When a spacetime has boundaries, the entangling surface does not have to be necessarily compact and it may have boundaries as well. Then there appear a new, boundary, contribution to the entanglement entropy due to the intersection of the entangling surface with the boundary of the spacetime. We study the boundary contribution to the logarithmic term in the entanglement entropy in dimensions $d=3$ and $d=4$ when the entangling surface is orthogonal to the boundary. In particular, we compute a boundary term in the entropy of $\mathcal{N}=4$ supergauge multiplet at weak coupling. For gauge fields we use a prescription which is consistent with the positive area law. The boundary term is compared with the holographic calculation of the entropy based on the Ryu-Takayanagi proposal adapted appropriately to the present situation. We find a complete agreement between these two calculations provided the boundary conditions imposed on the gauge multiplet preserve $1/2$ of the supersymmetry and the extension of the boundary into the anti--de Sitter bulk is a minimal hypersurface.When a spacetime has boundaries, the entangling surface does not have to be necessarily compact and it may have boundaries as well. Then there appear a new, boundary, contribution to the entanglement entropy due to the intersection of the entangling surface with the boundary of the spacetime. We study the boundary contribution to the logarithmic term in the entanglement entropy in dimensions $d=3$ and $d=4$ when the entangling surface is orthogonal to the boundary. In particular, we compute a boundary term in the entropy of $\mathcal{N}=4$ supergauge multiplet at weak coupling. For gauge fields we use a prescription which is consistent with the positive area law. The boundary term is compared with the holographic calculation of the entropy based on the Ryu-Takayanagi proposal adapted appropriately to the present situation. We find a complete agreement between these two calculations provided the boundary conditions imposed on the gauge multiplet preserve $1/2$ of the supersymmetry and the extension of the boundary into the anti--de Sitter bulk is a minimal hypersurface.

arXiv:1703.04186Physical review. D/Physical review. D.May 31, 201719 citations

Holographic calculations of Rényi entropy

Ling-Yan Hung, Robert Cobb Myers, Michael Smolkin +1 more

arXiv:1110.1084Journal of High Energy PhysicsDec 1, 2011251 citations

Relative entropy and holography

David D. Blanco, Horacio Casini, Ling-Yan Hung +1 more

arXiv:1305.3182Journal of High Energy PhysicsAug 1, 2013374 citations

Holographic Calculation of Boundary Entropy

Tatsuo Azeyanagia, Andreas Karchb, Tadashi Takayanagia +1 more

We use the holographic proposal for calculating entanglement entropies to determine the boundary entropy of defects in strongly coupled two-dimensional conformal field theories. We study several examples including the Janus solution and show that the boundary entropy extracted from the entanglement entropy as well as its more conventional definition via the free energy agree with each other. Maybe somewhat surprisingly we find that, unlike in the case of a conformal field theory with boundary, the entanglement entropy for a generic region in a theory with defect carries detailed information about the microscopic details of the theory. We also argue that the g-theorem for the boundary entropy is closely related to the strong subadditivity of the entanglement entropy.We use the holographic proposal for calculating entanglement entropies to determine the boundary entropy of defects in strongly coupled two-dimensional conformal field theories. We study several examples including the Janus solution and show that the boundary entropy extracted from the entanglement entropy as well as its more conventional definition via the free energy agree with each other. Maybe somewhat surprisingly we find that, unlike in the case of a conformal field theory with boundary, the entanglement entropy for a generic region in a theory with defect carries detailed information about the microscopic details of the theory. We also argue that the g-theorem for the boundary entropy is closely related to the strong subadditivity of the entanglement entropy.

