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

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.