  We study the positivity properties of the leading Regge trajectory in
higher-dimensional, unitary, conformal field theories (CFTs). These conditions
correspond to higher spin generalizations of the averaged null energy condition
(ANEC). By studying higher spin ANEC, we will derive new bounds on the
dimensions of charged, spinning operators and prove that if the
Hofman-Maldacena bounds are saturated, then the theory has a higher spin
symmetry. We also derive new, general bounds on CFTs, with an emphasis on
theories whose spectrum is close to that of a generalized free field theory. As
an example, we consider the Ising CFT and show how the OPE structure of the
leading Regge trajectory is constrained by causality. Finally, we use the
analytic bootstrap to perform additional checks, in a large class of CFTs, that
higher spin ANEC is obeyed at large and finite spin. In the process, we
calculate corrections to large spin OPE coefficients to one-loop and higher in
holographic CFTs.
