Aspects of Quantum Integrable Systems

In classical mechanics, an integrable system is a model that admits a complete set of integrals of motion that are in involution. This notion can be extended to include classical integrable field theories. A natural question is how to lift these notions to the quantum setting. Despite the extensive literature on the subject, these models are still not very well understood. As an example, in the first section of this work, we present a new class of integrable quantum field theories which develop an unusual behaviour that may be interpreted as a Hagedorn transition. The principal reason for the enigmatic nature of these models is the absence of a universal mathematical framework to describe them. One potential solution may be provided by quantum affine Gaudin models, which we examine in greater detail in the second part of this work. In particular, we introduce the first non-trivial Hamiltonian of quartic order for the affine sl2 Gaudin model, as well as the next-to-leading order expression for all higher Hamiltonians. Furthermore, we provide new insights into the double-loop version of the Feigin-Frenkel homomorphism, which is expected to be a crucial component in the construction of the Bethe ansatz for these models.