Holographic Complexity as a Probe of Boundary Entropy in AdS/BCFT

Fabiano F. Santos

REVIEW 5 major objections 6 minor 50 references

Holographic Complexity as a Probe of Boundary Entropy in AdS/BCFT

T0 review · 5 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Under a relative renormalization, holographic complexity difference equals the boundary entropy $\log g$ divided by $\pi\hbar$.

desk verdict A competent survey of AdS/BCFT complexity that hides a circular core: the log g result is put in by hand via the counterterm, not derived. read the letter →

arxiv 2608.10348 v1 pith:3BFSYSNQ submitted 2026-08-11 hep-th

Fabiano F. Santos This is my paper · ORCID

classification hep-th

keywords holographiccomplexityAdS/BCFTend-of-the-worldbraneboundaryentropyAffleck-LudwigWheeler-DeWittactiontopologicalquantumfieldtheoryBTZblackhole

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Within the AdS/BCFT correspondence, this paper tries to show that the Complexity=Action prescription, after a relative renormalization, turns the difference in holographic complexity between two boundary conditions into a direct measurement of the universal boundary entropy, $S_{\rm bdry}=\log g$. The same quantity emerges as the finite term in the Ryu-Takayanagi entropy of an interval ending on the end-of-the-world brane and in the Euclidean on-shell action, so the paper claims a single interface datum appears in three holographic observables. To organize the calculation, the paper reads the gluing rules of topological quantum field theory — state preparation, contraction, cobordism composition — as a dictionary for the metric-dependent action contributions in AdS/BCFT, with the end-of-the-world brane playing the role of an interface state. If the claim is right, holographic complexity becomes a probe of the Affleck-Ludwig entropy, and the late-time complexity growth of a BTZ black hole carries an explicit boundary-condition term alongside the thermal contribution.

What carries the argument

The load-bearing structure is the Wheeler-DeWitt (WDW) patch action of the Complexity=Action proposal, split into bulk, boundary, joint, null-counterterm, and end-of-the-world brane pieces. The joint term uses the angle formula $a=\log|k\cdot\bar{k}/2|$ at the intersection of WDW null sheets with the brane, and the finite boundary/joint counterterm of Eq. (3.77) is chosen, with scheme constants $I_{\rm scheme}$ and $c_j$, so that its tension-dependent part is exactly minus the joint difference of Eq. (3.78). That engineered cancellation removes all metric-dependent pieces and leaves the brane-tension sector, which the Euclidean calculation identifies with $\log g$. The TQFT axioms (factorization, gluing as contraction over interface indices, boundary state as an overlap with the vacuum) provide the organizing language: the end-of-the-world brane is the bulk image of a boundary state $|B\rangle$, and the complexity difference measures the vacuum overlap $g=\langle 0|B\rangle$.

What would settle it

Compute $\Delta C(T)$ with a different local, covariant finite counterterm for the brane-joint intersection — for example one that omits the $c_j$ constant — and check whether the $\log(t/L)$ and $\log\sin\theta_0$ terms still cancel; any residual metric dependence would show that Eq. (3.79) is an artifact of the subtraction. A second check is to evaluate the BTZ late-time growth with explicit affine parametrizations of the $J_{Q,\pm}$ joints and verify whether $\frac{1}{\pi\hbar}(2M+S_{\rm bdry})$ survives without the same counterterm choice.

Watch

Extended reading notes

Core claim

The paper's central discovery is the equality (3.79): after defining the finite boundary/joint counterterm (3.77) and subtracting a reference configuration, the total complexity difference equals $S_{\rm bdry}/(\pi\hbar)=\log g/(\pi\hbar)$. Equivalently, in the planar BTZ extension, the late-time growth rate becomes $\frac{dC_A}{dt}=\frac{1}{\pi\hbar}(2M+S_{\rm bdry})$, where $2M=T_H S_{BH}$ is the standard thermal complexity growth and $S_{\rm bdry}$ is the interface contribution controlled by the brane tension. The same $S_{\rm bdry}$ appears in the Euclidean action and in the Ryu-Takayanagi entropy of an interval ending at the boundary, so the paper identifies a single universal boundary datum $\log g$ living in three different observables. The TQFT reading is that the end-of-the-world brane is the bulk realization of a boundary state $|B\rangle$, and its overlap with the vacuum, $g=\langle 0|B\rangle$, is what complexity measures after the chosen renormalization.

Load-bearing premise

The load-bearing premise is that the finite boundary/joint counterterm (3.77), with its arbitrary scheme constants $I_{\rm scheme}$ and $c_j$, is the correct renormalization subtraction; if another natural counterterm is used, the metric-dependent joint terms do not cancel and the complexity difference no longer equals $\log g/(\pi\hbar)$.

