S and T matrices for the super $U(1,1)$ WZW model. Application to surgery and 3-manifold invariants based on the Alexander Conway polynomial

TL;DR

This paper explores the $U(1,1)$ super WZW model's $S$ and $T$ matrices, revealing new representations of the modular group and applying these to compute Alexander invariants for 3-manifolds and links, uncovering rich topological and arithmetic insights.

Contribution

It introduces new finite-dimensional representations of the modular group for the super $U(1,1)$ WZW model at integer levels and applies these to topological invariants of 3-manifolds and links.

Findings

Some $S$ matrix elements are infinite.

Invariants depend trivially on level but encode topological info.

Invariants of Seifert manifolds relate to first homology group.

Abstract

We carry on the study of the Alexander Conway invariant from the quantum field theory point of view started in \cite{RS91}. We first discuss in details $S$ and $T$ matrices for the $U(1,1)$ super WZW model and obtain, for the level $k$ an integer, new finite dimensional representations of the modular group. These have the remarkable property that some of the $S$ matrix elements are infinite. Moreover, typical and atypical representations as well as indecomposable blocks are mixed: truncation to maximally atypical representations, as advocated in some recent papers, is not consistent. The main topological application of this work is the computation of Alexander invariants for 3-manifolds and for links in 3-manifolds. Invariants of 3-manifolds seem to depend trivially on the level $k$, but still contain interesting topological information. For Seifert manifolds for instance, they coincide with the order of the first homology group. Examples of invariants of links in 3-manifolds are given. They exhibit interesting arithmetic properties.