Unified quantum invariants for integral homology spheres associated with simple Lie algebras

TL;DR

This paper constructs a unified quantum invariant for integral homology spheres that encapsulates all Witten-Reshetikhin-Turaev invariants associated with simple Lie algebras, providing a comprehensive analytic framework.

Contribution

It introduces an invariant in cyclotomic completion that generalizes the $sl_2$ case, unifying quantum invariants across all roots of unity without using finite-dimensional representations.

Findings

Invariant $J_M$ determines all quantum invariants at roots of unity.

WRT invariants are algebraic integers.

Invariants are determined by the Ohtsuki series and Le-Murakami-Ohtsuki invariant.

Abstract

For each finite dimensional, simple, complex Lie algebra $\mathfrak g$ and each root of unity $\xi$ (with some mild restriction on the order) one can define the Witten-Reshetikhin-Turaev (WRT) quantum invariant $\tau_M^{\mathfrak g}(\xi)\in \mathbb C$ of oriented 3-manifolds $M$. In the present paper we construct an invariant $J_M$ of integral homology spheres $M$ with values in the cyclotomic completion $\widehat {\mathbb Z [q]}$ of the polynomial ring $\mathbb Z [q]$, such that the evaluation of $J_M$ at each root of unity gives the WRT quantum invariant of $M$ at that root of unity. This result generalizes the case ${\mathfrak g}=sl_2$ proved by the first author. It follows that $J_M$ unifies all the quantum invariants of $M$ associated with $\mathfrak g$, and represents the quantum invariants as a kind of "analytic function" defined on the set of roots of unity. For example, $\tau_M(\xi)$ for all roots of unity are determined by a "Taylor expansion" at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. It follows that WRT quantum invariants $\tau_M(\xi)$ for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at $q=1$, and hence by the Le-Murakami-Ohtsuki invariant. Another consequence is that the WRT quantum invariants $\tau_M^{ \mathfrak g}(\xi)$ are algebraic integers. The construction of the invariant $J_M$ is done on the level of quantum group, and does not involve any finite dimensional representation, unlike the definition of the WRT quantum invariant. Thus, our construction gives a unified, "representation-free" definition of the quantum invariants of integral homology spheres.