Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Quantum Dilogarithms and New Integrable Lattice Models in Three Dimensions

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read The paper constructs new integrable 3D lattice models from quantum dilogarithms: the local weights satisfy the tetrahedron equation, and the infinite-lattice partition function of the real-line model is calculated exactly in closed form.

desk verdict Announces a genuinely new family of 3D integrable models built from quantum dilogarithms, but all load-bearing proofs are deferred to two unpublished companion papers. read the letter →

arxiv 2512.23338 v2 pith:375F6WOS submitted 2025-12-29 math-ph cond-mat.stat-mechhep-thmath.MP

Vladimir V. Bazhanov , Rinat M. Kashaev , Vladimir V. Mangazeev , Sergey M. Sergeev This is my paper · ORCID
classification math-phcond-mat.stat-mechhep-thmath.MP MSC 82B2081R12
keywords integrablelatticemodelstetrahedronequationquantumdilogarithmcommutingtransfermatricespartitionfunctionpersiteLobachevskyinteraction-round-a-cubemodelpentagonidentity
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

This paper tries to establish that quantum dilogarithms — functions obeying an inversion relation and a five-term "pentagon" identity — are a factory for exactly solvable three-dimensional lattice models. It gives a universal recipe: any such dilogarithm produces both a vertex-model R-matrix and an interaction-round-a-cube weight satisfying the tetrahedron equation, the 3D analogue of the Yang–Baxter equation. Commuting layer-to-layer transfer matrices follow, making the models integrable. For the real-line dilogarithm, the partition function per site on an infinite lattice is computed exactly, written in terms of Lobachevsky functions, with a neat value exp(8η²G/π) at the symmetric point, where G is Catalan's constant.

What carries the argument

The central object is a quantum dilogarithm φ(x) on a self-dual locally compact Abelian group S, subject to the inversion relation φ(x)φ(−x)=φ(0)²G(x) and the quantum pentagon identity (a five-term integral identity). These two identities alone produce the R-matrix (2.19) and, via a 3D "propagation through the vertex" technique, the cube weights (2.28). A third property, Fourier self-duality (2.15) with the consistency condition γ²φ(0)⁶G(η)=1, is needed for the simplified weights and for the exact partition-function calculation; the "crossing parameter" η is generally outside S, so shifted values are understood by analytic continuation.

What would settle it

Evaluate both sides of the vertex-type tetrahedron equation (2.22) numerically for the real-line model with a generic non-zero choice of the six spectral parameters, approximating L²(R) by a finite basis of localized functions; a nonzero difference would disprove the claimed identity. Similarly, a direct finite-lattice computation of the partition function (4.11) by transfer-matrix or Monte Carlo methods that disagrees with exp(8η²G/π) would falsify the exact-result claim.

Watch

Extended reading notes

Core claim

The central claim is that the R-matrix defined in eq. (2.19) satisfies the vertex-type tetrahedron equation (2.22) for arbitrary values of the six spectral parameters, with the proof relying only on the inversion and pentagon identities of the quantum dilogarithm. When the dilogarithm is Fourier self-dual, the weights simplify to eq. (2.23), and the equivalent interaction-round-a-cube weights (2.28) satisfy the corresponding tetrahedron relation (2.31). These relations imply that layer-to-layer transfer matrices commute in all three spin models (S = R, R × Z_N, T × Z). For the real-line example, the partition function per site of both the vertex and IRC models is calculated exactly and shown

Load-bearing premise

The simplified weights and the exact partition functions assume the quantum dilogarithm is Fourier self-dual (2.15) with a consistent crossing parameter η satisfying γ²φ(0)⁶G(η)=1, and that expressions like φ(x±η) make sense by analytic continuation even though η generally lies outside the spin set; this is fully established only for the real-line example, and cited rather than proven for the other two.

