Research Projects

This proposal focuses on the classification of quantum entanglement in quantum field theories, a long-standing issue. Recent developments suggest that topological quantum field theory (TQFT) may be a way to study entanglement, particularly in relation to the 3d Chern-Simons theory. TQFT is closely connected to knot theory, which provides a way to compute quantum states and study their entanglement properties. The Chern-Simons theory is defined on a three-manifold with boundary, and the Chern-Simons partition function corresponds to a quantum state. This state lives in the Hilbert space specific to the manifold's boundary. The Chern-Simons theory is an example of a finite-dimensional quantum mechanical system, as the Hilbert spaces are finite-dimensional. One crucial aspect of Chern-Simons theory is that the Hilbert space admits a tensor decomposition for disconnected boundaries. This allows for the study of entanglement entropy by tracing out a subset of Hilbert spaces, known as'multi-boundary entanglement.' Current studies are limited to genus 0 or genus 1 boundaries. The project aims to study and improve the current understanding of multi-boundary entanglement in Chern-Simons theory. The first part explores the entanglement properties of quantum states on genus 2 surfaces, aiming to understand how genus plays a role in determining these features. The second part aims to give the multi-boundary entanglement entropy a geometrical interpretation by showing that entropy may capture information about the volume of a specific manifold.