Demystifying integrable QFTs in AdS: No-go theorems for higher-spin charges

Higher-spin conserved currents and charges feature prominently in integrable 2d QFTs in flat space. Motivated by the question of integrable field theories in AdS space, we consider the consequences of higher-spin currents for QFTs in AdS$_2$, and find that their effect is much more constraining than in flat space. Specifically, it is impossible to preserve: (a) any higher-spin charges when deforming a free field of generic mass by interactions (even boundary-localized), or (b) any spin-4 charges when deforming a CFT by a Virasoro primary. Therefore, in these settings, there are no integrable theories in AdS with higher-spin conserved charges. Along the way, we explain how higher-spin charges lead to integer spacing in the spectrum of primaries, sum rules on the OPE data, and constraints on correlation functions. We also explain a key difference between AdS and flat space: in AdS one cannot `partially' conserve a higher-spin current along particular directions, since the AdS isometries imply full conservation. Finally, we describe the consequences of higher-spin symmetry breaking on the spectrum of long-range models.