2D CFTs: Large charge is not enough to control the dynamics

Abstract

In this note we study two-dimensional conformal field theories at large global charge. Since the large-charge sector decouples from the dynamics, it does not control the dynamics and an effective field theory construction that works in higher-dimensional theories fails. It is however possible to use large charge in a double-scaling limit when another controlling parameter is present. We find some general features of the spectrum of models that admit a nonlinear sigma model description in a Wentzel-Kramers-Brillouin approximation and use the large-charge sector of the solvable $SU(2{)}_{k}$ Wess-Zumino-Witten model to argue the regimes of applicability of both the large-$Q$ expansion and the double-scaling limit.