Knot state asymptotics II: Witten conjecture and irreducible representations
Publications Mathématiques de l'IHÉS, Tome 121 (2015), pp. 323-361.

This article pursues the study of the knot state asymptotics in the large level limit initiated in Charles and Marché (Knot state asymptotics I. Abelian representations and the A–J conjecture, 2011). As a main result, we prove the Witten asymptotic expansion conjecture for the Dehn fillings of the figure eight knot.

The state of a knot is defined in the realm of Chern-Simons topological quantum field theory as a holomorphic section on the SU 2 -character manifold of the peripheral torus.

In the previous paper, we conjectured that the knot state concentrates on the character variety of the knot with a given asymptotic behavior on the neighborhood of the abelian representations. In the present paper we study the neighborhood of irreducible representations. We conjecture that the knot state is Lagrangian with a phase and a symbol given respectively by the Chern-Simons and Reidemeister torsion invariants. We show that under some mild assumptions, these conjectures imply the Witten conjecture on the asymptotic expansion of WRT invariants of the Dehn fillings of the knot.

Using microlocal techniques, we show that the figure eight knot state satisfies our conjecture starting from q-differential relations verified by the colored Jones polynomials. The proof relies on a differential equation satisfied by the Reidemeister torsion along the branches of the character variety, a phenomenon which has not been observed previously as far as we know.

DOI : 10.1007/s10240-015-0069-x
Mots clés : Modulus Space, Line Bundle, Toeplitz Operator, Heisenberg Group, State ASYMPTOTICS
Charles, L. 1 ; Marché, J. 1

1 Institut de Mathématiques de Jussieu, UMR 7586, Université Pierre et Marie Curie—Paris 6 75005 Paris France
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     title = {Knot state asymptotics {II:} {Witten} conjecture and irreducible representations},
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Charles, L.; Marché, J. Knot state asymptotics II: Witten conjecture and irreducible representations. Publications Mathématiques de l'IHÉS, Tome 121 (2015), pp. 323-361. doi : 10.1007/s10240-015-0069-x. http://archive.numdam.org/articles/10.1007/s10240-015-0069-x/

[A] J. E. Andersen, The Witten invariant of finite order mapping tori I, | arXiv

[AH06] Andersen, J. E.; Hansen, S. K. Asymptotics of the quantum invariants for surgeries on the figure 8 knot, J. Knot Theory Ramif., Volume 15 (2006), pp. 479-548 | 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

[C03] 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 mapping class group, Trans. Am. Math. Soc., | arXiv

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

[CM11] Charles, L.; Marché, J. Knot state asymptotics I. Abelian representations and the A-J conjecture, Publ. Math. (2015)

[Fra35] Franz, W. Über die Torsion einer überdeckung, J. Reine Angew. Math., Volume 173 (1935), pp. 245-254

[Fre92] Freed, D. S. Reidemeister torsion, spectral sequences and Brieskorn spheres, J. Reine Angew. Math., Volume 429 (1992), pp. 75-89 | MR | Zbl

[Fre95] Freed, D. S. Classical Chern-Simons Theory, Adv. Math., Volume 113 (1995), pp. 237-303 | DOI | MR | Zbl

[FG91] Freed, D.; Gompf, R. Computer calculation of Witten’s 33-manifold invariant, Commun. Math. Phys., Volume 141 (1991), pp. 79-117 | DOI | MR | Zbl

[Ha05] S. K. Hansen, Analytic asymptotic expansions of the Reshetikhin-Turaev invariants of Seifert 3-manifolds for SU(2), | arXiv

[HT01] Hansen, S. K.; Takata, T. Quantum invariants of Seifert 3-manifolds and their asymptotic expansions, Invariants of Knots and 3-Manifolds (2002), pp. 69-87 (electronic)

[Hi05] Hikami, H. On the quantum invariant for the Brieskorn homology spheres, Int. Math. J., Volume 16 (2005), pp. 661-685 | DOI | MR | Zbl

[HK98] Hodgson, C. D.; Kerckhoff, S. P. Rigidity of hyperbolic cone-manifolds and hyperbolic surgery, J. Differ. Geom., Volume 48 (1998), pp. 1-60 | MR | 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

[JW93] Jeffrey, L. C.; Weitsman, J. Half density quantization of the moduli space of flat connections and Witten’s semiclassical manifold invariants, Topology, Volume 32 (1993), pp. 509-529 | DOI | MR | Zbl

[K91] Klassen, E. P. Representations of knot groups in SU(2), Trans. Am. Math. Soc., Volume 326 (1991), pp. 795-828 | MR | Zbl

[LZ99] Lawrence, R.; Zagier, D. Modular forms and quantum invariants of 3-manifolds, Asian J. Math., Volume 3 (1999), pp. 93-107 | MR | Zbl

[Mi62] Milnor, J. A duality theorem for Reidemeister torsion, Ann. Math., Volume 76 (1962), pp. 134-147 | DOI | MR

[Mi66] Milnor, J. Whitehead torsion, Bull. Am. Math. Soc., Volume 72 (1966), pp. 358-426 | DOI | MR | Zbl

[Mu08] Murakami, H. An introduction to the volume conjecture and its generalizations, Acta Math. Vietnam., Volume 33 (2008), pp. 219-253 | MR | Zbl

[O01] Ohtsuki, T. Problems on invariants of knots and 3-manifolds, Invariants of knots and 3-manifolds (2002), pp. 377-572 (i–iv)

[P97] Porti, J. Torsion de Reidemeister pour les variétés hyperboliques (1997)

[RSW89] Ramadas, T. R.; Singer, I. M.; Weitsman, J. Some comments on Chern-Simons gauge theory, Commun. Math. Phys., Volume 126 (1989), pp. 409-420 | DOI | MR | 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

[R96] Rozansky, L. Residue formulas for the large k asymptotics of Witten’s invariants of Seifert manifolds. The case of SU(2), Commun. Math. Phys., Volume 178 (1996), pp. 27-60 | DOI | MR | Zbl

[Tu02] Turaev, V. Torsions of 3-Manifolds (2002) | DOI

[W64] Weil, A. Remarks on the cohomology of groups, Ann. Math., Volume 80 (1964), pp. 149-157 | DOI | MR | Zbl

[W64] Witten, E. Quantum field theory and the Jones polynomial, Commun. Math. Phys., Volume 121 (1989), pp. 351-399 | DOI | MR | Zbl

[W91] Witten, E. On quantum Gauge theories in two dimensions, Commun. Math. Phys., Volume 141 (1991), pp. 153-209 | DOI | MR | Zbl

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