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

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 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. 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 Picard-Lefschetz theory for Laplace integrals.

3d-3d correspondence for mapping tori

“…Originally inspired by physics, the methods of computing Z(M 3 ) on the M 3 side of the 3d-3d correspondence use structural properties that can be given precise mathematical meaning: resurgent analysis [45,[59][60][61], q-difference equations that come from quantization of A-polynomial curves [23], etc. During the past several years, these methods allowed to compute Z(M 3 ) for much more general examples of 3-manifolds, including the hyperbolic ones, and, more importantly, allowed to formulate the structural properties -such as gluing formulae -as a set of axioms that define Z as a (decorated) TQFT.…”

“…where Stab Z 2 (a) is the stabiliser of a under the action of the Weyl group: a → −a. The fact that (3.6) is satisfied with all these ingredients (3.10)-(3.13) is a classic result of [65]; see also [60,[66][67][68][69] for recent work on WRT invariants of mapping tori. Aiming to understand the precise definition/characterization of almost abelian flat connections on general 3-manifolds, in the rest of this section we extend the notion of the two sets (3.2) and (3.3) to plumbed 3-manifolds with b 1 > 0 and 0-surgeries on some knots.…”

“…Secondly, it can be obtained via a general formula (3.19) for plumbings with loops which, in turn, can be derived by extending the arguments in [54], as we explain below. Alternatively, it can be derived using either the resurgent analysis [45,[59][60][61] or, in some cases, using surgery formulae and q-difference equations that come from quantization of A-polynomial curves [23].…”

Resurgence analysis of quantum invariants of Seifert fibered homology spheres

“…This quantization approach to quantum invariants is also considered by other authors in the works of [25][26][27]67]. By building on the work of the first author [2] and Toeplitz operator theory [70], the quantization approach allowed us in [9] to reduce the proof of the asymptotic conjecture to an application of stationary phase approximation applied to oscillatory integrals (1.20) . This is possible connected to Gukov and Witten's theory of brane quantization [56], in which complexification plays a central role, and where Hitchin's moduli space of Higgs bundles [64] plays the role of a complexification of  Flat (Σ, SU(𝑛)).…”

Geometric quantization of symplectic maps and Witten's asymptotic conjecture

“…In that case, Andersen establishes the asymptotic expansion of the Witten-Reshetikhin-Turaev invariant of the mapping torus Σ f in [2, Th.1.1], and Andersen and Himpel compute the first coefficient in [4, Th.1.6] using (1.8). Recently, Andersen and Petersen announced in [5,Th.1.2] the asymptotic expansion of the Witten-Reshetikhin-Turaev invariant of a general mapping torus, but do not compute the first coefficient. We recover all these results as consequences of the results of Section 4 and Section 5.…”