Knot state asymptotics I: AJ conjecture and Abelian representations

Charles, L.; Marché, J.

doi:10.1007/s10240-015-0068-y

Charles, L.¹ ; Marché, J.¹

¹ Institut de Mathématiques de Jussieu, UMR 7586, Université Pierre et Marie Curie-Paris 6 75005 Paris France

Publications Mathématiques de l'IHÉS, Tome 121 (2015), pp. 279-322.

Résumé

Consider the Chern-Simons topological quantum field theory with gauge group ${SU}_{2}$ and level $p$ . 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 ${SU}_{2}$ -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.

DOI : 10.1007/s10240-015-0068-y

Mots-clés : Modulus Space, Toeplitz Operator, Heisenberg Group, State ASYMPTOTICS, Holomorphic Section

Affiliations des auteurs :

Charles, L. ¹ ; Marché, J. ¹

¹ Institut de Mathématiques de Jussieu, UMR 7586, Université Pierre et Marie Curie-Paris 6 75005 Paris France

@article{PMIHES_2015__121__279_0,
     author = {Charles, L. and March\'e, J.},
     title = {Knot state asymptotics {I:} {AJ} conjecture and {Abelian} representations},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {279--322},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {121},
     year = {2015},
     doi = {10.1007/s10240-015-0068-y},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-015-0068-y/}
}

TY  - JOUR
AU  - Charles, L.
AU  - Marché, J.
TI  - Knot state asymptotics I: AJ conjecture and Abelian representations
JO  - Publications Mathématiques de l'IHÉS
PY  - 2015
SP  - 279
EP  - 322
VL  - 121
PB  - Springer Berlin Heidelberg
PP  - Berlin/Heidelberg
UR  - http://archive.numdam.org/articles/10.1007/s10240-015-0068-y/
DO  - 10.1007/s10240-015-0068-y
LA  - en
ID  - PMIHES_2015__121__279_0
ER  -

%0 Journal Article
%A Charles, L.
%A Marché, J.
%T Knot state asymptotics I: AJ conjecture and Abelian representations
%J Publications Mathématiques de l'IHÉS
%D 2015
%P 279-322
%V 121
%I Springer Berlin Heidelberg
%C Berlin/Heidelberg
%U http://archive.numdam.org/articles/10.1007/s10240-015-0068-y/
%R 10.1007/s10240-015-0068-y
%G en
%F PMIHES_2015__121__279_0

Charles, L.; Marché, J. Knot state asymptotics I: AJ conjecture and Abelian representations. Publications Mathématiques de l'IHÉS, Tome 121 (2015), pp. 279-322. doi : 10.1007/s10240-015-0068-y. http://archive.numdam.org/articles/10.1007/s10240-015-0068-y/

Bibliographie
Cité par

[A05] Andersen, J. E. Deformation quantization and geometric quantization of Abelian moduli spaces, Commun. Math. Phys., Volume 255 (2005), pp. 727-745 | DOI | Zbl

[BNG96] Bar-Natan, D.; Garoufalidis, S. On the Melvin-Morton-Rozansky conjecture, Invent. Math., Volume 125 (1996), pp. 103-133 | DOI | MR | Zbl

[BHMV95] Blanchet, C.; Habegger, N.; Masbaum, G.; Vogel, P. Topological quantum field theories derived from the Kauffman bracket, Topology, Volume 34 (1995), pp. 883-927 | DOI | MR | Zbl

[C03a] Charles, L. Berezin-Toeplitz operators, a semi-classical approach, Commun. Math. Phys., Volume 239 (2003), pp. 1-28 | DOI | MR | Zbl

[C03b] Charles, L. Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators, Commun. Partial Differ. Equ., Volume 28 (2003), pp. 1527-1566 | DOI | MR | Zbl

[C06] Charles, L. Symbolic calculus for Toeplitz operators with half-forms, J. Symplectic Geom., Volume 4 (2006), pp. 171-198 | DOI | MR | Zbl

[C10a] Charles, L. On the quantization of polygon spaces, Asian J. Math., Volume 14 (2010), pp. 109-152 | DOI | MR | Zbl

[C10b] L. Charles, Asymptotic properties of the quantum representations of the modular group, Trans. Am. Math. Soc. (2010), | arXiv