Editorial extensions

If this is right

  • Three concrete 3D lattice models are produced, with spins on the real line, on R×Z_N, and on T×Z; all have continuous families of commuting layer-to-layer transfer matrices.
  • The same quantum-dilogarithm input yields both vertex-type and interaction-round-a-cube solutions of the tetrahedron equation, and the two formulations give exactly equal partition functions on the infinite lattice.
  • For the real-line model the partition function per site has a closed form in Lobachevsky functions, depending on the modular parameter only through an overall factor; at the symmetric point it reduces to exp(8η²G/π).
  • The field-dependent R-matrix (2.36) produces a three-parameter family of commuting transfer matrices, so the models can be extended by external fields without losing integrability.
  • The structure mirrors the known discrete N-state solutions of the tetrahedron equation, and the continuous-spin solutions should generate infinite families of solutions of the Yang–Baxter equation by treating one lattice direction as an auxiliary space.

Reading between the lines

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

  • If the tetrahedron equation holds for all six parameters, then by treating one of the three Hilbert spaces as auxiliary, each R-matrix yields an infinite family of Yang–Baxter R-matrices; this should produce new integrable 2D models parameterized by the continuous spin set, a consequence the paper only gestures at.
  • The b-dependence of the partition function (only an overall coefficient) suggests a universality: the free-energy shape is governed by the spherical-triangle angles rather than the microscopic spin values; one could test whether the same shape appears in the R×Z_N and T×Z models.
  • The quasiclassical limit b→0 is said to connect to circular quadrilateral lattices; if so, the stationary configurations of these lattice models should realize these classical discrete geometries, offering a geometric falsifier for the model class.
  • Because the proof of the tetrahedron equation and the partition-function calculations are deferred to companion papers, an independent derivation of (2.22) from (2.11)–(2.12) would significantly increase confidence; this is my suggestion, not the paper's claim.
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.

Referee Report

3 major / 4 minor

Summary. The paper proposes a general scheme for constructing integrable three-dimensional lattice models from quantum dilogarithms on Pontryagin self-dual locally compact Abelian groups. It defines a vertex R-matrix (2.19), a simplified vertex weight (2.23), IRC weights (2.28), and a field-dependent vertex R-matrix (2.33), and claims that these satisfy the Zamolodchikov tetrahedron equation (2.22), the IRC tetrahedron relation (2.31), and the deformed version (2.37), yielding commuting layer-to-layer transfer matrices. Three quantum dilogarithm examples are reviewed: Faddeev, Andersen–Kashaev, and Woronowicz. For the Faddeev case the paper reports exact infinite-lattice partition functions per site, Eqs. (4.8), (4.13), and (4.15), with the symmetric-point value exp(8η²G/π). The paper explicitly states that the proofs of the tetrahedron equations and partition-function derivations are deferred to companion papers [29] and [33].

Significance. If the claims hold, this is a substantial contribution: it unifies a large class of tetrahedron-equation solutions under the quantum-dilogarithm pentagon identity, extends integrable 3D models to new spin sets S = R, R×Z_N, and T×Z, and produces exact free energies depending on the modular parameter only through an overall coefficient. The construction is parameter-free in the sense that no fitting is involved; the claimed results are derived from identities (2.11)–(2.16). The algebraic check at the symmetric point in Sec. 4.1, reducing (4.8) to exp(8η²G/π) via Λ(π/4) = G/2, is correct. However, the central integrability and exact-solvability claims are not demonstrated in the submitted text; they are announced and deferred to unpublished companions, and the required Fourier self-duality and analytic continuation for the new examples are imported from references. As submitted, the paper functions as a research announcement rather than a self-contained proof.

