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On the Witten–Reshetikhin–Turaev invariants of torus bundles

Jørgen Ellegaard Andersen

Centre for Quantum Geometry of Moduli Spaces, Faculty of Science, Aarhus University, DK-8000, Denmark

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and

Søren Fuglede Jørgensen

Department of Mathematics, Uppsala University, Box 480, SE-75106 Uppsala, Sweden

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https://doi.org/10.1142/S0218216515500558Cited by:0 (Source: Crossref)

Abstract

By methods similar to those used by L. Jeffrey [L. C. Jeffrey, Chern–Simons–Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Commun. Math. Phys.147 (1992) 563–604], we compute the quantum SU(N)-invariants for mapping tori of trace 2 homeomorphisms of a genus 1 surface when N = 2, 3 and discuss their asymptotics. In particular, we obtain directly a proof of a version of Witten's asymptotic expansion conjecture for these 3-manifolds. We further prove the growth rate conjecture for these 3-manifolds in the SU(2) case, where we also allow the 3-manifolds to contain certain knots. In this case we also discuss trace -2 homeomorphisms, obtaining — in combination with Jeffrey's results — a proof of the asymptotic expansion conjecture for all torus bundles.

Keywords:

AMSC: 57M25, 57M27, 53D50

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