Correlation functions in finite temperature CFT and black hole singularities

Abstract

We compute thermal 2-point correlation functions in the black brane AdS5 background dual to 4d CFT’s at finite temperature for operators of large scaling dimension. We find a formula that matches the expected structure of the OPE. It exhibits an exponentiation property, whose origin we explain. We also compute the first correction to the two-point function due to graviton emission, which encodes the proper time from the event horizon to the black hole singularity.

Thermal correlation functions in CFT and factorization

“…This was studied in general in [1] (see also [2]) for the case of the 2-point function, finding an elegant expansion in terms of Gegenbauer polynomials, whose coefficients incorporate both the OPE data as well as the non-zero vacuum expectation values in the thermal background. For strongly coupled theories with a gravity dual this structure naturally arises from the computation of the 2-point function in the black brane background, as recently shown for operators of large dimension in [3]. 1 It is interesting to note that, through the Eigenstate Thermalization Hypothesis, the thermal 2-point function is related to Heavy-Light-Light-Heavy 4-point correlators at T = 0 (e.g.…”

“…As a result, the value of the bulk cubic coupling enters only as an overall constant in the final result. Just as in [3], we will focus on correlators of operators with large dimension. In this case, the propagator can be computed in the WKB approximation, where it is represented as the exponential of the geodesic length of the trajectory for a particle moving between the boundary point where the operator is inserted up to the point of interaction.…”

Tidal excitation as mixing in thermal CFT

“…The high dimension operator in [1] mixes with multistress tensors, which are directly associated with strong gravity. The two-point correlator in [2] detects the presence of strong gravity by its modification of the propagation of massive bulk scalars by strong gravity, again represented by multistress tensors in the boundary. The tidal excitation mechanism crucial for removing a bulk-cone singularity (see [9]) in [3] explicitly diffuses the original signal into other signal modes.…”

“…As will be seen below, these operators are formally gauge singlet operators constructed with gauge covariant derivatives, but as emphasised above, the gauge fields in the formal expressions just prepares the operators for the requisite projection integrals that tunes their internal structure to thermal equilibrium. 2 In this first study of mixing in singlet models/models with higher spin symmetry, we confine our attention to two of the simplest operators that can mix, the ground state scalar and the energy density, which is the time-time component of the stress-energy tensor, dual to bulk gravitons and undoubtedly important in gauge-gravity duality.…”

“…One theme is to apply thermal field theory to the problem. Another is to look for field theory probes that uniquely focus on the bulk regions of interest, be it the singularity [1,2], the region just outside the photon sphere [3], or the region just inside, which support evanescent modes [4][5][6][7][8]. The filtering of information to focus on the respective phenomenon is achieved by, a) studying particular higher dimension operators that only couple in regions of high curvature, b) noting corrections to the propagation of particles from the higher curvature regions, c) finding depletion of elastic scattering in black hole geometries by tidal excitations of the scattered particles, or d) extraction of zero frequency spectral densities on the boundary, corresponding to the propensity of an operator to excite the thermal background (the black hole) by a given momentum.…”