45 research outputs found
Exact results for the entanglement entropy and the energy radiated by a quark
We consider a spherical region with a heavy quark in the middle. We compute
the extra entanglement entropy due to the presence of a heavy quark both in
Super Yang Mills and in the Chern-Simons matter
theory (ABJM). This is done by relating the computation to the expectation
value of a circular Wilson loop and a stress tensor insertion. We also give an
exact expression for the Bremsstrahlung function that determines the energy
radiated by a quark in the ABJM theory.Comment: 23+12 pages, 8 figures. V2: references added. V3: references added.
V4: small comments and references adde
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
Bulk locality from modular flow
We study the reconstruction of bulk operators in the entanglement wedge in
terms of low energy operators localized in the respective boundary region. To
leading order in , the dual boundary operators are constructed from the
modular flow of single trace operators in the boundary subregion. The
appearance of modular evolved boundary operators can be understood due to the
equality between bulk and boundary modular flows and explicit formulas for bulk
operators can be found with a complete understanding of the action of bulk
modular flow, a difficult but in principle solvable task. We also obtain an
expression when the bulk operator is located on the Ryu-Takayanagi surface
which only depends on the bulk to boundary correlator and does not require the
explicit use of bulk modular flow. This expression generalizes the geodesic
operator/OPE block dictionary to general states and boundary regions.Comment: 36 pages, 2 figure
Entropy, Extremality, Euclidean Variations, and the Equations of Motion
We study the Euclidean gravitational path integral computing the Renyi
entropy and analyze its behavior under small variations. We argue that, in
Einstein gravity, the extremality condition can be understood from the
variational principle at the level of the action, without having to solve
explicitly the equations of motion. This set-up is then generalized to
arbitrary theories of gravity, where we show that the respective entanglement
entropy functional needs to be extremized. We also extend this result to all
orders in Newton's constant , providing a derivation of quantum
extremality. Understanding quantum extremality for mixtures of states provides
a generalization of the dual of the boundary modular Hamiltonian which is given
by the bulk modular Hamiltonian plus the area operator, evaluated on the
so-called modular extremal surface. This gives a bulk prescription for
computing the relative entropies to all orders in . We also comment on how
these ideas can be used to derive an integrated version of the equations of
motion, linearized around arbitrary states.Comment: 37 pages; v2: typos fixed and new references added; v3: new
references and minor clarifications adde
Inside Out: Meet The Operators Inside The Horizon
Based on the work of Heemskerk, Marolf, Polchinski and Sully (HMPS), we study
the reconstruction of operators behind causal horizons in time dependent
geometries obtained by acting with shockwaves on pure states or thermal states.
These geometries admit a natural basis of gauge invariant operators, namely
those geodesically dressed to the boundary along geodesics which emanate from
the bifurcate horizon at constant Rindler time. We outline a procedure for
obtaining operators behind the causal horizon but inside the entanglement wedge
by exploiting the equality between bulk and boundary time evolution, as well as
the freedom to consider the operators evolved by distinct Hamiltonians. This
requires we carefully keep track of how the operators are gravitationally
dressed and that we address issues regarding background dependence. We compare
this procedure to reconstruction using modular flow, and illustrate some formal
points in simple cases such as AdS and AdS.Comment: 48 pages, 14 figure