Editorial extensions

If this is right

Relative holographic complexity in AdS$_3$/BCFT$_2$ becomes a direct observable for the Affleck-Ludwig boundary entropy $\log g$.
The late-time complexity growth of a planar BTZ black hole with an end-of-the-world brane reads $\frac{1}{\pi\hbar}(2M+S_{\rm bdry})$, so complexity is sensitive to how the boundary theory is terminated.
The end-of-the-world brane is interpreted as the bulk realization of a boundary state $|B\rangle$, making TQFT gluing a structural dictionary for AdS/BCFT sewing.
The same universal boundary entropy appears in the Euclidean action, the Ryu-Takayanagi entropy of intervals ending on the brane, and the renormalized CA complexity, unifying three holographic computations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural testable extension: apply the same relative subtraction to rotating or higher-dimensional black holes with branes, where $\log g$ can depend on temperature; the scheme dependence of the counterterm would be visible there as a temperature-dependent residue.
The paper leaves implicit that the equality (3.79) is scheme-relative; any claim that complexity measures boundary entropy should be accompanied by the renormalization convention, and different conventions may yield different 'universal' terms.
If the dictionary is sound, it suggests a reverse use: measurements of complexity growth in tensor-network models of holography could constrain the effective brane tension of the boundary condition.
A covariant counterterm built from the induced geometry of the brane alone would be the natural competitor to Eq. (3.77); checking whether $\log g$ is the only scheme-independent residue would sharpen the proposal.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: the advertised result—relative CA complexity in AdS/BCFT equals log g/πħ—is imposed by the renormalization scheme, not derived. The reader's stress-test is right, and the problems are visible in the text without needing the cited literature. I would not send this out in its current form.

What is actually useful: the paper is a competent survey of the standard AdS/BCFT machinery. The Euclidean disk action, the RT calculation for intervals ending on the brane, the list of WDW joint terms, and the BTZ late-time growth 2M/πħ are all collected carefully, with the right references. As a summary of known material, it is readable and honest.

The soft spots are load-bearing. First, the tension is inconsistent: Eq. (3.33) says T_brane = sinθ/(8πGL), but Eq. (3.47) uses T_brane = sinθ/L to get K−T = sinθ/L. You cannot have both. This feeds directly into I_Q and S_bdry. Second, Eq. (3.43) for the bulk difference does not follow from the preceding integral; the actual θ dependence produces a 1/ε UV term, not the claimed 1/z_IR with Δx_R. The way the UV divergence is dropped is not justified.

Third, and most important, the central equality (3.79) is manufactured. The finite counterterm in Eq. (3.77) is explicitly defined to be minus the joint term, so the cancellation (3.78) is an identity by construction. No locality, covariance, or null-counterterm principle fixes that particular form; therefore ΔC = log g/πħ is a scheme choice, not a prediction. Fourth, even if you accept the counterterm, the advertised identification fails: Eq. (3.53) gives S_bdry = L/(2G) tanθ, whereas Eq. (3.19) gives log g = L/(4G) arctanh(T). These are not equal for generic θ, so ΔC = log g/πħ is unsupported on its own equations. The conclusion also contains a dimensionally wrong sentence, dI_WDW/dt = 2M/πħ = πħ T_H S_BH, which mixes 1/time with action.

Who is this for? Someone who wants a compact review of AdS/BCFT complexity ingredients might skim it, but they should not rely on the new claim. The TQFT dictionary is presented as an analogy and never produces a new formula. There may be a salvageable shorter paper here if the author fixes the tension, recomputes the bulk difference, and motivates the counterterm with an independent principle. As is, I would desk reject rather than spend referee time; the central derivation fails on its own terms.

Referee Report

5 major / 6 minor

Summary. The manuscript develops a dictionary between TQFT sewing/composition principles and AdS/BCFT, interpreting the end-of-the-world brane as an interface carrying boundary-state data. It computes the Affleck–Ludwig boundary entropy log g from the Euclidean action in Poincaré AdS3 with a constant-tension brane, and then applies the Complexity=Action proposal to the Wheeler–DeWitt patch. The central technical claim is that, after a 'relative renormalization' prescription, the complexity difference between two brane tensions equals exactly the universal boundary entropy divided by πℏ, ΔC(T)=S_bdry/(πℏ)=log g/(πℏ). This is extended to a planar BTZ black hole with an EOW brane, where the late-time complexity growth is claimed to be dC_A/dt=(1/πℏ)(2M+S_bdry).

