Note on Type $III_1$ Algebras in $ c= 1$ String Theory and Bulk Causal Diamonds

T. Banks

REVIEW 3 major objections 5 minor 1 cited by

Note on Type $III_1$ Algebras in $ c= 1$ String Theory and Bulk Causal Diamonds

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Applying the standard holographic limiting-algebra construction to the c=1 string yields free fermions on a half line, whose full algebra is Type I and whose finite causal diamonds are Type $\mathrm{III}_1$ — despite the model having no…

desk verdict A short, honest interpretive note applying LL operator algebras to c=1 string theory; the Type I classification of the full algebra rests on an unproved Rindler-vacuum identification. read the letter →

arxiv 2504.15076 v1 pith:RO6QMEUC submitted 2025-04-21 hep-th gr-qc

T. Banks This is my paper · ORCID

classification hep-thgr-qc

keywords c=1stringtheoryType0BmatrixmodelvonNeumannalgebrasIII_1factorsRindlervacuumcausaldiamondsdoublescalinglimittensornetworkrenormalizationgroup

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

The paper tries to pin down what the holographic limiting-algebra construction actually detects by running it on the one exactly soluble string theory, the double-scaled Type 0B matrix model at c=1. It argues that the limiting operator algebra is exactly the algebra of massless relativistic fermion fields on a half line; in the state relevant to the matrix model this is the Rindler-vacuum wedge algebra, hence Type I, while subalgebras belonging to finite causal diamonds are Type $\mathrm{III}_1$. Because the model is integrable and contains no black-hole excitations, the paper concludes that Type $\mathrm{III}_1$ algebras in 1+1 dimensional gravity are not a signature of horizons. It further argues that adding interactions to many copies of the model leaves the algebra type unchanged, and that an infrared cutoff replaces infinite-dimensional algebras with finite-dimensional ones in which causal diamonds are defined by time-dependent embedding maps.

What carries the argument

The load-bearing object is the exact scattering transform that takes nonrelativistic fermions in an upside-down oscillator, the double-scaled matrix model, to a pair of massless Dirac fermion fields on a half line. In the transformed variables the state is the Rindler vacuum, so the full field algebra is Type $\mathrm{I}$, the simplest von Neumann factor type, while the subalgebra of any finite interval is Type $\mathrm{III}_1$, the factor type with no trace that is characteristic of local algebras in quantum field theory. The second mechanism is a renormalization-group-style family of unitary embeddings of smaller Hilbert spaces into larger ones, modeled on tensor network renormalization, which the paper uses to give finite systems sharp causal diamonds after an infrared cutoff.

What would settle it

Compute the modular Hamiltonian of the half-line fermion algebra in the exact state defined by the double-scaled matrix model, using the explicit scattering states from the paper's transform. If that modular operator is not the one belonging to the Rindler vacuum, the full algebra is not Type $\mathrm{I}$ and the central classification fails; if instead the commutant of a finite-interval subalgebra turns out to be trivial in the $N=\infty$ limit, the claimed Type $\mathrm{III}_1$ structure of causal diamonds would not appear.

Watch

Extended reading notes

Core claim

The central claim is that the limiting algebra of the double-scaled Type 0B matrix model is the algebra of relativistic fermion fields on a half line in the Rindler vacuum: a Type $\mathrm{I}$ von Neumann algebra. Any subalgebra restricted to a finite spatial interval, which the paper identifies with a bulk causal diamond, is Type $\mathrm{III}_1$. The same classification holds for M non-interacting or interacting copies, because the algebra is a finite tensor product and the interaction does not change its Murray-von Neumann type. Hence, in this exactly soluble setting, the appearance of Type $\mathrm{III}_1$ has nothing to do with black hole horizons; it reflects the infinite-dimensionality of the boundary Hilbert space. The paper extends this into a general picture in which sharp causal diamonds are defined not by infinite-dimensional subalgebras but by a sequence of time-dependent unitary embeddings of smaller Hilbert spaces into larger ones, guided by an entropy-area formula and analogous to tensor network renormalization.

Load-bearing premise

The load-bearing premise is that the exact dictionary between the double-scaled matrix model and free fermions on a half line places the model in the special state where the full fermion algebra is that of a Rindler wedge with no entanglement between the two sides; if the state differs, the full algebra need not be of the simplest von Neumann type and the finite-region conclusion would need separate proof.