Aug 18, 201699 citations

Holographic entanglement entropy with conformal boundary conditions

Elena Cáceres, Hare Krishna, Harita Palani Balaji +1 more

We study holographic entanglement entropy in 3-dimensional AdS gravity with conformal boundary conditions which fix the conformal class of the boundary metric and its extrinsic curvature, $K$ , while leaving the Weyl mode dynamical. Extending Lewkowycz-Maldacena-Dong's replica construction, we derive the corresponding holographic entanglement entropy formula. We show that the fluctuating Weyl mode does not contribute additional entropy. The entropy of the full boundary is therefore the Bekenstein-Hawking entropy, and the entropy of a boundary subregion continues to obey the Ryu-Takayanagi prescription, namely, the area of a minimal surface divided by $4G_N$ . We carry out explicit calculations for global AdS, rotating and non-rotating BTZ geometries. For an interval in AdS $_3$ we find that the $K$ -dependent holographic entanglement entropy is governed by $c_m=\frac{3 \ell}{2 G_N}.$ For a thermal state at high conformal temperatures we find that, the entropy is governed by $c_{\rm eff}=\frac{3 \ell}{2 G_N} \frac{K \ell- \sqrt{K^2 \ell^2-4}}{2}$ , the same effective central charge that governs the Cardy-like density of states, in agreement with previous results in the literature. Finally, we also compute the entanglement entropy directly from the conjectured dual boundary theory - a holographic CFT coupled with time-like Liouville theory and deformed by a marginal $T \bar{T}$ like operator - and find $S_{EE} =\frac{c_{\rm eff}}{3} \ln\!\left(\frac{2 R \sinϕ_0}ε\right) .$ This result for the state with no operator insertions (vacuum state ), provides an independent boundary realization of $c_{\mathrm{eff}}$ while clarifying that this state is not the state dual to global AdS.We study holographic entanglement entropy in 3-dimensional AdS gravity with conformal boundary conditions which fix the conformal class of the boundary metric and its extrinsic curvature, $K$, while leaving the Weyl mode dynamical. Extending Lewkowycz-Maldacena-Dong's replica construction, we derive the corresponding holographic entanglement entropy formula. We show that the fluctuating Weyl mode does not contribute additional entropy. The entropy of the full boundary is therefore the Bekenstein-Hawking entropy, and the entropy of a boundary subregion continues to obey the Ryu-Takayanagi prescription, namely, the area of a minimal surface divided by $4G_N$. We carry out explicit calculations for global AdS, rotating and non-rotating BTZ geometries. For an interval in AdS$_3$ we find that the $K$-dependent holographic entanglement entropy is governed by $c_m=\frac{3 \ell}{2 G_N}.$ For a thermal state at high conformal temperatures we find that, the entropy is governed by $c_{\rm eff}=\frac{3 \ell}{2 G_N} \frac{K \ell- \sqrt{K^2 \ell^2-4}}{2}$, the same effective central charge that governs the Cardy-like density of states, in agreement with previous results in the literature. Finally, we also compute the entanglement entropy directly from the conjectured dual boundary theory - a holographic CFT coupled with time-like Liouville theory and deformed by a marginal $T \bar{T}$ like operator - and find $S_{EE} =\frac{c_{\rm eff}}{3} \ln\!\left(\frac{2 R \sinϕ_0}ε\right) .$ This result for the state with no operator insertions (vacuum state ), provides an independent boundary realization of $c_{\mathrm{eff}}$ while clarifying that this state is not the state dual to global AdS.

arXiv:2608.06277Aug 6, 20260 citations

Aspects of holographic entanglement entropy

Shinsei Ryu, Tadashi Takayanagi

arXiv:hep-th/0605073Journal of High Energy PhysicsAug 21, 20061,962 citations

d=3

and

d=4

when the entangling surface is orthogonal to the boundary. In particular, we compute a boundary term in the entropy of

\mathcal{N}=4

supergauge multiplet at weak coupling. For gauge fields we use a prescription which is consistent with the positive area law. The boundary term is compared with the holographic calculation of the entropy based on the Ryu-Takayanagi proposal adapted appropriately to the present situation. We find a complete agreement between these two calculations provided the boundary conditions imposed on the gauge multiplet preserve

1/2

of the supersymmetry and the extension of the boundary into the anti--de Sitter bulk is a minimal hypersurface.

We study holographic entanglement entropy in 3-dimensional AdS gravity with conformal boundary conditions which fix the conformal class of the boundary metric and its extrinsic curvature,

K

, while leaving the Weyl mode dynamical. Extending Lewkowycz-Maldacena-Dong's replica construction, we derive the corresponding holographic entanglement entropy formula. We show that the fluctuating Weyl mode does not contribute additional entropy. The entropy of the full boundary is therefore the Bekenstein-Hawking entropy, and the entropy of a boundary subregion continues to obey the Ryu-Takayanagi prescription, namely, the area of a minimal surface divided by

4G_N

. We carry out explicit calculations for global AdS, rotating and non-rotating BTZ geometries. For an interval in AdS

_3

we find that the

K

-dependent holographic entanglement entropy is governed by

c_m=\frac{3 \ell}{2 G_N}.

For a thermal state at high conformal temperatures we find that, the entropy is governed by

c_{\rm eff}=\frac{3 \ell}{2 G_N} \frac{K \ell- \sqrt{K^2 \ell^2-4}}{2}

, the same effective central charge that governs the Cardy-like density of states, in agreement with previous results in the literature. Finally, we also compute the entanglement entropy directly from the conjectured dual boundary theory - a holographic CFT coupled with time-like Liouville theory and deformed by a marginal

T \bar{T}

like operator - and find

S_{EE} =\frac{c_{\rm eff}}{3} \ln\!\left(\frac{2 R \sinϕ_0}ε\right) .

This result for the state with no operator insertions (vacuum state ), provides an independent boundary realization of

c_{\mathrm{eff}}

while clarifying that this state is not the state dual to global AdS.