[C11] L. Charles, Torus knot state asymptotics, | arXiv

[CM11] Charles, L.; Marché, J. Knot state asymptotics. II. Witten conjecture and irreducible representations, Publ. Math. (2015)

[CCGLS94] Cooper, D.; Culler, M.; Gillet, H.; Long, D.; Shalen, P. Plane curves associated to character varieties of 3-manifolds, Invent. Math., Volume 118 (1994), pp. 47-84 | DOI | MR | Zbl

[D74] Duistermaat, J. J. Oscillatory integrals, Lagrange immersions and unfolding of singularities, Commun. Pure Appl. Math., Volume 27 (1974), pp. 207-281 | DOI | MR | Zbl

[FK91] Frohman, C. D.; Klassen, E. P. Deforming representations of knot groups in SU₂, Comment. Math. Helv., Volume 66 (1991), pp. 340-361 | DOI | MR | Zbl

[G98] Garoufalidis, S. Applications of TQFT invariants in low dimensional topology, Topology, Volume 37 (1998), pp. 219-224 | DOI | MR | Zbl

[G04] Garoufalidis, S. On the characteristic and deformation varieties of a knot, Proceedings of the Casson Fest (2004), pp. 91-309

[GL05] Garoufalidis, S.; Le, T. T. Q. The colored Jones function is q-holonomic, Geom. Topol., Volume 9 (2005), pp. 1253-1293 | DOI | MR | Zbl

[GL] Garoufalidis, S.; Le, T. T. Q. Asymptotics of the colored Jones function of a knot, Geom. Topol., Volume 15 (2011), pp. 2135-2180 | DOI | MR | Zbl

[GS10] Garoufalidis, S.; Sun, X. The non-commutative A-polynomial of twist knots, J. Knot Theory Ramif., Volume 19 (2010), pp. 1571-1595 | DOI | MR | Zbl

[GU10] R. Gelca and A. Uribe, From classical theta functions to topological quantum field theory, | arXiv

[GM10] Gilmer, P.; Masbaum, G. Maslov index, Lagrangians, mapping class groups and TQFT, Forum Math., Volume 25 (2013), pp. 1067-1106 | MR | Zbl

[Ha01] Habiro, K. On the quantum sl(2) invariants of knots and integral homology spheres, Invariant of Knots and 3-Manifolds (2002), pp. 55-68

[Hi04] Hikami, K. Difference equation of the colored Jones polynomial for torus knot, Int. J. Math., Volume 15 (2004), pp. 959-965 | DOI | MR | Zbl

[Hö90] Hörmander, L. The Analysis of Linear Partial Differential Operators, I (1990) | DOI | Zbl

[J92] Jeffrey, L. C. Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Commun. Math. Phys., Volume 147 (1992), pp. 563-604 | DOI | MR | Zbl

[KM04] Kronheimer, P. B.; Mrowka, T. S. Witten’s conjecture and property P, Geom. Topol., Volume 8 (2004), pp. 295-310 | DOI | MR | Zbl

[Le06] Le, T. Q. T. The colored Jones polynomial and the A-polynomial of knots, Adv. Math., Volume 207 (2006), pp. 782-804 | DOI | MR | Zbl

[Ma03] Masbaum, G. Skein-theoretical derivation of some formulas of Habiro, Algebr. Geom. Topol., Volume 3 (2003), pp. 537-556 | DOI | MR | Zbl

[Mo95] Morton, H. C. The coloured Jones function and Alexander polynomial for torus knots, Math. Proc. Camb. Philos. Soc., Volume 117 (1995), pp. 129-135 | DOI | MR | Zbl

[Mu83] Mumford, D. Tata Lectures on Theta, I (1983) | Zbl

[RT91] Reshetikhin, N.; Turaev, V. G. Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math., Volume 103 (1991), pp. 547-597 | DOI | MR | Zbl

[R98] Rozansky, L. The universal R-matrix, Burau representation and the Melvin-Morton expansion of the colored Jones polynomial, Adv. Math., Volume 134 (1998), pp. 1-31 | DOI | MR | Zbl

[S96] Sorger, C. La formule de Verlinde, Astérisque, Volume 237 (1996), pp. 87-114 | Numdam | MR

[W89] Witten, E. Quantum field theory and the Jones polynomial, Commun. Math. Phys., Volume 121 (1989), pp. 351-399 | DOI | MR | Zbl

Cité par Sources :