Holographic Schwinger-Keldysh effective field theories

Abstract

We construct a holographic dual of the Schwinger-Keldysh effective action for the dissipative low-energy dynamics of relativistic charged matter at strong coupling in a fixed thermal background. To do so, we use a mixed signature bulk spacetime whereby an eternal asymptotically anti-de Sitter black hole is glued to its Euclidean counterpart along an initial time slice in a way to match the desired double-time contour of the dual field theory. Our results are consistent with existing literature and can be regarded as a fully-ab initio derivation of a Schwinger-Keldysh effective action. In addition, we provide a simple infrared effective action for the near horizon region that drives all the dissipation and can be viewed as an alternative to the membrane paradigm approximation.

Figures (3)

• Figure 1: The Schwinger-Keldysh contour at finite temperature. Time flows forward from $t=0$ to $t=+\infty$ and back to where the initial state is defined. The latter is represented by an imaginary time segment identified along the circles with period $\beta=1/T$.
• Figure 2: On the left-hand side (a): an eternal AdS black hole where the arrows indicate the flow of time. We cut along the an initial time slice at $t=0$ and a late time slice at $t=\hat{t}$ (dashed lines), and we keep the region in between them. On the right-hand side (b): a Euclidean black hole in AdS with period $\beta$. We cut along finite time slices at $\tau=0$ and $\tau = \beta-\sigma$ (dashed lines). For simplicity we have depicted the spacetimes in $2+1$ dimensions in the global time and radial coordinate.
• Figure 3: The mixed signature bulk spacetime containing two Lorentzian regions, ${\cal M}_R$ and ${\cal M}_L$, with future horizons identified and one Euclidean black hole ${\cal M}_E$ glued smoothly together along finite time slices. Notice that in order to simplify the drawing we flipped the left segment with respect to the way it appears in fig. \ref{['fig.spacetime']}, so that the left time increases now upwards.