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 work develops a quantum-field-theoretic framework for Alexander-Conway invariants via the supergroup WZW model U(1,1), constructing regularized, finite-dimensional modular representations and deriving S and T matrices that act on an extended set of typical, atypical, and indecomposable representations. Through surgery, the authors extract multivariable Alexander invariants for links in S^3 and extend to invariants of links in 3-manifolds, with Seifert manifolds yielding invariants tied to the first homology order. The study clarifies when Verlinde-type formulas apply and provides explicit cabling and torus-link formulas, highlighting the arithmetic structure of Alexander theory embedded in a Chern–Simons/WZW framework. Overall, the paper bridges topology and QFT in the GL(1,1) setting, reveals regularization-driven doubling phenomena, and points to broader extensions to other supergroups and connections to Casson-type invariants.

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.