Membrane Paradigm, Gravitational $\mathrm{\Theta }$-Term and Gauge/Gravity Duality

Following the membrane paradigm, we explore the effect of the gravitational $\mathrm{\Theta }$-term on the behavior of the stretched horizon of a black hole in (3+1)-dimensions. We reformulate the membrane paradigm from a quantum path-integral point of view where we interpret the macroscopic properties of the horizon as effects of integrating out the region inside the horizon. The gravitational $\mathrm{\Theta }$-term is a total derivative, however, using our framework we show that this term affects the transport properties of the horizon. In particular, the horizon acquires a third order parity violating, dimensionless transport coefficient which affects the way localized perturbations scramble on the horizon. Then we consider a large-N gauge theory in (2+1)-dimensions which is dual to an asymptotically AdS background in (3+1)-dimensional spacetime to show that the $\mathrm{\Theta }$-term induces a non-trivial contact term in the energy-momentum tensor of the dual theory. As a consequence, the dual gauge theory in the presence of the $\mathrm{\Theta }$-term acquires the same third order parity violating transport coefficient.