Significance. If the central equality were established, it would be a novel and significant result: holographic complexity would directly probe the Affleck–Ludwig entropy and would sharpen the holographic meaning of EOW branes as boundary states. The paper also provides a useful organizational framing of AdS/BCFT in TQFT language, and it correctly identifies that WDW action decompositions require boundary, joint, and counterterm terms beyond the bulk contribution. However, the advertised result is not derived: the key equality is imposed by hand through a finite counterterm chosen to cancel the joint terms, and there are internal inconsistencies in the brane-tension conventions and in the identification of S_bdry with log g. Because the central claim is load-bearing and the derivation is not self-consistent, the manuscript does not currently support its conclusions.

major comments (5)

[§3.2, Eqs. (3.32)–(3.47)] The brane tension is defined inconsistently. Eq. (3.32) together with Eq. (3.33) gives T_brane = sinθ0/(8πGL), but Eq. (3.47) evaluates KQ − T_brane = sinθ0/L, which requires T_brane = sinθ0/L, missing the factor 8πG. This error propagates directly into I_Q in Eq. (3.50) and hence into S_bdry in Eq. (3.53), so the numerical content of the boundary-entropy extraction is not derived.
[§3.2, Eq. (3.43)] The claimed bulk difference does not follow from the preceding integral. Direct evaluation of Eq. (3.42) gives ΔI_bulk = L τ tanθ0/(4πG)(1/ε − 1/z_IR), including a 1/ε divergence and a factor tanθ0, whereas Eq. (3.43) states ΔI_bulk = τ Δx_R L/(8π z_IR G), which has neither the divergence nor the correct θ0 dependence. This discrepancy affects the subsequent renormalized-difference computations.
[§3.2, Eqs. (3.75)–(3.79)] The central equality ΔC(T)=log g/(πℏ) is manufactured by the renormalization prescription. Eq. (3.77) defines the finite boundary/joint counterterm I_ct^(Q),fin(θ0) to be exactly the negative of the joint term inside the brackets of Eq. (3.75); Eq. (3.78) then gives ΔI_ct = −ΔI_joint by construction. No independent principle—locality, covariance, power counting, or the Lehner et al. null-boundary counterterm—fixes this finite counterterm. Any other natural counterterm would leave metric-dependent pieces in ΔC, so the claimed universality is an artifact of the subtraction rather than a consequence of the CA prescription.
[§3.1 vs. §3.2] The paper identifies S_bdry in Eq. (3.53) with log g_B from Eq. (3.18), but the two expressions are not equal as functions of the tension. Eq. (3.18) gives log g = R/(4G_N) arctanh(RT), while Eq. (3.53) gives S_bdry = L/(2G) tanθ0 with T = sinθ0/(8πGL) (or sinθ0/L if Eq. (3.47) is used). Using the Brown–Henneaux relation c=3R/(2G_N), the prefactors differ by a factor of two and the functional forms differ (arctanh vs. tan), so even after the counterterm cancellation the advertised equality ΔC = log g/(πℏ) is unsupported.
[§3.2, Eqs. (3.70)–(3.73)] The decomposition into boundary and metric complexity is not well defined. Eq. (3.71) states ΔC_total = ΔC_bdry + ΔC_metric, while the sentence after Eq. (3.73) reads 'we define the boundary complexity as ΔC_bdry ≡ ΔC_bdry − ΔC_metric', which is circular. Moreover, Eq. (3.72) replaces the previously derived joint difference Eq. (3.68) with a new expression containing log g and a factor 1/(4πt) without derivation; the two expressions are not equivalent. These issues make the central computation internally inconsistent.

minor comments (6)

[Introduction, bullet list] The fourth bullet in the Introduction breaks off as '•a further extend this analysis to the thermal BTZ setting', which is grammatically incomplete and suggests an editing error.
[§2, Eq. (2.2)] Eq. (2.2) contains duplicated inner products, '⟨ψΣ,α|ψΣ,β|ψΣ,α|ψΣ,β⟩', which appears to be a typesetting artifact and should be corrected to ⟨ψΣ,α|ψΣ,β⟩=δαβ.
[§3.2, notation] The symbol T is used both for the brane tension (e.g., Eq. (3.32)) and for sinθ0 (Eq. (3.33) defines T≡sinθ0), causing ambiguity in later equations such as Eq. (3.74) and Section 4.
[§3.2, Eqs. (3.59)–(3.60)] The approximate expression z*(t)=t/(tanθ0+1) is not derived and appears dimensionally inconsistent since t has length units while tanθ0 is dimensionless; the subsequent joint-angle formula in Eq. (3.60) should be justified more carefully.
[§4.1, brane contribution] In the BTZ section, the brane contribution to the complexity growth is stated as dI_Q/dt = L/(2G) sinh^{-1}[cotθ0] without a derivation connecting it to the profile X'(r) given earlier; the θ0 dependence also does not match the vacuum expression S_bdry ∝ tanθ0, so the relation to boundary entropy is unclear.
[Throughout] The paper contains numerous typographical and rendering errors (e.g., 'Z=e −IE', 'gB =⟨0|BB⟩', misaligned subscripts, repeated words), which impede readability and should be corrected in a revision.

