Beyond $a=c$: Gravitational Couplings to Matter and the Stress Tensor OPE

TL;DR

The paper establishes that in large N CFTs with a substantial higher-spin gap, the TT OPE becomes universal and isolated, with ⟨TT𝒪⟩ suppressed as Δ_gap^{-2}, which implies a local Einstein gravity dual and ties bulk derivative suppression to the CFT gap. By employing spinning conformal Regge theory and unitarity constraints, the authors derive both large-gap results and general conformal collider bounds, extending them to a wide class of ⟨T𝒪₁𝒪₂⟩ couplings with spins ≤2 and to parity-even/odd structures. They show holographic implications, including no consistent truncation to Gauss–Bonnet gravity unless higher-spin effects are accounted for, and relate Δ_gap to AdS bulk derivatives via a precise counting correspondence. The work also demonstrates that all three tensor structures of ⟨TTT⟩ remain nonzero in interacting CFTs when coupled to a general spectrum, highlighting the rich dynamical content constrained by unitarity and analyticity. Overall, the paper provides a systematic framework connecting CFT data, Regge theory, and bulk gravity to constrain matter couplings to gravity and to reveal the deep role of the higher-spin gap in holography.

Abstract

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, $\langle TT{\cal O}\rangle=0$, where ${\cal O}\neq T$ is a single-trace primary. We show that this follows from a suppression of $\langle TT{\cal O}\rangle$ by powers of the higher spin gap, $Δ_{\rm gap}$, dual to the bulk mass scale of higher spin particles, and explain why $\langle TT{\cal O}\rangle$ is a more sensitive probe of $Δ_{\rm gap}$ 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 $\langle T{\cal O}_1{\cal O}_2\rangle$ where ${\cal O}_i$ have spin $s_i\leq 2$, we are led to propose that $1/Δ_{\rm gap}$ 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 $\langle TTT\rangle$ in the free-field basis are nonzero in interacting CFTs.

Figures (4)

• Figure 1: In AdS, the cubic coupling of a scalar to two gravitons is suppressed as $M_{\rm HS}^{-2}$, where $M_{\rm HS}\sim \Delta_{\rm gap}$ is the mass scale of higher spin particles.
• Figure 2: A prototypical leading Regge trajectory, here depicted in $d=4$. Recall that $i\nu\equiv \Delta-{d\over2}$. We have indicated the locations of the stress tensor, $T$, and the lightest spin-four operator, ${\cal O}_4$ which defines $\Delta_{\rm gap}$ through $j(-i(\Delta_{\rm gap}-2))=4$.
• Figure 3: In the Regge limit, the connected four-point function $\langle \Psi\phi\phi\Psi\rangle$ is dominated by Reggeon exchange in the $\Psi\Psi \rightarrow \phi\phi$ channel. Unitarity bounds the three-point couplings.
• Figure 4: CFT three-point functions $\langle{T{\cal O} j(\nu)}\rangle$, computed holographically from AdS three-point diagrams like the one shown, are "extremal" when $\tau_{\cal O} = \Delta_T+\Delta(\nu)+2n$.