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arXiv · 1508.06595

From Quantum Bäcklund Transforms to Topological Quantum Field Theory

Christian Korff
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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 $\hbar $. 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.

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Christian Korff. 2016-02-03. From Quantum Bäcklund Transforms to Topological Quantum Field Theory. https://doi.org/10.1088/1751-8113%2F49%2F10%2F104001

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