Quantum aspects of spacetime Modular Berry phases and entanglement islands

Holography has proven to be a powerful guiding principle in the study of quantum gravity, offering a duality between a lower-dimensional quantum field theory and a higher-dimensional gravitational theory. This dissertation explores two topics in holography: modular Berry phases and entanglement islands. It examines the role of geometric phases in the AdS/CFT correspondence, showing that the modular Berry phase in the CFT is holographically dual to the bulk symplectic form in the AdS space. Additionally, it investigates the island formula and its potential application to cosmology, focusing on the possible existence of islands in Friedmann-Robertson-Walker (FRW) cosmologies with thermal radiation, cosmological constant, and non-zero spatial curvature.