Real-time finite-temperature correlators from AdS/CFT

TL;DR

This paper develops a method using AdS/CFT correspondence to compute real-time finite-temperature 3-point correlators in ${ m N}=4$ super Yang-Mills theory, introducing a novel prescription involving the Penrose diagram and Schwinger-Keldysh propagators.

Contribution

It proposes a new prescription for calculating real-time finite-temperature correlators in AdS/CFT, utilizing the Penrose diagram and adapting Veltman's circling rules to Witten diagrams.

Findings

Derived a general expression for 3-point real-time Green's functions.

Validated the prescription with KMS identities and zero-temperature limit.

Obtained a simple causal form for retarded 3-point functions.

Abstract

In this paper we use AdS/CFT ideas in conjunction with insights from finite temperature real-time field theory formalism to compute 3-point correlators of ${\cal N}{=}4$ super Yang-Mills operators, in real time and at finite temperature. To this end, we propose that the gravity field action is integrated only over the right and left quadrants of the Penrose diagram of the Anti de Sitter-Schwarzschild background, with a relative sign between the two terms. For concreteness we consider the case of a scalar field in the black hole background. Using the scalar field Schwinger-Keldysh bulk-to-boundary propagators, we give the general expression of a 3-point real-time Green's correlator. We then note that this particular prescription amounts to adapting the finite-temperature analog of Veltman's circling rules to tree-level Witten diagrams, and comment on the retarded and Feynman scalar bulk-to-boundary propagators. We subject our prescription to several checks: KMS identities, the largest time equation and the zero-temperature limit. When specializing to a particular retarded (causal) 3-point function, we find a very simple answer: the momentum-space correlator is given by three causal (two retarded and one advanced) bulk-to-boundary propagators, meeting at a vertex point which is integrated from spatial infinity to the horizon only. This result is expected based on analyticity, since the retarded n-point functions are obtained by analytic continuation from the imaginary time Green's function, and based on causality considerations.