• crossed product
• generalized entropy
• Geometric Modular Flow conjecture
• two-dimensional CFT
• von Neumann algebras

In this thesis, we study recent developments in gravitational entropy. We focus on the crossed product construction of local algebras in quantum field theory on a curved background. In particular, we investigate the Geometric Modular Flow conjecture, which allows us to study the entropy of generic regions of spacetime without any specific symmetry. The conjecture states that, given any quantum field theory, any region $\mathcal{U}$, and any Cauchy slice $\Sigma$ for the region, it is possible to find a state, referred to as the geometric state, whose modular flow for the algebra $\mathcal{A}(\mathcal{U})$ induces a diffeomorphism on $\Sigma$. Given a theory specified by an action $S$, we tackle the conjecture by explicitly preparing the geometric state with a Euclidean path integral weighted by a modified action $S + S'$. First, we find the conditions that $S'$ must satisfy. Then, we apply our construction to the case of two-dimensional conformal field theories coupled to dynamical gravity. Every two-dimensional metric $g$ is conformally flat and can be written as $g = e^{2w} \eta$. If the conformal factor $w$ is a degree of freedom of the theory, i.e., if we path integrate over it, then the obstruction to a geometric state is given by the trace anomaly of the CFT. We eliminate the trace anomaly by choosing $S'$ to be the Polyakov effective action, thus proving the conjecture in this setting.