I’ll try to present informally the connection between certain algebro-geometric data and quantum integrable systems, as we first accidentally discovered with P. Di Francesco and A. Knutson about 12 years ago, and was then rediscovered and expanded much further by Nekrasov and Shatashvili in the physics literature, and by Okounkov et al in the mathematics literature. After an overview of the simplest model in which this strategy applies (rational 5-vertex model), we shall discuss the more complicated case of the Temperley–Lieb loop model, how it relates to solutions of the quantum Knizhnik–Zamolodchikov equation, and ultimately the application to the ground state of the model with loop weight 1 and the Razumov–Stroganov correspondence.

LPTHE