Circularity Check

1 steps flagged · score 8.0 of 10

Central equality ΔC=log(g)/(πℏ) is manufactured by the finite counterterm (3.77), which is defined to cancel the joint difference; no independent principle fixes it.

self definitional [Sec. 3.2, Eqs. (3.75)-(3.79)]
"the necessary and sufficient condition for the metric contribution to vanish is ∆I(Q) ct =−∆I (Q) joint. ... We define the finite boundary/joint counterterm as follows: I(Q),fin ct (θ0) =− L(1 + tanθ 0)/(8πGt) (log(tsinθ 0/L)+c j)+I scheme, (3.77) ... Therefore ... we immediately conclude that ∆I (Q) ct =−∆I (Q) joint. Consequently, the total complexity difference is entirely associated with the boundary sector: ∆C(T) = ∆C bdry(T) = Sbdry/πℏ = log(g)/πℏ ,(3.79)."

The counterterm in Eq. (3.77) is defined, with arbitrary constants c_j and I_scheme, to be the negative of the joint expression in Eq. (3.75). The paper first states that the metric contribution vanishes iff ΔI_ct^(Q)=−ΔI_joint^(Q), then simply imposes this condition by defining I_ct^(Q),fin. Eq. (3.78) therefore contains no new information; it is the definition rewritten as a difference. Eq. (3.79) follows identically, so the advertised 'extraction' of log(g)/(πℏ) is an artifact of the chosen subtraction, not a derived prediction. No independent principle—locality, covariance, null-reparametrization invariance of the Lehner counterterm, or matching to the independent log-g computation in Eq. (3.18)—selects this counterterm.

full rationale

The central derivation reduces to the definition of the renormalization counterterm. Eq. (3.79), which is the paper's main result for the vacuum AdS/BCFT case, is obtained by defining I_ct^(Q),fin in Eq. (3.77) so that its difference cancels ΔI_joint^(Q). The paper explicitly states the target condition 'the necessary and sufficient condition for the metric contribution to vanish' and then implements it by hand. Because I_scheme and c_j are free, and because the log(t sinθ0/L) structure is copied from Eq. (3.75), the cancellation is an identity. A different natural counterterm would leave a metric-dependent residue, so the advertised universality is not demonstrated. The same issue propagates to the thermal claim: the final dC_A/dt=(1/πℏ)(2M+S_bdry) depends on the same identification of S_bdry with the boundary entropy, which is not separately derived. The TQFT dictionary and the citations [6,7] do not supply an independent fixing of the counterterm. The score is 8 rather than 10 because the brane-action and joint computations preceding the subtraction are genuine; what is circular is the final scheme-dependent step that turns those computations into ΔC=log(g)/(πℏ).

Assumptions & free parameters 6 free parameters · 8 assumptions · 0 invented entities

The new part of the paper rests on the CA conjecture, the null-boundary joint prescription, and two ad hoc choices: the finite joint counterterm in Eq. (3.77) and the period tau = 2 pi z_max in Eq. (3.52). The first makes the final cancellation an identity; the second converts action differences into the quoted boundary entropy. Standard results such as Brown-Henneaux and RT are imported as inputs. No new particles, forces, or conserved quantities are introduced.

free parameters (6)

Brane angle theta_0 / tension T = T = sin(theta_0) in the dimensionless convention; T_brane = sin(theta_0)/(8 pi G L) or sin(theta_0)/L in different…
The family of BCFT boundary conditions is scanned by theta_0, and all final claims, including S_bdry, log g, and Delta C, are functions of this parameter. It is a physical input, but the two conflicting tension conventions in Eqs. (3.33) and (3.47) make its value ambiguous.
Temporal period tau = 2 pi z_max = tau = 2 pi z_max
In Eq. (3.52), the action difference is converted into S_bdry by the ad hoc choice tau = 2 pi z_max, with no derivation from the Lorentzian WDW patch. Changing this period rescales the claimed boundary entropy.
Joint counterterm constant c_j = unspecified
Introduced in Eqs. (3.75)-(3.77) to define the finite joint counterterm. Keeping it fixed is essential for the cancellation, but its value is never fixed by a principle.
Scheme constant I_scheme = unspecified, theta_0-independent
Added by hand in Eq. (3.77). It is advertised as not affecting differences, but it is part of the ad hoc renormalization that makes Delta I_ct = -Delta I_joint.
UV and IR cutoffs epsilon, z_max = epsilon -> 0, z_max ~ L
The 1/epsilon and 1/z_max terms are kept or dropped selectively. The final S_bdry = L/(2G) tan(theta_0) uses z_max together with the tau choice, so the regulator handling affects the quoted result.
Null counterterm scale c_t = unspecified
From the Lehner et al. counterterm in Eq. (3.62). The paper states its contribution is absorbed into Delta C_metric, but that absorption is part of the scheme that makes the joint cancellation work.

