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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Asymptotic expansions of the Witten-Reshetikhin-Turaev invariants of mapping tori I


Authors: Jørgen Ellegaard Andersen and William Elbæk Petersen
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 57M27; Secondary 53D50, 53C23
DOI: https://doi.org/10.1090/tran/7740
Published electronically: December 28, 2018
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Abstract: In this paper we engage in a general study of the asymptotic expansion of the Witten-Reshetikhin-Turaev invariants of mapping tori of surface mapping class group elements. We use the geometric construction of the Witten-Reshetikhin-Turaev topological quantum field theory via the geometric quantization of moduli spaces of flat connections on surfaces. We identify assumptions on the mapping class group elements that allow us to provide a full asymptotic expansion. In particular, we show that our results apply to all pseudo-Anosov mapping classes on a punctured torus and show by example that our assumptions on the mapping class group elements are strictly weaker than hitherto successfully considered assumptions in this context. The proof of our main theorem relies on our new results regarding asymptotic expansions of oscillatory integrals, which allows us to go significantly beyond the standard transversely cut out assumption on the fixed point set. This makes use of the Picard-Lefschetz theory for Laplace integrals.


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Additional Information

Jørgen Ellegaard Andersen
Affiliation: Center for Quantum Geometry of Moduli Spaces, Department of Mathematics, University of Aarhus, Aarhus DK-8000, Denmark
Email: andersen@qgm.au.dk

William Elbæk Petersen
Affiliation: Center for Quantum Geometry of Moduli Spaces, Department of Mathematics, University of Aarhus, Aarhus DK-8000, Denmark
Email: william@qgm.au.dk

DOI: https://doi.org/10.1090/tran/7740
Received by editor(s): April 6, 2018
Received by editor(s) in revised form: October 22, 2018
Published electronically: December 28, 2018
Additional Notes: This work is supported in part by the center of excellence grant (Center for Quantum Geometry of Moduli Spaces) from the Danish National Research Foundation (DNRF95)
Article copyright: © Copyright 2018 American Mathematical Society