Free fermions in classical and quantum integrable models

Abstract

This thesis studies the role of free fermions in certain classical and quantum integrable models. The classical models studied are the KP and BKP hierarchies of partial differential equations. We review the presence of free fermions in the theory of these hierarchies, a topic which was established in the work of the Kyoto school. The quantum models studied are descendents and relatives of the XYZ model, or its lattice equivalent, the eight-vertex model. We give a number of new results in the context of these models, revealing the presence of fermions in typical quantities such as Bethe eigenvectors, partition functions and scalar products.