Conformal field theory and quantum many-body physics

Finding general principles that go beyond the well developed thermodynamic theory of equilibrium is one of the most important challenges of current research. As thermodynamics is a theory for extensive quantities such as the energy, and its fluctuations in space, one looks for a parallel theory that asks about quantities that are spatially transported, say from left to right, over time: currents and their fluctuations. In this series of lectures, I will develop a number of topics to answer some of these questions. I will restrict to the context of two-dimensional spacetime for simplicity, going over the basic ideas and exact results in conformal field theory (CFT) and integrable systems. Topics covered will be: (1) Nonequilibrium steady states in many-body quantum systems that admit ballistic transport. (2) Fluctuations of transported quantities at large times using the largedeviation framework. (3) The relation between static and dynamic fluctuations, and the entanglement entropy and its growth after quenches. As time permits, I will overview hydrodynamic techniques, twist fields techniques, and, in CFT, conformal welding techniques for large-deviation functions.