The Holographic Shape of Entanglement and Einstein's Equations

We study shape-deformations of the entanglement entropy and the modular Hamiltonian for an arbitrary subregion and state (with a smooth dual geometry) in a holographic conformal field theory. More precisely, we study a double-deformation comprising of a shape deformation together with a state deformation, where the latter corresponds to a small change in the bulk geometry. Using a purely gravitational identity from the Hollands-Iyer-Wald formalism together with the assumption of equality between bulk and boundary modular flows for the original, undeformed state and subregion, we rewrite a purely CFT expression for this double deformation of the entropy in terms of bulk gravitational variables and show that it precisely agrees with the Ryu-Takayanagi formula including quantum corrections. As a corollary, this gives a novel, CFT derivation of the JLMS formula for arbitrary subregions in the vacuum, without using the replica trick. Finally, we use our results to give an argument that if a general, asymptotically AdS spacetime satisfies the Ryu-Takayanagi formula for arbitrary subregions, then it must necessarily satisfy the non-linear Einstein equation.Comment: 37 pages, 3 figure

Higher Spin Fronsdal Equations from the Exact Renormalization Group

We show that truncating the exact renormalization group equations of free $U(N)$ vector models in the single-trace sector to the linearized level reproduces the Fronsdal equations on $AdS_{d+1}$ for all higher spin fields, with the correct boundary conditions. More precisely, we establish canonical equivalence between the linearized RG equations and the familiar local, second order differential equations on $AdS_{d+1}$, namely the higher spin Fronsdal equations. This result is natural because the second-order bulk equations of motion on $AdS$ simply report the value of the quadratic Casimir of the corresponding conformal modules in the CFT. We thus see that the bulk Hamiltonian dynamics given by the boundary exact RG is in a different but equivalent canonical frame than that which is most natural from the bulk point of view.Comment: 34 pages, 4 figures; v2: typos fixed, better abstrac

Topics in quantum field theory and holography

In the first part of this thesis, we will study free fermions as models for topological insulators, on gravitational backgrounds which include both torsion and curvature, in d = 2 + 1 and d = 4 + 1 dimensions. We compute the parity-odd effective actions for these systems, and use these effective actions to deduce the structure of anomalies (in particular, the torsional contributions) in the edge states which live on the boundary between two different bulk phases. We also give intrinsic, microscopic derivations of these torsional anomalies by considering Hamiltonian spectral flow for edge states in the presence of torsion. All of these calculations fit perfectly within the well-known framework of anomaly inflow, and extend the framework to include torsional contributions. Furthermore, our condensed-matter-inspired setup provides natural resolutions to some previously ill-understood ultraviolet divergences in intrinsic edge calculations of torsional anomalies. In the second part of this thesis, we consider the Bosonic and Fermionic U(N) vector models close to their free fixed points, with single-trace deformations turned on. We derive the higher-spin holographic duals corresponding to these vector models by first formulating these theories in terms of the geometry of infinite jet bundles, and then interpreting the renormalization group equations for single-trace deformations as Hamilton's equations of motion on a one-higher dimensional emergent spacetime. We evaluate the resulting bulk on-shell action explicitly, and show that it reproduces all the correlation functions of the vector models. Furthermore, we show that the linearized bulk equations of motion contain the Fronsdal equations of motion on Anti-de Sitter space, thus proving equivalence with Vasiliev higher-spin theories to linearized order. The bulk theory we derive is consistent with the known AdS/CFT framework, and gives a concrete boundary to bulk implementation of AdS/CFT as a geometrization of the renormalization group