From quantum Backlund transforms to topological quantum field theory

We derive the quantum analogue of a Backlund transformation for the quantized Ablowitz-Ladik chain, a space discretization of the nonlinear Schrodinger equation. The quantization of the Ablowitz-Ladik chain leads to the q-boson model. Using a previous construction of Baxter's Q-operator for this model by the author, a set of functional relations is obtained which matches the relations of a one-variable classical Backlund transform to all orders in.. We construct also a second Q-operator and show that it is closely related to the inverse of the first. The multi-Backlund transforms generated from the Q-operator define the fusion matrices of a 2D TQFT and we derive a linear system for the solution to the quantum Backlund relations in terms of the TQFT fusion coefficients.