Editorial extensions

If this is right

In the double-scaled Type 0B model, the appearance of Type $\mathrm{III}_1$ algebras in finite regions cannot be read as evidence of a black hole horizon, because the model is integrable and has no black-hole excitations.
Adding a non-integrable four-fermion interaction to a large number of copies leaves the operator algebra a finite tensor product of the original algebras, so the Murray-von Neumann type is unchanged even when the model develops metastable, horizon-like excitations.
An infrared cutoff that moves away from double scaling makes the Hilbert spaces finite dimensional; sharp causal diamonds are then defined by time-dependent unitary embeddings of smaller into larger Hilbert spaces, not by infinite-dimensional subalgebras.
Finite-area causal diamonds in these cutoff lattice models are finite-dimensional Type $\mathrm{I}$ algebras, which the paper argues makes the proposed Type $\mathrm{II}_1$ description of finite-area diamonds non-universal for finite N.
The c=1 algebras are more like AdS/CFT boundary algebras than bulk horizon algebras: they encode the infinite size of the boundary Hilbert space, and their apparent bulk causal structure is an artifact of the infinite-volume limit.

Reading between the lines

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

An implication the paper leaves implicit is that the limiting-algebra criterion for horizons should be supplemented by a different invariant, such as the structure of half-sided modular inclusions or the behavior of the modular Hamiltonian, rather than the Murray-von Neumann class alone.
The exact operator-algebraic dictionary suggests a concrete test in the interacting multi-copy model: the metastable excitations should leave the modular flow and commutant structure of finite-region algebras essentially unchanged, since the interaction alters the state but not the algebra type.
One could make the finite-N causal-diamond proposal quantitative by requiring the low-lying spectrum of the N-fermion double-well model to match that of the $N{+}1$-fermion model while the well separation follows the double-scaled Fermi level; the large-N limit of the resulting diamonds could then be checked against the exact scattering data.
If Type $\mathrm{III}_1$ algebras also appear in other integrable 1+1 dimensional string models, the paper's conclusion would generalize to a broad statement that in two dimensions infinite-dimensional operator algebras encode boundary volume rather than local gravitational physics.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Colleague,

Short version: this note applies the Leutheusser-Liu operator-algebra framework to the c=1 string, argues the limiting algebra is fermions on a half line, and concludes that Type III_1 in this model has nothing to do with horizons. That's a genuinely useful contribution to an active debate. But the load-bearing claim that the full algebra is Type I rests on an identification of the physical vacuum with the Rindler vacuum, which is asserted rather than shown.

What's new: the specific application of LL to the double-scaled 0B model, and the interpretation that the Type III_1 subalgebras encode infinite-volume/boundary structure rather than black hole horizons. That is not in the cited literature. The paper also frames the TNRG as a time-dependent flow of embeddings, which is a suggestive analogy, though largely a reprise of Banks's earlier work.

What it does well: it uses exact results—Moore's double-scaled field theory, the AKK transform, Maldacena-Seiberg's fermion formulation—so the algebraic description is on solid ground. It is honest about its assumptions; Section 3 explicitly lists three conditions needed to connect the commutant to bulk causal diamonds. The paper's own caveats are more clearly stated than in many speculative notes.

The soft spots: the main one is the vacuum identification. After Eq. (2.1) the paper says the AKK-transformed Hilbert space is "more like that of the Rindler vacuum" and therefore the full algebra is Type I. That's not a consequence of the AKK/Moore dictionary. The dictionary fixes the algebra, but the MvN type of the weak closure depends on the physical state. Scattering states being pure does not select the Rindler vacuum. If the state is the Minkowski vacuum, the full half-line fermion algebra is Type III_1, not Type I. The paper needs to prove the state identification or soften the claim. Also, the finite-region Type III_1 claim is asserted without a locality argument; the AKK kernel is nonlocal, so a finite bulk region need not correspond to a finite interval in the fermion field. That's a second gap, though the conclusion is plausible in either vacuum because finite interval subalgebras of a chiral fermion are Type III_1 regardless.

Minor issues: reference [20] is missing from the bibliography; there's a typo "unitary unitary transformation" in the conclusions.

