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 $\Sigma$ in the dual string theory expansion was found to exhibit a triple decomposition into the parts $\Sigma^\pm$ corresponding to the forward and backward segments of the closed time contour, and $\Sigma^\wedge$ which corresponds to the instant in time where the two segments meet. After the Keldysh rotation, we find that the worldsheet surface $\Sigma$ of the dual string theory undergoes a very different natural decomposition: $\Sigma$ consists of a "classical" part $\Sigma^{\textrm{cl}}$, and a "quantum embellishment" part $\Sigma^{\textrm{qu}}$. We show that both parts of $\Sigma$ 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 $\Sigma$ 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.