of the Melvin-Morton-Rozansky theorem from the two conjectures.