Who this is for: people working on operator algebras in holography, especially the LL program and whether Type III_1 is a horizon diagnostic. The paper is a speculation, but an honest one.

Recommendation: worth sending to referees. It is short, thought-provoking, and the main gap is fixable—either prove the vacuum identification or explicitly condition the Type I claim on it. I would engage with it.

Referee Report

3 major / 5 minor

Summary. The paper applies the Leutheusser-Liu algebraic procedure to the double-scaled Type 0B (c = 1) matrix model and claims that the resulting N = ∞ operator algebra is the algebra of a pair of massless Dirac fermion fields on a half line. Relying on the Moore and AKK/Maldacena-Seiberg results, the author asserts that the full half-line algebra is Type I because the relevant state is "more like" the Rindler vacuum, while subalgebras associated to finite regions of the linear dilaton spacetime are Type III_1. The author uses this example to argue that Type III_1 algebras need not signal black-hole horizons, and extends the discussion to M copies of the model, to interactions that break integrability, and to a tensor-network/tensor-network-renormalization-group picture of finite causal diamonds. The paper also criticizes Type II_1 descriptions of finite-area diamonds as non-universal for finite N.

Significance. If the central claims are correct, the paper provides an explicit exactly solvable model in which a Leutheusser-Liu limiting algebra is Type I while its local subalgebras are Type III_1 in a horizonless setting, sharpening the distinction between the emergence of Type III_1 structure and the presence of black-hole horizons. The manuscript is valuable as a concise research note: it relies on nontrivial exact results (Moore; Alexandrov, Kazakov and Kostov; Maldacena and Seiberg), it states several of its assumptions explicitly, and it makes a concrete, falsifiable proposal for how finite causal diamonds should be defined through tensor-network embedding maps. The main weaknesses are that the identification of the state as the Rindler vacuum is asserted rather than derived, and the finite-region Type III_1 claim is not supported by a locality argument despite the nonlocal character of the AKK transform.

major comments (3)

[§2, after Eq. (2.1)] The claim that the full algebra of the AKK-transformed model is Type I rests on the assertion that the Hilbert space is "more like that of the Rindler vacuum" and that scattering states are pure. This is load-bearing: the AKK/Moore dictionary determines the operator algebra and the dynamics, but the Murray-von Neumann type of the weak closure in the GNS representation is determined by the physical state, which must be supplied separately. Scattering states being pure shows only that a Fock space description exists; it does not select the Rindler vacuum. If the matrix-model ground state maps to the Minkowski vacuum of the half-line fermion theory, the full half-line algebra is Type III_1, not Type I. A derivation of the state identification, or at least an explicit computation of the relevant two-point function and the spectrum on the positive-frequency Rindler subspace, is needed. Footnote 1 attributes the point to Leutheusser and Liu, but private communication is not a substitute for a proof.
[§2, paragraph beginning "However the Hilbert space..."] The sentence "Any restriction of the algebra to a finite region of the linear dilaton space-time is Type III_1" is asserted without a derivation. Because the transform in Eq. (2.1) is nonlocal, a finite region in the bulk/linear-dilaton coordinates need not coincide with a finite interval in the half-line fermion field coordinate. The paper does not define the relevant notion of "finite region" in the transformed variables, nor does it prove that the finite-region algebras are isomorphic to those of finite intervals of a relativistic fermion field. This matters because the paper's headline conclusion (Type III_1 without black-hole horizons) depends directly on this finite-region claim. If the claim is meant only for finite intervals in the fermion coordinate, it should be stated that way and justified by standard algebraic QFT; if it is meant for finite regions in the spacetime of the c = 1 string, a separate locality argument is required.
[§3, list of three assumptions] The paper explicitly says that identifying the commutant of a finite-time boundary algebra with the algebra of a bulk causal diamond requires three further assumptions: the TNRG Hamiltonian approximates K_0 + P_0, the lattice notion of smeared single-trace operators is local, and sharp causal diamonds exist in finite tensor networks and coincide with the LL diamonds in the N → ∞ limit. Since this set of assumptions underlies the central interpretive claim that Type III_1 algebras encode bulk causal structure in a way analogous to AdS/CFT boundary algebras, the manuscript should clearly mark this part as conjectural. The Abstract and Conclusions state these as conclusions rather than as consequences of a proof, which overstates the strength of the argument.