assumptions (8)

standard math Atiyah-Segal TQFT axioms: Hilbert space per closed manifold, factorization under disjoint union, gluing by inner product (Eqs. 2.1-2.8).
Used as the organizational framework in Section 2; no new mathematical content is added.
domain assumption AdS/BCFT correspondence with an EOW brane and Neumann condition K_ab - (K - T) h_ab = 0 (Eqs. 3.9, 3.16, 3.32).
The brane and its tension determine the geometry; this is imported from Takayanagi [3] and Fujita et al. [4].
domain assumption Brown-Henneaux central charge c = 3R/(2G_N).
Used to convert R/(4G_N) into c/6 in Eqs. (3.18)-(3.19); it is a standard input.
domain assumption Complexity equals Action: C = I_WDW/(pi hbar), with growth dC/dt = (1/pi hbar) dI_WDW/dt (Eqs. 3.26, 4.1).
The entire complexity interpretation rests on this unproven conjecture from Refs. [16,24]; the paper does not test it.
domain assumption Ryu-Takayanagi formula S = A/(4G_N) applied in BCFT with a boundary term (Eq. 3.21).
Used for the consistency check in Eq. (3.23); it is a standard holographic input.
domain assumption Null boundary joint and counterterm prescription of Lehner et al. (Eqs. 3.37-3.38, 3.62).
Needed to define the WDW action on null boundaries; the paper imports the prescription without re-deriving it.
ad hoc to paper The finite joint counterterm in Eq. (3.77) is an admissible renormalization of the WDW action.
This is the load-bearing choice: it alone makes Delta I_ct = -Delta I_joint and hence Delta C = log g/(pi hbar). No independent principle fixes it.
ad hoc to paper tau = 2 pi z_max is the correct period for converting the action difference to boundary entropy in the Lorentzian Poincare patch.
This choice is asserted in Section 3.2 without derivation; it converts Delta I_total into S_bdry.

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Pith. "Pith review of Holographic Complexity as a Probe of Boundary Entropy in AdS/BCFT." pith.science (2026). https://pith.science/paper/3BFSYSNQ

@misc{pith2026260810348,
  author       = {Pith},
  title        = {Pith review of: Holographic Complexity as a Probe of Boundary Entropy in AdS/BCFT},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3BFSYSNQ}},
  note         = {Machine review of arXiv:2608.10348}
}

read the original abstract

This work establishes a formal bridge between the organizational principles of TQFTs and the AdS/BCFT correspondence, interpreting end of the world EOW branes as physical interfaces carrying boundary-state data. By developing a dictionary that maps cobordism composition to holographic sewing, we show that the Euclidean action and the Ryu and Takayanagi prescription consistently isolate the universal boundary entropy (log g). Applying the Complexity-Action proposal, we demonstrate that a relative renormalization prescription extracts this same universal contribution from the Wheeler and DeWitt action. Finally, we extend the analysis to BTZ black holes, where the late-time complexity growth is shown to encode both the thermal interior dynamics and the universal information of the boundary conditions via the brane tension.

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Works this paper leans on

50 extracted references · 13 canonical work pages

[1]

J. M. Maldacena,The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys.38, 1113 (1999) [Adv. Theor. Math. Phys.2, 231 (1998)] [hep-th/9711200]

arXiv 1999
[2]

Witten,Anti-de Sitter space and holography,Adv

E. Witten,Anti-de Sitter space and holography,Adv. Theor. Math. Phys.2(1998), 253-291 [arXiv:hep-th/9802150 [hep-th]]

arXiv 1998
[3]

Takayanagi,Holographic Dual of BCFT, Phys

T. Takayanagi,Holographic Dual of BCFT, Phys. Rev. Lett.107, 101602 (2011) doi:10.1103/PhysRevLett.107.101602 [arXiv:1105.5165 [hep-th]]

arXiv 2011
[4]

Fujita, T

M. Fujita, T. Takayanagi and E. Tonni,Aspects of AdS/BCFT, JHEP1111, 043 (2011), [arXiv:1108.5152 [hep-th]]

arXiv 2011
[5]

Ryu and T

S. Ryu and T. Takayanagi,Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96, 181602 (2006) [arXiv:hep-th/0603001 [hep-th]]

arXiv 2006
[6]

F. F. Santos, B. Pourhassan, E. N. Saridakis, O. Sokoliuk, A. Baransky and E. O. Kahya,Holographic boundary conformal field theory within Horndeski gravity, JHEP12, 217 (2025) doi:10.1007/JHEP12(2024)217 [arXiv:2410.18781 [hep-th]]

arXiv 2025
[7]

