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

S. K. Hansen, T. Takata

TL;DR

The authors derive explicit RT-invariant formulas for Seifert 3-manifolds associated to any complex simple Lie algebra $\mathfrak g$, expressed in terms of Seifert invariants and Lie-theoretic data. A central technical contribution is a generalized SL$(2,\mathbb Z)$-representation theory result for $\mathcal{R}_r^{\mathfrak g}$, used to compute RT-invariants via continued-fraction decompositions and Gauss-sum reciprocity. They obtain comprehensive formulas for all Seifert manifolds, with substantial simplifications for coprime cases and lens spaces, including explicit lens space invariants and their large-$r$ asymptotics that align with Andersen’s asymptotic expansion conjecture (AEC). Furthermore, a rational surgery formula is established, enabling computation of RT-invariants under rational surgeries in a modular-categorical framework. Overall, the work confirms AEC in the lens-space setting for arbitrary $\mathfrak g$ and provides a versatile toolkit for RT-invariants of Seifert manifolds via SL$(2,\mathbb Z)$ representations and Gauss-sum reciprocity.

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.

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

TL;DR

The authors derive explicit RT-invariant formulas for Seifert 3-manifolds associated to any complex simple Lie algebra , expressed in terms of Seifert invariants and Lie-theoretic data. A central technical contribution is a generalized SL-representation theory result for , used to compute RT-invariants via continued-fraction decompositions and Gauss-sum reciprocity. They obtain comprehensive formulas for all Seifert manifolds, with substantial simplifications for coprime cases and lens spaces, including explicit lens space invariants and their large- asymptotics that align with Andersen’s asymptotic expansion conjecture (AEC). Furthermore, a rational surgery formula is established, enabling computation of RT-invariants under rational surgeries in a modular-categorical framework. Overall, the work confirms AEC in the lens-space setting for arbitrary and provides a versatile toolkit for RT-invariants of Seifert manifolds via SL representations and Gauss-sum reciprocity.

Abstract

We derive formulas for the Reshetikhin-Turaev invariants of all oriented Seifert manifolds associated to an arbitrary complex finite dimensional simple Lie algebra in terms of the Seifert invariants and standard data for . 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.

Paper Structure

This paper contains 6 sections, 22 theorems, 209 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Formulas for the representations of $\text{\rm SL}(2,\mathbb{Z})$.
  3. Seifert manifolds
  4. The RT--invariants of the Seifert $3$--manifolds
  5. The case of lens spaces
  6. A rational surgery formula for the invariant $\tau_{r}^{\mathfrak g}$

Key Result

Proposition 2.1

The map $g_{\lambda_{0},\lambda_{1}}^{a,b,k} \co X \to \mathbb{C}$ is invariant under the action of the affine Weyl group $W^{\text{\rm aff}}_{k}$. Moreover, $g_{\lambda_{0},\lambda_{1}}^{a,b,k}(\lambda)=0$ for any $\lambda \in X \cap H^{k}$.

Figures (2)

  • Figure 1: Surgery presentation of $(\text{\rm o} ;g \, | \, b;(\alpha_{1},\beta_{1}),\ldots,(\alpha_{n},\beta_{n}))$
  • Figure 2: Surgery presentation of $(\text{\rm n} ; g \, | \, b;(\alpha_{1},\beta_{1}),\ldots,(\alpha_{n},\beta_{n}))$

Theorems & Definitions (35)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • ...and 25 more