minor comments (5)

[References] Reference [20] is empty in the bibliography; it should either be filled or removed.
[§3.2, paragraph on TNRG embedding maps] The text reads "unitary embedding map of the Hilbert space of the N-th model into the N+ ﬁrst"; this should be "N+1st".
[§4, Conclusions] The phrase "full unitary unitary transformation" contains a duplicated word and should be corrected.
[Throughout] The notation for Murray-von Neumann types is inconsistent: the text alternates between "Type III 1" and "Type III_1", and "Type I M" appears instead of, e.g., "Type I_M" or "Type I_∞". Please standardize.
[§2, footnote 1] The acknowledgment to S. Leutheusser and H. Liu in footnote 1 would be more appropriately placed in the Acknowledgments section.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the Type I/III_1 classification derives from external exact results, with the Rindler-vacuum identification as an unproved premise rather than a circular reduction.

full rationale

The main derivation chain is external: Moore's double-scaled fermion algebra, the AKK/Maldacena-Seiberg mapping to massless Dirac fermions on a half line, and the standard von Neumann algebra fact that the Rindler-wedge algebra in the Rindler vacuum is Type I while finite-region subalgebras are Type III_1. None of these inputs is fitted or renamed in the present paper, and the paper does not predict a quantity that was used as a fit parameter. The key step that could be challenged is the assertion in Section 2, immediately after Eq. (2.1), that 'the Hilbert space of the AKK transformed double scaled matrix model is more like that of the Rindler vacuum'; this is an unproved physical identification, and if the true state were the Minkowski vacuum the full algebra would be Type III_1. That is a rigor gap, not circularity: the Type I conclusion is a genuine consequence of the stated Rindler-vacuum premise, not a restatement of it. The paper itself flags a related limitation in Section 2.1 ('There is no small parameter that justifies this approximation'). Self-citations to the author's earlier work appear in the M-copy black-hole speculation [5], the JT-gravity model [8], and the tensor-network/holographic-space-time interpretation [10,22,23]; these are peripheral to the Murray-von Neumann classification of the c=1 model and do not force the central claim. Accordingly, the paper is essentially self-contained against external benchmarks for its main algebraic conclusion; score 2 reflects only minor self-reference in interpretive material.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central algebraic classification depends on external exact-solution results (Moore, AKK, Maldacena-Seiberg) and standard algebraic QFT facts, not on parameters fitted here. The load-bearing physical premise is the identification of the model's Hilbert space with a Rindler-like vacuum; this is attributed to Leutheusser and Liu in footnote 1 and is not independently checked in the paper. The causal-diamond interpretation rests on three assumptions the paper itself lists in Section 3. No free parameters are fitted and no new entities are introduced.

assumptions (5)

domain assumption The double-scaled Type 0B matrix model is exactly equivalent to a pair of massless Dirac fermions on a half line via the AKK transform.
Invoked for the central classification in Section 2; the paper relies on results of Moore [3], Alexandrov-Kazakov-Kostov [6], and Maldacena-Seiberg [7] rather than rederiving them.
domain assumption The limiting operator algebra is the algebra of smeared fermion fields of the upside-down oscillator problem.
Taken as input from Moore [3] in the first paragraph of Section 2.
standard math Local subalgebras of free relativistic fermions in open regions are Type III_1, and Rindler wedge algebras in the vacuum are Type III_1.
Used implicitly when asserting that finite-region subalgebras are Type III_1 in Section 2; not proven in the paper.
domain assumption The Hilbert space of the AKK-transformed model is like the Rindler vacuum, with pure scattering states, so the full field algebra is Type I.
Stated in Section 2 after eq. (2.1); attributed to Leutheusser and Liu in footnote 1, and is the questionable premise behind the Type I claim.
domain assumption The commutant of a finite-time boundary algebra coincides with the algebra of a bulk causal diamond under three listed conditions: TNRG Hamiltonian approximates K0+P0; smeared single-trace operators are local lattice gauge-invariant operators; sharp causal diamonds exist in finite tensor networks…
Section 3, paragraph beginning 'In order to justify...'; the paper explicitly labels these as further assumptions.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Note on Type $III_1$ Algebras in $ c= 1$ String Theory and Bulk Causal Diamonds." pith.science (2026). https://pith.science/paper/RO6QMEUC

