Quantum field theory for the multi-variable Alexander-Conway polynomial

Abstract

We investigate in this paper the quantum field theory description of the multi-variable Alexander polynomial (Δ). We first study the WZW model on the GL(1, 1). It presents a number of interesting features including non-compactness, non-simplicity, 1/ k 2 quantum corrections, and logarithms in the current blocks. We compute the four-point functions, determine the scalar product of current blocks and make contact with the U q gl(1, 1) quantum group. We then discuss the gl(1,1) Chern-Simons model and recover the eight relations that uniquely determine Δ, as shown recently by Murakami. We discuss also how Δ can be computed by link surgery. The analysis runs into some difficulties due to the facts that Δ is zero in the braid group approach unless a strand is open, and that Δ vanishes for a split link. We propose some formal solutions to these. In appendices we give the full set of U q gl(1, 1) 6 j coefficients, we deal with details of the free field representations of gl(1, 1) (1), and we explain how the Burau matrix can be computed via monodromy and contour representation of a link, establishing the connection between quantum field theory and a recent paper by Moody.