Gauge Theory and Quantum Integrability

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In 1988 Sklyanin wrote down the equation governing the integrability of quantum systems with boundary, which was subsequently referred to as the reflection equation. This equation plays a central role in exactly solvable models with boundary in statistical mechanics, and in the factorised scattering of particles off a boundary in 2 dimensional integrable QFT. Since then many solutions to the reflection equation have been written down, and its associated algebraic structures have been explored. Despite this a simple classification of its solutions has proved elusive, and much of the research concerning its structure is opaque. Using ideas developed by Costello, Witten, and Yamazaki in their construction of quantum integrable systems from gauge theory I have generated solutions to the reflection equation in a number of simple cases and am now in a position to apply the technique in full generality. Furthermore, this approach makes transparent the origin of many of the algebraic structures present in quantum integrable systems with boundary and gives them a natural interpretation in terms of configurations of exotic Wilson lines in a particular quantum field theory. In the future it is my intention to further explore Costello's theory in a number of different contexts. In particular I believe there exists a connection between this theory and the description of classical integrable systems as symmetry reductions of the self-dual Yang-Mills equations. The hope is that in an analogous way quantum integrable systems arise as symmetry reductions of the self-dual supersymmetric Yang-Mills QFT, which itself has a natural interpretation on twistor space.