@misc{pith2026250415076,
  author       = {Pith},
  title        = {Pith review of: Note on Type $III_1$ Algebras in $ c= 1$ String Theory and Bulk Causal Diamonds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RO6QMEUC}},
  note         = {Machine review of arXiv:2504.15076}
}

abstract

We argue that the Leutheusser-Liu procedure of isolating a von Neumann algebra in the $N = \infty$ limit of string theories, leads to the algebra of relativistic fermion fields on a half line for the $c = 1$ string theory. This is a Type $I$ von Neumann algebra, since it is the algebra of the Rindler wedge in the Rindler vacuum state. Subalgebras of finite regions are Type $III_1$. The argument uses the elegant results of Moore and of Alexandrov, Kazakov and Kostov. This model is well known to be integrable and have no black hole excitations. We have speculated that adding an interaction invisible in perturbation theory to a large finite number, $M$, of copies of the model, produces a non-integrable model with meta-stable excitations having all of the properties of linear dilaton black holes. The algebra of fields is the tensor product of $M$ copies of the $c = 1$ model's algebra, whether or not we add the non-integrable interaction. We argue that the infinite dimensional $c = 1$ algebras are analogous to those of the boundary field theory in AdS/CFT, even though they appear to encode bulk causal structure. An IR cutoff on the boundary renders them finite and causal structure must be formulated in terms of an analog of the Tensor Network Renormalization Group. This is a time dependent Hamiltonian flow, embedding smaller Hilbert spaces into larger ones. It is the analog of one sided modular inclusion in quantum field theory.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Diamonds in the Bulk and Large-$N$ Scaling in AdS/CFT
hep-th 2026-01 conditional novelty 6.0 of 10

Bulk field algebras of causal diamonds in AdS/CFT require a double-scaling limit, so sub-AdS-radius distances are not described by bulk QFT.

Reference graph

Works this paper leans on

22 extracted references · 3 canonical work pages · cited by 1 Pith paper

[1]

Subregion-subalgebra duality: emer gence of space and time in holography,

S. Leutheusser and H. Liu,“Subregion-subalgebra duality: emer gence of space and time in holography,” [arXiv:2212.13266 [hep-th]]; S. A. W. Leutheus ser and H. Liu, “Emergent Times in Holographic Duality,” Phys. Rev. D 108, no.8, 086020 (2023) doi:10.1103/PhysRevD.108.086020 [arXiv:2112.1 2156 [hep-th]]; S. Leutheusser and H. Liu,“Causal connectability be...

arXiv 2023
[2]

A New hat for the c=1 matrix model,

M. R. Douglas, I. R. Klebanov, D. Kutasov, J. M. Maldacena, E. J . Martinec and N. Seiberg,“A New hat for the c=1 matrix model,” [arXiv:hep-th/03071 95 [hep-th]]

work page
[3]

Double scaled ﬁeld theory at c = 1,

G. W. Moore, “Double scaled ﬁeld theory at c = 1,” Nucl. Phys. B 368, 557-590 (1992) doi:10.1016/0550-3213(92)90214-V

doi:10.1016/0550-3213(92)90214-v 1992
[4]

Evanescent black holes,

C. G. Callan, Jr., S. B. Giddings, J. A. Harvey and A. Strominger, “ Evanescent black holes,” Phys. Rev. D 45, no.4, R1005 (1992) doi:10.1103/PhysRevD.45.R1005 [arXiv:hep-th/9111056 [hep-th]]

arXiv 1992
[5]

Holographic Space-time Models in 1 + 1 Dimensions,

T. Banks, “Holographic Space-time Models in 1 + 1 Dimensions,” [arXiv :1506.05777 [hep-th]]; T. Banks, Microscopic Models of Linear Dilaton Gravity and T heir Semi- classical Ap- proximations, [arXiv:hep-th/2005.09479]

arXiv 2005
[6]

Time depend ent backgrounds of 2- D string theory,

S. Y. Alexandrov, V. A. Kazakov and I. K. Kostov, “Time depend ent backgrounds of 2- D string theory,” Nucl. Phys. B 640, 119-144 (2002) doi:10.1016/S0550-3213(02)00541-2 [arXiv:hep-th/0205079 [hep-th]]

arXiv 2002
[7]

