  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.
