Reshetikhin-Turaev invariants of Seifert 3-manifolds for classical simple Lie algebras, and their asymptotic expansions

TL;DR

This paper derives explicit formulas for Reshetikhin-Turaev invariants of Seifert 3-manifolds related to simple Lie algebras and analyzes their asymptotic behavior for lens spaces, confirming a conjecture by Andersen.

Contribution

It provides a comprehensive formula for these invariants for all Seifert manifolds associated with simple Lie algebras and determines their asymptotic expansions for lens spaces.

Findings

Formulas for invariants in terms of Seifert invariants and Lie algebra data

Asymptotic expansions for lens spaces at large quantum levels

Results agree with Andersen's asymptotic expansion conjecture

Abstract

We derive formulas for the Reshetikhin-Turaev invariants of all oriented Seifert manifolds associated to an arbitrary complex finite dimensional simple Lie algebra $\mathfrak g$ in terms of the Seifert invariants and standard data for $\mathfrak g$. A main corollary is a determination of the full asymptotic expansions of these invariants for lens spaces in the limit of large quantum level. Our results are in agreement with the asymptotic expansion conjecture due to J. E. Andersen.