Flux-vacua in two dimensional str ing theory,

J. M. Maldacena and N. Seiberg,“Flux-vacua in two dimensional str ing theory,” JHEP 09, 077 (2005) doi:10.1088/1126-6708/2005/09/077 [arXiv:hep-th/ 0506141 [hep-th]]

work page doi:10.1088/1126-6708/2005/09/077 2005
[8]

JT gravity coupled to fermio ns,

T. Banks, P. Draper and B. Zhang, “JT gravity coupled to fermio ns,” Adv. Theor. Math. Phys. 27, no.2, 483-522 (2023) doi:10.4310/ATMP.2023.v27.n2.a2 [arXiv:2205 .07382 [hep-th]]

doi:10.4310/atmp.2023.v27.n2.a2 2023

Show all 22 references

[9]

Tensor Network Renormalization,

G. Evenbly and G. Vidal,“Tensor Network Renormalization,” Phys. R ev. Lett. 115, no.18, 180405 (2015) doi:10.1103/PhysRevLett.115.180405; G. E venbly and 11 G. Vidal, “Tensor Network Renormalization Yields the Multiscale Entan gle- ment Renormalization Ansatz,” Phys. Rev. Lett...

2015 doi
[10]

Holographic Space-time Models of Ant i-deSitter Space- times,

T. Banks and W. Fischler, “Holographic Space-time Models of Ant i-deSitter Space- times,” [arXiv:1607.03510 [hep-th]]

arXiv
[11]

The Ryu–Takayanagi Formula from Quantum Error Correction,

D. Harlow, “The Ryu–Takayanagi Formula from Quantum Error Correction,” Commun. Math. Phys. 354, no.3, 865-912 (2017) doi:10.1007/s00220-017-2904-z [arXiv:16 07.03901 [hep-th]]

2017 doi
[12]

Holographic quantum error- correcting codes: Toy models for the bulk/boundary correspond ence,

F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, “Holographic quantum error- correcting codes: Toy models for the bulk/boundary correspond ence,” JHEP 06, 149 (2015) doi:10.1007/JHEP06(2015)149 [arXiv:1503.06237 [hep-th]]

2015 arXiv
[13]

Towards a derivation of h olographic entangle- ment entropy,

H. Casini, M. Huerta and R. C. Myers, “Towards a derivation of h olographic entangle- ment entropy,” JHEP 05, 036 (2011) doi:10.1007/JHEP05(2011)036 [arXiv:1102.0440 [hep-th]]

2011 arXiv
[14]

Gravitational Thermodynamics of Causal Diamonds in (A)dS,

T. Jacobson and M. Visser, “Gravitational Thermodynamics of Causal Diamonds in (A)dS,” SciPost Phys. 7, no.6, 079 (2019) doi:10.21468/SciPostPhys.7.6.079 [arXiv:1812.01596 [hep-th]]

2019 arXiv
[15]

The Holographic bound in anti-de Sitt er space,

L. Susskind and E. Witten, “The Holographic bound in anti-de Sitt er space,” [arXiv:hep-th/9805114 [hep-th]]

arXiv
[16]

A n algebra of ob- servables for de Sitter space,

V. Chandrasekaran, R. Longo, G. Penington and E. Witten, “A n algebra of ob- servables for de Sitter space,” JHEP 02, 082 (2023) doi:10.1007/JHEP02(2023)082 [arXiv:2206.10780 [hep-th]]

2023 arXiv
[17]

Generalized entrop y for general sub- regions in quantum gravity,

K. Jensen, J. Sorce and A. J. Speranza, “Generalized entrop y for general sub- regions in quantum gravity,” JHEP 12, 020 (2023) doi:10.1007/JHEP12(2023)020 [arXiv:2306.01837 [hep-th]]

2023 arXiv
[18]

Ad S dynamics from conformal ﬁeld theory,

T. Banks, M. R. Douglas, G. T. Horowitz and E. J. Martinec, “Ad S dynamics from conformal ﬁeld theory,” [arXiv:hep-th/9808016 [hep-th]]

arXiv
[19]

Hologra phic representation of local bulk operators,

A. Hamilton, D. N. Kabat, G. Lifschytz and D. A. Lowe, “Hologra phic representation of local bulk operators,” Phys. Rev. D 74, 066009 (2006) doi:10.1103/PhysRevD.74.066009 [arXiv:hep-th/0606141 [hep-th]]. [20]

