Universal lowest-twist in CFTs from holography

Abstract

We probe the conformal block structure of a scalar four-point function in d ≥ 2 conformal field theories by including higher-order derivative terms in a bulk gravitational action. We consider a heavy-light four-point function as the boundary correlator at large central charge. Such a four-point function can be computed, on the gravity side, as a two-point function of the light operator in a black hole geometry created by the heavy operator. We consider analytically solving the corresponding scalar field equation in a near-boundary expansion and find that the multi-stress tensor conformal blocks are insensitive to the horizon boundary condition. The main result of this paper is that the lowest-twist operator product expansion (OPE) coefficients of the multi-stress tensor conformal blocks are universal: they are fixed by the dimension of the light operators and the ratio between the dimension of the heavy operator and the central charge C T . Neither supersymmetry nor unitary is assumed. Highertwist coefficients, on the other hand, generally are not protected. A recursion relation allows us to efficiently compute universal lowest-twist coefficients. The universality result hints at the potential existence of a higher-dimensional Virasoro-like symmetry near the lightcone. While we largely focus on the planar black hole limit in this paper, we include some preliminary analysis of the spherical black hole case in an appendix.Aside from making the expressions more compact, the variable σ defined above is convenient since it parameterizes the angle away from the lightcone of the path made when one varies 32 This fact is obvious for n = 0, since Q 0 (w) = 1. Taking Q 2n (w) ∼ w γn at small w, the field equation requires γ n = γ n−1 − 2 for the contributions from subsequent Q 2n (w)s to cancel against each other.33 In our convention, a n,n = (−4) J 2 c OPE (τ min , J) in d = 4.

Leading multi-stress tensors and conformal bootstrap

“…Furthermore, from (1.12) and (3.18), we find the OPE coefficients of twist-six triplestress tensor operators as a finite sum (for details see Section 3.4). Two such OPE coefficients for twist-6 triple-stress tensors were calculated holographically in [37] and agree with our results.…”

“…It is now clear that the correlator (3.16) can be written as a double sum over functions f 9+2(n+m) . We can thus write the stress tensor sector of the correlator in the lightcone limit at O(µ 3 ) as agree with the expression obtained holographically in [37].…”

“…In [37] the authors considered holographic CFTs dual to gravitational theories defined by the Einstein-Hilbert Lagrangian plus higher derivative terms and a scalar field minimally coupled to gravity in AdS d+1 . Interpreting the scalar propagator in an asymptotically…”

“…AdS d+1 black hole background as a heavy-heavy-light-light four point function, enabled the authors of [37] to extract the OPE coefficients of a few multi-stress tensor operators from holography (see also [38][39][40] for related work). Ref.…”

“…Ref. [37] also argued that the OPE coefficients of the leading, minimal-twist multi-stress operators are universal -they do not depend on the higher derivative terms in the Lagrangian. Their results agree with the general expressions obtained in this paper, upon substitution of the relevant quantum numbers.…”