Universal Aspects of Entanglement in Quantum Field Theory and String Theory

We apply path integral methods to derive universal aspects of quantum entanglement in quantum field theory and string theory. These methods were originally used to understand vacuum entanglement between half spaces in Lorentz-invariant quantum field theories. In this work we begin by generalizing these results to spherical subregion in conformally invariant field theories and deriving a first law for entanglement entropy. For holographic CFT's, we also provide a simple derivation of the bulk entanglement first law. Next we address the question of entanglement and Hilbert space factorization in string theory by studying a simple two dimensional string theory dual to two-dimensional Yang Mills. Even though this is a closed string theory, we find that open strings appear upon restriction to a spatial subregion, as first noted by Susskind and Uglum. We show that the entangling surface acts as an ''entanglement brane'' on which open strings end, and host open string edge modes that are responsible for the entanglement entropy in the Hartle-Hawking vacuum. Finally, we elaborate on the extended Hilbert space factorization of Chern Simons theory and show how this arises naturally from a proper regularization of the entangling surface in the Euclidean path integral. The regularization amounts to stretching the entangling surface into a co-dimension one surface which hosts edge modes of the Chern Simons theory when quantized on a spatial subregion.