F. F. Santos, B. Pourhassan and E. N. Saridakis,Black hole entropy and complexity growth in Horndeski gravity within the AdS/BCFT framework, JHEP06, 096 (2026) doi:10.1007/JHEP06(2026)096 [arXiv:2509.23430 [hep-th]]

arXiv 2026
[8]

S. H. Simon,Topological Quantum, Oxford University Press, 2023, ISBN 978-0-19-888672-3

work page 2023

Show all 50 references

[9]

Carqueville and I

N. Carqueville and I. Runkel,Introductory lectures on topological quantum field theory, Banach Center Publ.114, 9-47 (2018) doi:10.4064/bc114-1 [arXiv:1705.05734 [math.QA]]

2018 arXiv
[10]

Dedushenko,Gluing

M. Dedushenko,Gluing. Part I. Integrals and symmetries, JHEP04, 175 (2020) doi:10.1007/JHEP04(2020)175 [arXiv:1807.04274 [hep-th]]

2020 arXiv
[11]

Nishioka, T

T. Nishioka, T. Takayanagi and Y. Taki,Topological pseudo entropy, JHEP09, 015 (2021) doi:10.1007/JHEP09(2021)015 [arXiv:2107.01797 [hep-th]]

2021 arXiv
[12]

Kawamoto, S

T. Kawamoto, S. M. Ruan and T. Takayanagi,Gluing AdS/CFT, JHEP07, 080 (2023) doi:10.1007/JHEP07(2023)080 [arXiv:2303.01247 [hep-th]]

2023 arXiv
[13]

Melnikov,A Tutorial on Knots and Quantum Mechanics, [arXiv:2503.08846 [quant-ph]]

D. Melnikov,A Tutorial on Knots and Quantum Mechanics, [arXiv:2503.08846 [quant-ph]]

arXiv
[14]

Melnikov,Connectomes as holographic states, Nucl

D. Melnikov,Connectomes as holographic states, Nucl. Phys. B1010, 116757 (2025) doi:10.1016/j.nuclphysb.2024.116757 [arXiv:2312.16683 [hep-th]]

2025
[15]

Melnikov,Connectomes and properties of quantum entanglement, JHEP07, 015 (2023) doi:10.1007/JHEP07(2023)015 [arXiv:2302.08548 [hep-th]]

D. Melnikov,Connectomes and properties of quantum entanglement, JHEP07, 015 (2023) doi:10.1007/JHEP07(2023)015 [arXiv:2302.08548 [hep-th]]

2023 arXiv
[16]

A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao,Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett.116, no.19, 191301 (2016) – 31 – [arXiv:1509.07876 [hep-th]]

2016 arXiv
[17]

Lehner, R

L. Lehner, R. C. Myers, E. Poisson, and R. D. Sorkin,Gravitational Action with Null Boundaries, Phys. Rev. D94(2016) 084046, [arXiv:1609.00207 [gr-qc]]

2016 arXiv
[18]

Stanford and L

D. Stanford and L. Susskind,Complexity and Shock Wave Geometries, Phys. Rev. D 90, no.12, 126007 (2014) doi:10.1103/PhysRevD.90.126007 [arXiv:1406.2678 [hep-th]]

2014 arXiv
[19]

Swingle,Entanglement Renormalization and Holography, Phys

B. Swingle,Entanglement Renormalization and Holography, Phys. Rev. D86, 065007 (2012) doi:10.1103/PhysRevD.86.065007 [arXiv:0905.1317 [cond-mat.str-el]]

2012 arXiv
[20]

Evenbly and G

G. Evenbly and G. Vidal,Tensor Network States and Geometry, J. Statist. Phys. 145, no.4, 891-918 (2011) doi:10.1007/s10955-011-0237-4

2011 doi
[21]

Hartman and J

T. Hartman and J. Maldacena,Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP05, 014 (2013) [arXiv:1303.1080 [hep-th]]

2013 arXiv
[22]

Susskind,Entanglement is not enough, Fortsch

L. Susskind,Entanglement is not enough, Fortsch. Phys.64, 49-71 (2016) doi:10.1002/prop.201500095 [arXiv:1411.0690 [hep-th]]

2016 arXiv
[23]

W. Tang, L. Chen, W. Li, X. C. Xie, H. H. Tu and L. Wang,Universal Boundary Entropies in Conformal Field Theory: A Quantum Monte Carlo Study, Phys. Rev. B 96, 115136 (2017) doi:10.1103/PhysRevB.96.115136 [arXiv:1708.04022 [cond-mat.str-el]]

2017 arXiv
[24]

A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao,Complexity, action, and black holes, Phys. Rev. D93, no.8, 086006 (2016) [arXiv:1512.04993 [hep-th]]

