Holographic Schwinger-Keldysh field theory of SU(2) diffusion

We construct effective field theory for SU(2) isospin charge diffusion, based on holographic Schwinger-Keldysh contour arXiv:2008.01269. The model consists of a probe gauge in doubled Schwarzschild-AdS$_5$ geometry. Accurate to first order derivative expansion, we analytically compute the action up quartic hydrodynamical fields. contains both non-Gaussianity noises and nonlinear interactions between dynamical variables. Moreover, captures thermal quantum fluctuations, which perfectly satisfy Kubo-Martin-Schwinger (KMS) symmetry at level. Interestingly, KMS symmetry, is crucial formulating non-equilibrium many-body system, found have nice interpretation.