Exact Results in Quantum Theory

I will analyse quench processes in two dimensional quantum field theories with infinite number of conservation laws which also include fermionic charges that close a N=1 supersymmetric algebra. While in general the quench protocol induces a breaking of supersymmetry, we show that there are particular initial states which ensure the persistence of supersymmetry also for the dynamics out of equilibrium. We will discuss the conditions that identify such states and, as application, we present the significant case of the Sine-Gordon model at its supersymmetric point. I will also address the issue of the Generalized Gibbs Ensemble in the presence of fermionic conserved charges. If time allows we will also discuss the N=2 SUSY case where it is possible to compute exactly the infinite-time limit of certain correlation functions.