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 CT. Neither supersymmetry nor unitary is assumed. Higher-twist 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.
Highlights
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
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 CT
When applied to four-point functions of local CFT operators, the irreducible representations of the algebra, known as Virasoro Conformal Blocks, capture the thermal properties of black holes, the information paradox associated with late-time decay of correlators in semiclassical gravity [6,7,8], the universal properties of Renyi and entanglement entropy [9,10,11,12] leading to a proof of Ryu-Takayanagi formula [13] in AdS3/CFT2, the maximal growth of chaos in gravity [14, 15], the nonperturbative resolution of some perturbative violations of unitarity [16, 17], and more, all without appealing to a gravitational Lagrangian
Summary
We will warm up with an analysis of the holographic heavy-light four-point function at leading order in the OPE. We consider a simpler case where all other matter fields decouple or can be integrated out, and that self-interactions of φ can be neglected.9 At infinite c, this approximation is appropriate and was adopted in the analysis of the d = 2 Virasoro vacuum block [7, 8], where the heavy operator OH enters only through the metric that it induces. Let us begin by trying to solve for the two-point function in a short-distance expansion, where the two light-operators approach each other, i.e. in their OPE limit. When these two operators are close, their correlator should depend only on the behavior of the bulk fields near the boundary, and in this limit the two-point function can be computed in a large r expansion. The variable w turns out to be a very convenient coordinate to use
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