Thermal effects in conformal field theories

Conformal Field Theories (CFTs) are special classes of quantum field theories that find applications ranging from critical phenomena to theories of quantum gravity via holography. Understanding thermal effects in CFTs is crucial: criticality is experimentally probed at finite temperature, and, from the holographic perspective, the study of thermal CFTs is dual to the study of black holes in Anti-de Sitter space. In this thesis, we explore the kinematics and dynamics of finite-temperature CFTs by analyzing broken and unbroken symmetries and adapting various analytical and numerical bootstrap approaches to finite-temperature correlation functions. These methods are non-perturbatively valid and can be tested against exactly solvable models, such as free theories and two-dimensional systems, as well as compared with perturbative calculations. The main applications discussed in this thesis concern one- and two-point functions and the free energy density in the O(N) models in three dimensions, with particular focus on the 3d Ising, XY, and Heisenberg models (N = 1,2,3).