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

Abstract

We carry on (in a self-contained fashion) the study of the Alexander-Conway invariant from the quantum field theory point of view started earlier. We investigate for that purpose various aspects of WZW models on supergroups. 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 (we show how to properly handle such divergences). 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. Using our approach, multivariable Alexander invariants for links in S 3 can now be fully computed by surgery. Examples of torus and cable knots are discussed. Consistency with classical results provides independent checks of the solution of the U(1,1) WZW model. The main topological application of this work is the computation of Alexander invariants for 3-manifolds and more generally for links in 3-manifolds. Invariants of 3-manifolds themselves seem to depend trivially on the level k, but still contain interesting topological information. For Seifert manifolds for instance, they essentially coincide with the order (number of elements) of the first homology group. Examples of invariants of links in 3-manifolds are given. They exhibit interesting arithmetic properties.