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

Abstract

We extend our study of the large-N expansion of general non-equilibrium manybody 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 ± -defined as the values of M on the forward and backward segments of the closed time contour -for their linear combinations M cl and M 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 ± corresponding to the forward and backward segments of the closed time contour, and Σ ∧ 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 cl , and a "quantum embellishment" part Σ 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.

Baby universes and worldline field theories

“…Whether such models make sense, and how one might interpret off-shell string field theory in the language described above, deserve further investigation 24 . A relevant perspective may be offered by the worldsheet description of out-of-equilibrium dynamics, as recently investigated in [106,107].…”

Tropological sigma models

“…But what would string perturbation theory look like, when extended far from equilibrium? This question was addressed using the techniques of large-N dualities [15][16][17], leading to a perhaps surprising prediction: the genus expansion of string perturbation theory, familiar from the study of equilibrium states, should undergo a refinement, whereby the JHEP06(2024)135 string worldsheets Σ contributing to the such should decompose into three parts: one, Σ + , at least roughly corresponds to the evolution forward in time, another -Σ − -corresponds to the evolution backwards in time, and finally the "wedge region" Σ ∧ connecting Σ + to Σ − , corresponds to the time in the far future where the two branches of the Schwinger-Keldysh time contour meet. It is this wedge region Σ ∧ whose hypothetical worldsheet description remains quite enigmatic.…”

“…In the absence of a direct worldsheet construction, this question was addressed in [15][16][17] with the use of expected holographic duality to gauge theories in the large-N expansion, for which the formulation away from equilibrium using the Schwinger-Keldysh contour is known. The result found in [15][16][17] strongly suggests that in nonequilibrium string perturbation theory, the standard expression for observable amplitudes in terms of a sum over contributions from worldsheets Σ g of genus g, weighted by the appropriate power of the string coupling constant g s ,…”

Stringy ER = EPR

“…target space. The contour consists of Euclidean caps at either end, which ensure convergence of the functional integral, joined along a Lorentzian excursion in the middle [84][85][86][87]. Several such contours in the complex target time plane are pictured in figure 12.…”