In the study of integrability for various models, from spin chains to integrable field theories, some operators called $Q$-operators proved to be a key tool, for instance in order to deduce Bethe equations, which govern the spectrum of integrable theories. par This talk will on the one hand discuss the simple case of the $GL(K)$ and $GL(Kmid M)$ rational spin chains, for which an explicit expressions of Q-operators can be written in terms of differential operators. On the other hand, we will see how the rational tau-functions of the $KP$ integrable hierarchy are constructed. We will actually see that this second construction sheds light on the construction of $q$-operators, which raises hopes for similar analysis of other integrable models.

Imperial College, London