Quantum field theory and integrable lattice models

Integrable lattice models have played a central role in many topics in mathematics and physics, and were part of the motivation for the development of quantum groups. I’ll start by explaining the basic concepts of the theory of integrable lattice models, including the Yang-Baxter equation. Then, I’ll introduce the Atiyah-Segal-Witten axiomatic framework for quantum field theory, and explain a theorem of myself and Gwilliam which allows one to construct solutions of a
(modification) of these axioms starting with a classical Lagrangian.
Finally, I’ll show that integrable lattice models arise in a very simple and natural way from line operators in 4-dimensional supersymmetric quantum field theories. This construction gives rise to a whole new class of solutions to the Yang-Baxter equation. If time permits, I’ll discuss applications of these ideas to (generalizatons of) geometric Langlands.