From Quantum B"acklund Transforms to Topological Quantum Field Theory

We derive the quantum analogue of a B"acklund transformation for the quantised Ablowitz-Ladik chain, a space discretisation of the nonlinear Schr"odinger equation. The quantisation 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 B"acklund transform to all orders in $\hbar $. We construct also a second Q-operator and show that it is closely related to the inverse of the first. The multi-B"acklund 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 B"acklund relations in terms of the TQFT fusion coefficients.

💡 Research Summary

The paper investigates the quantum analogue of Bäcklund transformations for the quantised Ablowitz‑Ladik chain, which is the lattice discretisation of the nonlinear Schrödinger equation. The quantisation leads to the q‑boson model, whose algebraic structure is encoded in the q‑Heisenberg algebra Hₙ(q). The author builds on a previously constructed Baxter Q‑operator (denoted Q₊) and introduces a second operator Q₋, showing that the two are essentially inverses of each other in the infinite‑size limit.

A central achievement is the derivation of exact functional relations for the similarity transformation generated by Q₊. Defining transformed fields
\