Asymptotic properties of the quantum representations of the mapping class group

Charles, Laurent

doi:10.1090/tran6680

Asymptotic properties of the quantum representations of the mapping class group
HTML articles powered by AMS MathViewer

by Laurent Charles PDF

Trans. Amer. Math. Soc. 368 (2016), 7507-7531 Request permission

Abstract:

For any surface with genus $\geqslant 2$, the monodromy of Hitchin’s connection is a projective representation of the mapping class group of the surface. We establish two results on the large level limit of these representations. First we prove that these projective representations lift to asymptotic representations. Second we show that under an infinitesimal rigidity assumption the characters of these representations have an asymptotic expansion. This proves the Witten’s asymptotic conjecture for mapping tori of surface diffeomorphisms. Our result is not limited to Seifert manifolds and applies to hyperbolic manifolds.

References

Jørgen Ellegaard Andersen and Kenji Ueno, Construction of the Witten-Reshetikhin-Turaev TQFT from conformal field theory, Invent. Math. 201 (2015), no. 2, 519–559. MR 3370620, DOI 10.1007/s00222-014-0555-7
Jørgen Ellegaard Andersen, Asymptotic faithfulness of the quantum $\textrm {SU}(n)$ representations of the mapping class groups, Ann. of Math. (2) 163 (2006), no. 1, 347–368. MR 2195137, DOI 10.4007/annals.2006.163.347
Jørgen Ellegaard Andersen, The Witten-Reshetikhin-Turaev invariants of finite order mapping tori I, J. Reine Angew. Math. 681 (2013), 1–38. MR 3181488, DOI 10.1515/crelle-2012-0033
Jørgen Ellegaard Andersen, Niels Leth Gammelgaard, and Magnus Roed Lauridsen, Hitchin’s connection in metaplectic quantization, Quantum Topol. 3 (2012), no. 3-4, 327–357. MR 2928088, DOI 10.4171/qt/31
M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. MR 702806, DOI 10.1098/rsta.1983.0017
Bojko Bakalov and Alexander Kirillov Jr., Lectures on tensor categories and modular functors, University Lecture Series, vol. 21, American Mathematical Society, Providence, RI, 2001. MR 1797619, DOI 10.1090/ulect/021
Arnaud Beauville and Yves Laszlo, Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), no. 2, 385–419. MR 1289330
Nicole Berline, Ezra Getzler, and Michèle Vergne, Heat kernels and Dirac operators, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 298, Springer-Verlag, Berlin, 1992. MR 1215720, DOI 10.1007/978-3-642-58088-8
Jean-Michel Bismut and François Labourie, Symplectic geometry and the Verlinde formulas, Surveys in differential geometry: differential geometry inspired by string theory, Surv. Differ. Geom., vol. 5, Int. Press, Boston, MA, 1999, pp. 97–311. MR 1772272, DOI 10.4310/SDG.1999.v5.n1.a2
L. Charles, Semi-classical properties of geometric quantization with metaplectic correction, Comm. Math. Phys. 270 (2007), no. 2, 445–480. MR 2276452, DOI 10.1007/s00220-006-0155-5
L. Charles, A note on the Chern-Simons bundle and the mapping class group, http://people.math.jussieu.fr/~charles/Articles/Chern_Simons.pdf, 2010.
L. Charles and J. Marché, Knot state asymptotics I: AJ conjecture and Abelian representations, Publ. Math. Inst. Hautes Études Sci. 121 (2015), 279–322. MR 3349834, DOI 10.1007/s10240-015-0068-y
L. Charles and J. Marché, Knot state asymptotics II: Witten conjecture and irreducible representations, Publ. Math. Inst. Hautes Études Sci. 121 (2015), 323–361. MR 3349835, DOI 10.1007/s10240-015-0069-x