2016 arXiv
[25]

Harper, H

J. Harper, H. Kanda, T. Takayanagi and K. Tasuki,g Theorem from Strong Subadditivity, Phys. Rev. Lett.133, no.3, 031501 (2024) doi:10.1103/PhysRevLett.133.031501 [arXiv:2403.19934 [hep-th]]

2024 arXiv
[26]

Shimaji, T

T. Shimaji, T. Takayanagi and Z. Wei,Holographic Quantum Circuits from Splitting/Joining Local Quenches, JHEP03, 165 (2019) doi:10.1007/JHEP03(2019)165 [arXiv:1812.01176 [hep-th]]

2019 arXiv
[27]

F. F. Santos and H. Boschi-Filho,Geometric Josephson junction, JHEP01, 135 (2025) doi:10.1007/JHEP01(2025)135 [arXiv:2407.10008 [hep-th]]

2025 arXiv
[28]

F. F. Santos and H. Boschi-Filho,Building an AdS/BCFT Josephson junction within Horndeski gravity, [arXiv:2510.07165 [hep-th]]

arXiv
[29]

Ritz-Zwilling, J

A. Ritz-Zwilling, J. N. Fuchs, S. H. Simon and J. Vidal,Topological and nontopological degeneracies in generalized string-net models, Phys. Rev. B109, no.4, 045130 (2024) doi:10.1103/PhysRevB.109.045130 [arXiv:2309.00343 [cond-mat.other]]

2024 arXiv
[30]

Melnikov,Entanglement classification from a topological perspective, Phys

D. Melnikov,Entanglement classification from a topological perspective, Phys. Rev. D107, no.12, 126005 (2023) doi:10.1103/PhysRevD.107.126005 [arXiv:2208.13901 [quant-ph]]. – 32 –

2023 arXiv
[31]

H. Geng, L. Randall and E. Swanson,BCFT in a black hole background: an analytical holographic model, JHEP12, 056 (2022) [arXiv:2209.02074 [hep-th]]

2022 arXiv
[32]

D. Basu, H. Chourasiya, V. Raj and G. Sengupta,Reflected entropy in a BCFT on a black hole background, JHEP05, 054 (2024) doi:10.1007/JHEP05(2024)054 [arXiv:2311.17023 [hep-th]]

2024 arXiv
[33]

H. Z. Chen, R. C. Myers, D. Neuenfeld, I. A. Reyes and J. Sandor,Quantum Extremal Islands Made Easy, Part I: Entanglement on the Brane, JHEP10, 166 (2020) [arXiv:2006.04851 [hep-th]]

2020 arXiv
[34]

H. Z. Chen, R. C. Myers, D. Neuenfeld, I. A. Reyes and J. Sandor,Quantum Extremal Islands Made Easy, Part II: Black Holes on the Brane, JHEP12, 025 (2020) [arXiv:2010.00018 [hep-th]]

2020 arXiv
[35]

H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas and S. Shashi,Entanglement phase structure of a holographic BCFT in a black hole background, JHEP05, 153 (2022) [arXiv:2112.09132 [hep-th]]

2022 arXiv
[36]

H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas and S. Shashi,Inconsistency of islands in theories with long-range gravity, JHEP01, 182 (2022) [arXiv:2107.03390 [hep-th]]

2022 arXiv
[37]

Moradi, S

H. Moradi, S. F. Moosavian and A. Tiwari,Topological holography: Towards a unification of Landau and beyond-Landau physics, SciPost Phys. Core6, 066 (2023) doi:10.21468/SciPostPhysCore.6.4.066 [arXiv:2207.10712 [cond-mat.str-el]]

2023 arXiv
[38]

J. Wang, K. Ohmori, P. Putrov, Y. Zheng, Z. Wan, M. Guo, H. Lin, P. Gao and S. T. Yau,Tunneling Topological Vacua via Extended Operators: (Spin-)TQFT Spectra and Boundary Deconfinement in Various Dimensions, PTEP2018, no.5, 053A01 (2018) doi:10.1093/ptep/pty051 [arXiv:1801.054...

