Higher Spin Symmetry and Its Breaking in Conformal Field Theories

I will discuss Conformal Field Theories (CFTs) in spacetime dimensions d>2. First, assuming that the theory has higher spin conserved current I will demonstrate that the theory is essentially free. Secondly, I will consider higher spin symmetry to be broken at the quantum level. I will show that in this case certain observables are fixed by the broken higher spin symmetry. Finally, I will talk about the most general higher spin symmetry breaking scenario and explain how one can use unitarity and crossing to constrain the spectrum in the broken phase. This will lead us to certain predictions about the spectrum of general CFTs, in particular, models relevant for 3d phase transitions like 3d Ising model.