From Quantum Bäcklund Transforms to Topological Quantum Field Theory

Abstract

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

29 pages,4 figures (v3: published version)

References in corpus (5)

• Diagonalization of an Integrable Discretization of the Repulsive Delta Bose Gas on the Circle
• Equivariant Verlinde formula from fivebranes and vortices
• A q-difference Baxter's operator for the Ablowitz-Ladik chain
• Quantum Integrals for a Semi-Infinite $q$-Boson System with Boundary Interactions
• On the developments of Sklyanin's quantum separation of variables for integrable quantum field theories

Cited by in corpus (5)

• Pieri rules, vertex operators and Baxter Q-matrix
• An algebraic approach to discrete time integrability
• Discretizations of the generalized AKNS scheme
• Stochastic analysis & discrete quantum systems
• Exactly Solved Models and Beyond: a special issue in honour of R J Baxter's 75th birthday