2018 arXiv
[39]

Arcioni, M

G. Arcioni, M. Blau and M. O’Loughlin,On the boundary dynamics of Chern-Simons gravity, JHEP01, 067 (2003) doi:10.1088/1126-6708/2003/01/067 [arXiv:hep-th/0210089 [hep-th]]

2003 arXiv
[40]

Dordevi´ c and D

D. Dordevi´ c and D. Goˇ canin,Boundary terms, branes, and AdS/BCFT duality in first-order gravity, Phys. Rev. D109, no.8, 086026 (2024) doi:10.1103/PhysRevD.109.086026 [arXiv:2401.16306 [hep-th]]

2024 arXiv
[41]

Erdmenger, B

J. Erdmenger, B. Heß, R. Meyer and I. Matthaiakakis,Gibbons-Hawking-York boundary terms and the generalized geometrical trinity of gravity, Phys. Rev. D110, no.6, 066002 (2024) doi:10.1103/PhysRevD.110.066002 [arXiv:2304.06752 [hep-th]]

2024 arXiv
[42]

Terashima,The Brown-Henneaux’s central charge from the path integral boundary condition, Phys

H. Terashima,The Brown-Henneaux’s central charge from the path integral boundary condition, Phys. Lett. B499, 229-232 (2001) doi:10.1016/S0370-2693(01)00037-5 [arXiv:hep-th/0011010 [hep-th]]

2001 arXiv
[43]

Susskind,Computational Complexity and Black Hole Horizons,Fortsch

L. Susskind,Computational Complexity and Black Hole Horizons,Fortsch. Phys.64, 24-43 (2016) [arXiv:1402.5674 [hep-th]]; (Addendum)ibid.44-48 [arXiv:1403.5695 [hep-th]]. – 33 –

2016 arXiv
[44]

Lloyd,Ultimate physical limits to computation, Nature406, 1047-1054 (2000)

S. Lloyd,Ultimate physical limits to computation, Nature406, 1047-1054 (2000)

2000
[45]

Susskind,Black Holes and Complexity Classes, [arXiv:1802.02175 [hep-th]]

L. Susskind,Black Holes and Complexity Classes, [arXiv:1802.02175 [hep-th]]

arXiv
[46]

A. R. Brown, H. Gharibyan, H. W. Lin, L. Susskind, L. Thorlacius and Y. Zhao, Complexity of Jackiw-Teitelboim gravity, Phys. Rev. D99, no.4, 046016 (2019) [arXiv:1810.08741 [hep-th]]

2019 arXiv
[47]

A. R. Brown and L. Susskind,Second law of quantum complexity, Phys. Rev. D97, no.8, 086015 (2018) [arXiv:1701.01107 [hep-th]]

2018 arXiv
[48]

A. R. Brown and L. Susskind,Complexity geometry of a single qubit, Phys. Rev. D 100, no.4, 046020 (2019) [arXiv:1903.12621 [hep-th]]

2019 arXiv
[49]

A. R. Brown and L. Susskind,holographic wormhole traversed in a quantum computer, Nature612, no.7938, 41-42 (2022)

2022
[50]

J. M. Mag´ an, D. Melnikov and M. R. O. Silva,Black Holes in AdS/BCFT and Fluid/Gravity Correspondence, JHEP11, 069 (2014) doi:10.1007/JHEP11(2014)069 [arXiv:1408.2580 [hep-th]]. – 34 –

2014 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

J. M. Maldacena,The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys.38, 1113 (1999) [Adv. Theor. Math. Phys.2, 231 (1998)] [hep-th/9711200]

arXiv 1999

[2] [2]

Witten,Anti-de Sitter space and holography,Adv

E. Witten,Anti-de Sitter space and holography,Adv. Theor. Math. Phys.2(1998), 253-291 [arXiv:hep-th/9802150 [hep-th]]

arXiv 1998

[3] [3]

Takayanagi,Holographic Dual of BCFT, Phys

T. Takayanagi,Holographic Dual of BCFT, Phys. Rev. Lett.107, 101602 (2011) doi:10.1103/PhysRevLett.107.101602 [arXiv:1105.5165 [hep-th]]

arXiv 2011

[4] [4]

Fujita, T

M. Fujita, T. Takayanagi and E. Tonni,Aspects of AdS/BCFT, JHEP1111, 043 (2011), [arXiv:1108.5152 [hep-th]]

arXiv 2011

[5] [5]

Ryu and T

S. Ryu and T. Takayanagi,Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96, 181602 (2006) [arXiv:hep-th/0603001 [hep-th]]

arXiv 2006

[6] [6]

F. F. Santos, B. Pourhassan, E. N. Saridakis, O. Sokoliuk, A. Baransky and E. O. Kahya,Holographic boundary conformal field theory within Horndeski gravity, JHEP12, 217 (2025) doi:10.1007/JHEP12(2024)217 [arXiv:2410.18781 [hep-th]]

arXiv 2025

[7] [7]

F. F. Santos, B. Pourhassan and E. N. Saridakis,Black hole entropy and complexity growth in Horndeski gravity within the AdS/BCFT framework, JHEP06, 096 (2026) doi:10.1007/JHEP06(2026)096 [arXiv:2509.23430 [hep-th]]

arXiv 2026

[8] [8]

S. H. Simon,Topological Quantum, Oxford University Press, 2023, ISBN 978-0-19-888672-3

work page 2023