On the Witten asymptotic conjecture for Seifert manifolds

TL;DR

This paper provides a new proof of the Witten asymptotic conjecture for Seifert manifolds with specific conditions, using semiclassical analysis and character varieties to estimate invariants in the large level limit.

Contribution

It introduces a novel semiclassical analysis approach to prove the conjecture, connecting Witten-Reshetikhin-Turaev invariants with character varieties and symplectic geometry.

Findings

Witten-Reshetikhin-Turaev invariants expressed as scalar products of Lagrangian states

Asymptotic expansion terms related to character varieties and Chern-Simons invariants

Development of a singular stationary phase lemma for discrete sums

Abstract

We give a new proof of Witten asymptotic conjecture for Seifert manifolds with non vanishing Euler class and one exceptional fiber. Our method is based on semiclassical analysis on a two dimensional phase space torus. We prove that the Witten-Reshetikhin-Turaev invariant of a Seifert manifold is the scalar product of two Lagrangian states, and we estimate this scalar product in the large level limit. The leading order terms of the expansion are naturally given in terms of character varieties, the Chern-Simons invariants and some symplectic volumes. For the analytic part, we establish some singular stationary phase lemma for discrete oscillatory sums.