Laurent Charles, A Lefschetz fixed point formula for symplectomorphisms, J. Geom. Phys. 60 (2010), no. 12, 1890–1902. MR 2735276, DOI 10.1016/j.geomphys.2010.07.002
Laurent Charles, Asymptotic properties of the quantum representations of the modular group, Trans. Amer. Math. Soc. 364 (2012), no. 11, 5829–5856. MR 2946934, DOI 10.1090/S0002-9947-2012-05537-1
Daniel S. Freed, Classical Chern-Simons theory. I, Adv. Math. 113 (1995), no. 2, 237–303. MR 1337109, DOI 10.1006/aima.1995.1039
K. Hansen, Analytic asymptotic expansions of the Reshetikhin–Turaev invariants of Seifert $3$-manifolds for $SU(2)$, ArXiv Mathematics e-prints, October 2005.
Søren Kold Hansen and Toshie Takata, Quantum invariants of Seifert 3-manifolds and their asymptotic expansions, Invariants of knots and 3-manifolds (Kyoto, 2001) Geom. Topol. Monogr., vol. 4, Geom. Topol. Publ., Coventry, 2002, pp. 69–87. MR 2002604, DOI 10.2140/gtm.2002.4.69
Kazuhiro Hikami, Difference equation of the colored Jones polynomial for torus knot, Internat. J. Math. 15 (2004), no. 9, 959–965. MR 2106155, DOI 10.1142/S0129167X04002582
N. J. Hitchin, Flat connections and geometric quantization, Comm. Math. Phys. 131 (1990), no. 2, 347–380. MR 1065677
Lars Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79–183. MR 388463, DOI 10.1007/BF02392052
Lisa C. Jeffrey, Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Comm. Math. Phys. 147 (1992), no. 3, 563–604. MR 1175494
Lisa C. Jeffrey. On some aspects of Chern-Simons gauge theory. PhD thesis, Oxford, 1992.
Lisa C. Jeffrey, Symplectic quantum mechanics and Chern-Simons gauge theory. I, J. Math. Phys. 54 (2013), no. 5, 052304, 30. MR 3098927, DOI 10.1063/1.4804152
Lisa C. Jeffrey, Symplectic quantum mechanics and Chern-Simons gauge theory. II. Mapping tori of tori, J. Math. Phys. 54 (2013), no. 5, 052305, 11. MR 3098928, DOI 10.1063/1.4804154
Yves Laszlo, Hitchin’s and WZW connections are the same, J. Differential Geom. 49 (1998), no. 3, 547–576. MR 1669720
Ruth Lawrence and Don Zagier, Modular forms and quantum invariants of $3$-manifolds, Asian J. Math. 3 (1999), no. 1, 93–107. Sir Michael Atiyah: a great mathematician of the twentieth century. MR 1701924, DOI 10.4310/AJM.1999.v3.n1.a5
E. Meinrenken and C. Woodward, Hamiltonian loop group actions and Verlinde factorization, J. Differential Geom. 50 (1998), no. 3, 417–469. MR 1690736
E. Meinrenken and C. Woodward, Canonical bundles for Hamiltonian loop group manifolds, Pacific J. Math. 198 (2001), no. 2, 477–487. MR 1835519, DOI 10.2140/pjm.2001.198.477
T. R. Ramadas, I. M. Singer, and J. Weitsman, Some comments on Chern-Simons gauge theory, Comm. Math. Phys. 126 (1989), no. 2, 409–420. MR 1027504
S. Ramanan, The moduli spaces of vector bundles over an algebraic curve, Math. Ann. 200 (1973), 69–84. MR 325615, DOI 10.1007/BF01578292
N. Reshetikhin and V. G. Turaev, Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547–597. MR 1091619, DOI 10.1007/BF01239527
L. Rozansky, A large $k$ asymptotics of Witten’s invariant of Seifert manifolds, Comm. Math. Phys. 171 (1995), no. 2, 279–322. MR 1344728
G. M. Tuynman, Quantization: towards a comparison between methods, J. Math. Phys. 28 (1987), no. 12, 2829–2840. MR 917637, DOI 10.1063/1.527681
Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399. MR 990772