Keldysh Rotation in the Large-N Expansion and String Theory Out of Equilibrium

We extend our study of the large-N expansion of general non-equilibrium many-body systems with matrix degrees of freedom M, and its dual description as a sum over surface topologies in a dual string theory, to the Keldysh-rotated version of the Schwinger-Keldysh formalism. The Keldysh rotation trades the original fields M_pm - defined as the values of M on the forward and backward segments of the closed time contour - for their linear combinations M_{textrm{cl}} and M_{textrm{qu}}, known as the "classical" and "quantum" fields. First we develop a novel "signpost" notation for non-equilibrium Feynman diagrams in the Keldysh-rotated form, which simplifies the analysis considerably. Before the Keldysh rotation, each worldsheet surface Σ in the dual string theory expansion was found to exhibit a triple decomposition into the parts Σ^pm corresponding to the forward and backward segments of the closed time contour, and Σ^wedge which corresponds to the instant in time where the two segments meet. After the Keldysh rotation, we find that the worldsheet surface Σ of the dual string theory undergoes a very different natural decomposition: Σ consists of a "classical" part Σ^{textrm{cl}}, and a "quantum embellishment" part Σ^{textrm{qu}}. We show that both parts of Σ carry their own independent genus expansion. The non-equilibrium sum over worldsheet topologies is naturally refined into a sum over the double decomposition of each Σ into its classical and quantum part. We apply this picture to the classical limits of the quantum non-equilibrium system (with or without interactions with a thermal bath), and find that in these limits, the dual string perturbation theory expansion reduces to its appropriately defined classical limit.