Einstein gravity 3-point functions from conformal field theory

Abstract

We study stress tensor correlation functions in four-dimensional conformal field theories with large N and a sparse spectrum. Theories in this class are expected to have local holographic duals, so effective field theory in anti-de Sitter suggests that the stress tensor sector should exhibit universal, gravity-like behavior. At the linearized level, the hallmark of locality in the emergent geometry is that stress tensor three-point functions T T T , normally specified by three constants, should approach a universal structure controlled by a single parameter as the gap to higher spin operators is increased. We demonstrate this phenomenon by a direct CFT calculation. Stress tensor exchange, by itself, violates causality and unitarity unless the three-point functions are carefully tuned, and the unique consistent choice exactly matches the prediction of Einstein gravity. Under some assumptions about the other potential contributions, we conclude that this structure is universal, and in particular, that the anomaly coefficients satisfy a ≈ c as conjectured by Camanho et al. The argument is based on causality of a four-point function, with kinematics designed to probe bulk locality, and invokes the chaos bound of Maldacena, Shenker, and Stanford.

More on supersymmetric and 2d analogs of the SYK model

“…But given that we have already derived an expression for the four-point function, we can also simply continue that expression to the chaos region [45]. In our 2d setting, this is the same as the Regge limit studied more generally in [23,24,47] and recently in [25,[48][49][50][51]. Whereas the retarded kernel approach only gives the exponents λ L (p), this direct continuation approach will give the actual OTO correlator, including the coefficients of the contributions with different values of p. This section also might be useful as an example where the Regge limit can be worked out explicitly.…”

Bounds for OPE coefficients on the Regge trajectory

“…We want to consider the expansion of the correlation function in terms of t-channel conformal blocks, 11) where C ijk are the OPE coefficients and G ∆,J (z,z) is the conformal block associated to the exchange of a primary of dimension ∆ and spin J. For our purposes, it is more useful to consider the spectral representation [12,13,22]…”

“…Different kinematical limits focus on different subsets of the CFT data. One such example is the light-cone limit [1][2][3][4] which has recently been used to prove the conformal collider bounds [5] and the CEMZ bounds [6] on OPE coefficients of conserved currents and stress-tensors from the CFT side [7][8][9][10][11]. Here we study the Regge limit of CFT fourpoint correlators [12,13], which are dominated by the leading Regge trajectory, i.e.…”

Non-renormalization for non-supersymmetric black holes

“…For example, physical principles require the ratio c/a ∼ 1 for conformal field theory in a curved background, with precise "conformal collider" bounds easily excluding c = 0 [19][20][21]. Such apparent conflicts are simply due to the additional matter contributions that arise when we take dynamical gravity into account.…”

“…At one-derivative order, we have the symplectic invariant 21) and its complex conjugate, where we have recognized the auxiliary Weyl multiplet field T + µν from its two-derivative equation of motion (3.14). The only other possible symplectic invariant with one derivative is F + µν · X, which vanishes 22) using the explicit form (4.10) for the dual tensor G + I µν and the special geometry identity N IJ X I = F I .…”