Beyond a = c : gravitational couplings to matter and the stress tensor OPE

A bstract We derive constraints on the operator product expansion of two stress tensors in conformal field theories (CFTs), both generic and holographic. We point out that in large N CFTs with a large gap to single-trace higher spin operators, the stress tensor sector is not only universal, but isolated: that is, TTO =0 T T O = 0 , where O≠ T O ≠ T is a single-trace primary. We show that this follows from a suppression of TTO T T O by powers of the higher spin gap, Δ gap , dual to the bulk mass scale of higher spin particles, and explain why TTO T T O is a more sensitive probe of Δ gap than a − c in 4d CFTs. This result implies that, on the level of cubic couplings, the existence of a consistent truncation to Einstein gravity is a direct consequence of the absence of higher spins. By proving similar behavior for other couplings TO₁O₂ T O 1 O 2 where Oᵢ O i have spin s i ≤ 2, we are led to propose that 1 / Δ gap is the CFT “dual” of an AdS derivative in a classical action. These results are derived by imposing unitarity on mixed systems of spinning four-point functions in the Regge limit. Using the same method, but without imposing a large gap, we derive new inequalities on these three-point couplings that are valid in any CFT. These are generalizations of the Hofman-Maldacena conformal collider bounds. By combining the collider bound on TT couplings to spin-2 operators with analyticity properties of CFT data, we argue that all three tensor structures of 〈 TTT 〉 in the free-field basis are nonzero in interacting CFTs.