Large-N Expansion and String Theory Out of Equilibrium

We analyze the large-$N$ expansion of general non-equilibrium systems with fluctuating matrix degrees of freedom and $SU(N)$ symmetry, using the Schwinger-Keldysh formalism and its closed real-time contour with a forward and backward component. In equilibrium, the large-$N$ expansion of such systems leads to a sum over topologies of two-dimensional surfaces of increasing topological complexity, predicting the possibility of a dual description in terms of string theory. We extend this argument away from equilibrium, and study the universal features of the topological expansion in the dual string theory. We conclude that in non-equilibrium string perturbation theory, the sum over worldsheet topologies is further refined: Each worldsheet surface $\Sigma$ undergoes a triple decomposition into the part $\Sigma^+$ corresponding to the forward branch of the time contour, the part $\Sigma^-$ on the backward branch, and the part $\Sigma^\wedge$ that corresponds to the instant in the far future where the two branches of the time contour meet. The sum over topologies becomes a sum over the triple decompositions. We generalize our findings to the Kadanoff-Baym time contour relevant for systems at finite temperature, and to the case of closed and open, oriented or unoriented strings. Our results are universal, and follow solely from the features of the large-$N$ expansion without any assumptions about the worldsheet dynamics.