Bootstrapping CFTs with moduli spaces

Many SUSY CFTs have a continuous space of vacua, known as a moduli space. Understanding how moduli spaces manifest in CFT data is crucial for classifying and constraining SUSY CFTs, yet remains an open problem. Specifically, it is not clear how to determine whether a CFT has a moduli space based on the CFT data alone. In this talk we discuss this problem and make significant progress towards its resolution. First we discuss the case where there is a continuous global symmetry which is spontaneously broken on the moduli space, and find a universal necessary condition on CFT data in order for the CFT to have a moduli space. Then we discuss the generic case and use an analytic bootstrap equation to find perturbative sum rules which constrain the CFT data of a CFT with a moduli space. Based on work with G. Cuomo and L. Rastelli