2006 arXiv
[21]

M theor y as a matrix model: A conjecture,

T. Banks, W. Fischler, S. H. Shenker and L. Susskind, “M theor y as a matrix model: A conjecture,” Phys. Rev. D 55, 5112-5128 (1997) doi:10.1201/9781482268737- 37 [arXiv:hep-th/9610043 [hep-th]]

1997 arXiv
[22]

Hilbert Bundles and Holographic Space-time Models,

T. Banks, “Hilbert Bundles and Holographic Space-time Models,” [a rXiv:2306.07038 [hep-th]]; T. Banks, “Hilbert Bundles and Holographic Space-time: th e Hydrodynamic Approach to Gravity,” [arXiv:2502.04924 [hep-th]]. 12

arXiv
[23]

On the Impossibility of Precise Veriﬁcation of Models o f Quantum Gravity,

T. Banks, “On the Impossibility of Precise Veriﬁcation of Models o f Quantum Gravity,” [arXiv:2309.07203 [gr-qc]]. 13

arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Subregion-subalgebra duality: emer gence of space and time in holography,

S. Leutheusser and H. Liu,“Subregion-subalgebra duality: emer gence of space and time in holography,” [arXiv:2212.13266 [hep-th]]; S. A. W. Leutheus ser and H. Liu, “Emergent Times in Holographic Duality,” Phys. Rev. D 108, no.8, 086020 (2023) doi:10.1103/PhysRevD.108.086020 [arXiv:2112.1 2156 [hep-th]]; S. Leutheusser and H. Liu,“Causal connectability be...

arXiv 2023

[2] [2]

A New hat for the c=1 matrix model,

M. R. Douglas, I. R. Klebanov, D. Kutasov, J. M. Maldacena, E. J . Martinec and N. Seiberg,“A New hat for the c=1 matrix model,” [arXiv:hep-th/03071 95 [hep-th]]

work page

[3] [3]

Double scaled ﬁeld theory at c = 1,

G. W. Moore, “Double scaled ﬁeld theory at c = 1,” Nucl. Phys. B 368, 557-590 (1992) doi:10.1016/0550-3213(92)90214-V

doi:10.1016/0550-3213(92)90214-v 1992

[4] [4]

Evanescent black holes,

C. G. Callan, Jr., S. B. Giddings, J. A. Harvey and A. Strominger, “ Evanescent black holes,” Phys. Rev. D 45, no.4, R1005 (1992) doi:10.1103/PhysRevD.45.R1005 [arXiv:hep-th/9111056 [hep-th]]

arXiv 1992

[5] [5]

Holographic Space-time Models in 1 + 1 Dimensions,

T. Banks, “Holographic Space-time Models in 1 + 1 Dimensions,” [arXiv :1506.05777 [hep-th]]; T. Banks, Microscopic Models of Linear Dilaton Gravity and T heir Semi- classical Ap- proximations, [arXiv:hep-th/2005.09479]

arXiv 2005

[6] [6]

Time depend ent backgrounds of 2- D string theory,

S. Y. Alexandrov, V. A. Kazakov and I. K. Kostov, “Time depend ent backgrounds of 2- D string theory,” Nucl. Phys. B 640, 119-144 (2002) doi:10.1016/S0550-3213(02)00541-2 [arXiv:hep-th/0205079 [hep-th]]

arXiv 2002

[7] [7]

Flux-vacua in two dimensional str ing theory,

J. M. Maldacena and N. Seiberg,“Flux-vacua in two dimensional str ing theory,” JHEP 09, 077 (2005) doi:10.1088/1126-6708/2005/09/077 [arXiv:hep-th/ 0506141 [hep-th]]

work page doi:10.1088/1126-6708/2005/09/077 2005

[8] [8]

JT gravity coupled to fermio ns,

T. Banks, P. Draper and B. Zhang, “JT gravity coupled to fermio ns,” Adv. Theor. Math. Phys. 27, no.2, 483-522 (2023) doi:10.4310/ATMP.2023.v27.n2.a2 [arXiv:2205 .07382 [hep-th]]

doi:10.4310/atmp.2023.v27.n2.a2 2023