Towards the quantization of integrable non-linear sigma models

The class of integrable Non-Linear Sigma Models (NLSM) have many interesting applications in Quantum Field Theory, Condensed Matter Physics and String Theory. However, despite being integrable, their study still presents many challenges. In this thesis the quantization of integrable NLSM is considered within the framework of the Quantum Inverse Scattering Method. The main focus are the O(3) and O(4) NLSM and their integrable deformations. On these examples, we will encounter and discuss the long-standing conceptual challenges of quantizing NLSM in general. A key technical tool, that will allow us to make progress, is the so-called ODE/IQFT correspondence. Among the results presented in this thesis is a new approach to the problem with non-ultralocality; a study of the integrable structures in the O(3) model as well as its deformation; and a remarkable relation between the Casimir energies of the deformed O(4) model and certain solutions of the modified sinh-Gordon equation. The original obtained results of the thesis have been published in the papers, whose preprints can be found on the arXiv 1409.0449, 1706.09941, 1805.07417