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

Details 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 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

1992-03-25

arxiv.org/abs/hep-th/9203069v1

Nucl.Phys.B389:365-423,1993

Physics

High Energy Physics

High Energy Physics - Theory

43 pages + 37 figures (not included)

Scientific paper

10.1016/0550-3213(93)90326-K

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.

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