Comments on Witten Invariants of 3-Manifolds for SU(2) and $Z_m$

TL;DR

This paper computes Witten invariants for lens spaces with SU(2) and Z_m groups, revealing duality relations and effective manifold distinctions, and introduces a general calculation method for any group G.

Contribution

It provides explicit formulas for Witten invariants of lens spaces, observes a duality relation at large levels, and describes a universal calculation approach for any gauge group.

Findings

Witten invariants for SU(2) lens spaces are obtained for all p.

A duality relation between p and k is identified at high levels.

Z_m invariants effectively distinguish different 3-manifolds.

Abstract

The values of the Witten invariants, $I_W$, of the lens space $L(p, 1)$ for SU(2) at level $k$ are obtained for arbitrary $p$. A duality relation for $I_W$ when $p$ and $k$ are interchanged, valid for asymptotic $k$, is observed. A method for calculating $I_W$ for any group $G$ is described. It is found that $I_W$ for $Z_m$, even for $m = 2$, distinguishes 3-manifolds quite effectively.