Conformal conservation laws from compact representation theory

Fields satisfying conservation conditions are among the simplest spinning multiplets in conformal field theory. Paradoxically, their kinematics are significantly more difficult to fix than their non-conserved counterparts. This is because, in addition to the conformal Ward identities, the conservation laws must be imposed at each point, and such direct calculations get increasingly complicated as the spin grows. In this talk, after reviewing the conformal kinematics of conserved fields, I will uncover a duality with the representation theory of a compact group, which is significantly simpler to study. I will then sketch how to use this to classify all three-point structures of conserved tensors in arbitrary Lorentz representations. Beyond the conformal bootstrap, this work has applications within inflationary cosmology and higher-spin holography.