Quantum discrete integrable systems

In recent decades a lot of progress has been made in discrete integrable systems, in particular systems described by 2D partial difference equations on quadrilateral lattices of higher order and in 3 dimensions as well (lattice KP systems). The quantum theory is still lagging behind even though some early work in the 1990s established some of the algebraic stuctures behind 'inegrable quantum mappings' (Nijhoff/Capel) and 'quantum field theory on the space-time lattice' (Faddeev/Volkov), providing a construction of quantum invariants and quantum unitary operators. More recent work with Field (2005) and King (2018) formed first step via a Lagrangian approach and a Feynman path integral. A main issue in the canonical approach is to establish the relevant Hilbert spaces of the theory, while in the path integral approach the role of singularities is an impotant question for nonlinear quantum integrable models. The PhD project will endeavour to give answers to these questions alongside the further development of the Lagrangian multiform theory (Lobb/Nijhoff, 2009) at the quantum level. The project will focus on some key examples, such as the quantum lattice KdV system and its reductions, in order to push the theory forward, and then branch out to other examples of integrable quantum systems.