Integrability, Defects, and Flows in 2 and 4 Dimensional Quantum Field Theories

Abstract

This thesis investigates the interplay between integrability, defects, and renormalization group (RG) flows, in quantum field theories in 2 and 4 dimensions. In the first part, we study the analytic properties and integrable structure of the meson spectrum in the large~$N_c$ limit of two-dimensional QCD. In this regime, the integral equation governing meson masses is shown to be equivalent to a TQ-Baxter equation, characteristic of integrable models. This reformulation gives access to the analytic structure of the spectrum in the complex plane of the quark masses. We demonstrate that the description via a spectral curve persists in a broader class of "generalized QCD" models with BF-type interactions, enabling the identification of new infrared phases and critical points. The second part focuses on the symmetries and renormalization group flows of disturbed two-dimensional Conformal Field Theories (CFTs). We propose a new class of non-linear integral equations that encode the finite-size spectrum along RG flows between Virasoro minimal models, motivated by anomaly matching of non-invertible symmetries. Furthermore, we characterize a family of non-topological yet conserved defects in disturbed CFTs. These defects give rise to conserved charges along the RG flow, and extend topological symmetries. Applying this framework to minimal models, we establish the existence of infinite sets of non-local commuting charges, beyond previously known integrable deformations. The final part examines the Schur quantization of four-dimensional N=2 supersymmetric theories and its interplay with RG flows, along with related mathematical and physical problems. We construct the quantization of the complex Teichmüller space, which describes the outcome of Schur quantization for class S theories. The quantum analytic Langlands correspondence emerges as a special case of Schur quantization. In a particular limit, this correspondence yields a geometric characterization of the spectrum of the quantum Hitchin integrable system in terms of real opers. Using the Separation of Variables method for the Gaudin integrable model, we provide a proof of the analytic Langlands correspondence for genus-zero curves.