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.
Highlights
One of the most active lines of research on non-equilibrium phenomena of the past two decades concerns relativistic hydrodynamics
The symmetries of the microscopic Schwinger-Keldysh path integral, such as unitarity and CPT invariance, have been used to constrain the form of the low-energy SchwingerKeldysh effective action and to re-derive the conventional phenomenological formulation of hydrodynamics in some limiting regimes. It was shown in ref. [31], see refs. [32, 33], how the local second law of thermodynamics can be obtained as the Noether current of a symmetry of the effective action
We identified the holographic dual of the degrees of freedom relevant for the fluid dynamic behavior with Wilson lines extending between the conformal boundary and the cutoff hypersurface, much in the spirit of ref. [43], see refs. [44, 45]
Summary
Let us briefly review the main ingredients that enter the construction of Schwinger-Keldysh effective actions. In this work we are interested in the low-energy effective action Seff for charged matter at finite temperature in a fixed d-dimensional background In this case, the external sources are taken to be a pair of U(1) flavor fields AR μ and AL μ which live on two different spacetimes, dubbed as R and L, and transform as connections under two independent gauge transformations parameterized by ΛR and ΛL, AR μ → AR μ + ∂μΛR , AL μ → AL μ + ∂μΛL ,. Having defined the low-energy degrees of freedom, the most general Schwinger-Keldysh effective action Seff can be constructed requiring compatibility with the symmetries. See, e.g., refs. [20, 31]
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