Covariant phase space and the semi-classical Einstein equation

The covariant phase space formalism in general relativity is a covariant method for constructing the symplectic two-form, Hamiltonian and other conserved charges on the phase space of solutions to the Einstein equation with classical matter. In this note, we consider a generalization of this formalism to the semi-classical Einstein equation coupled to quantum matter. Given a family of solutions in semi-classical gravity, we define the semi-classical symplectic two-form -- a natural generalization of the classical sympelctic two-form -- as the sum of the gravitational symplectic form and the Berry curvature associated to the quantum state of matter. We show that the semi-classical symplectic two-form is independent of the Cauchy slice, and satisfies the quantum generalization of the classical Hollands-Iyer-Wald identity. For small perturbations, we also extend our discussion to gauge-invariantly defined subregions of spacetime, where the quantum contribution is replaced by the Berry curvature of certain special purifications involving the Connes cocycle. In the AdS/CFT context, the semi-classical symplectic form defined here is naturally dual to the Berry curvature in the boundary CFT.