Asymptotic expansions of the Witten-Reshetikhin-Turaev Invariants of Mapping Tori I

Jørgen Ellegaard Andersen and William Elbæk Petersen arxives new paper

2018.04.06 | Christine Dilling

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 TQFT 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. The proof of this 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. Our results apply to mapping classes of a punctured surface of genus at least one. In particular, we show that our results apply to all pseudo-Anasov mapping classes on a punctured torus and show by example that our assumptions on the mapping class group elements are strictly weaker than hithero successfully considered assumptions in this context.

 

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