major comments (3)
  1. [§2.3, Eq. (2.22); §2.4, Eq. (2.31)] The central claim that the R-matrix (2.19), and its simplified form (2.23), satisfies the vertex-type Zamolodchikov tetrahedron equation is not proved in the text; the proof is said to be in [29]. Similarly, the IRC tetrahedron relation (2.31) is asserted with a remark that it follows from transformation identities for generalized hypergeometric series, but no derivation or statement of those identities is given. Since commutativity of layer-to-layer transfer matrices and the word “integrable” rest entirely on these equations, the main theorem is not verifiable from the submitted manuscript. A revision should include complete proofs or at least a detailed derivation of the reduction to the basic properties (2.11)–(2.12), not merely a citation to an unpublished companion.
  2. [§2.2, Eqs. (2.15)–(2.17); §3.2, §3.3] The simplified weights (2.23), (2.28), (2.33) and all results of Sec. 4 depend on the Fourier self-duality (2.15) and its consistency constraint (2.16). For the Faddeev example this property is checked explicitly in Eqs. (3.7a)–(3.7b), but for the Andersen–Kashaev and Woronowicz examples the self-duality is imported from [31] and [32] without proof. Moreover, the paper itself notes after Eq. (2.17) that η ∉ S in general, so φ(x±η) requires an analytic continuation, but no continuation is specified for the new examples. If the self-duality or the analytic continuation fails on the relevant groups, the simplified weights and all partition-function formulas built on them are not well defined. This is a load-bearing gap, not a cosmetic one.
  3. [§4.1–§4.3, Eqs. (4.8), (4.13), (4.15)] The exact partition-function results are claimed but all derivations are said to be contained in the unpublished companion [33]. No functional-relation, factorization, or saddle-point argument is even sketched in the present text. In particular, the exact equality (4.13) between the vertex and IRC partition functions is asserted through a vertex–IRC equivalence whose details are also deferred. Since these formulas constitute the paper’s concrete exact-solvability output, the revision should provide the derivation or a rigorous outline that can be checked.
minor comments (4)
  1. [§1, §4.1, §5] Typographical issues: “Boltzmannn” in the introduction, “requred” in Sec. 1, “θ1, θ2, θ2” should presumably be “θ1, θ2, θ3” in Sec. 4.1, and “line accesses” in Sec. 5 should likely be “line angles.”
  2. [§2.3, Eq. (2.25); §2.5, Eq. (2.33)] The square roots and fractional powers in the normalization factors (2.25) and (2.30) and in the R-matrix (2.33) require a branch specification, especially for complex η or λ’s. The paper does not state the chosen branch, which is a clarity issue for a mathematically precise definition of the weights.
  3. [§2.2, Eqs. (2.19)–(2.20)] The notation for the Fourier-transformed dilogarithm is inconsistent: the text uses both ṡ and eφ with the same meaning. Please unify the notation (e.g., always use the tilde as defined in Eq. (2.13)).
  4. [References] Refs. [29] and [33] are listed as “to be published” with no version or preprint number. Since the present paper relies on them for its main proofs, the manuscript would be greatly improved by including arXiv identifiers or an appendix containing the relevant statements.

Circularity Check

3 steps flagged · score 4.0 of 10

No fitted or definitional circularity, but the central ZTE and partition-function claims rest on unpublished same-author companion papers, and the concrete models rely on an imported, unproved Fourier self-duality.

  1. self citation load bearing [§2.3, eqs. (2.19)–(2.22); see also §1 and §5]
    "We claim, that the R-matrix (2.19) satisfies the vertex-type Zamolodchikov tetrahedron equation ... The proof of the tetrahedron equation (2.22) is given in [29]. It does not require a specialization to a particular set of spin variables in (2.3). It is only based on the basic properties of the quantum dilogarithms, given by (2.11) and (2.12). [29] Bazhanov, V. V., Kashaev, R. M., Mangazeev, V. V., and Sergeev, S. M. ... to be published, 2026."

    This is the load-bearing step of the paper: the vertex solution (2.19) is claimed to satisfy the Zamolodchikov tetrahedron equation (2.22) for arbitrary λ_i, and the commuting transfer matrices of all the proposed models depend on this claim. The only support offered in this text is a citation to [29], an unpublished companion paper by the same four authors. No proof or reduction is exhibited here, and the cited source is not machine-checked or externally verified. This is self-citation load-bearing rather than a fit or definitional identity: the theorem is logically independent of premises (2.11)–(2.12), so the paper is not circular by construction, but its central integrability claim is not self-contained.

  2. self citation load bearing [§4.1–§4.3, eqs. (4.8), (4.13), (4.15)]
    "With this parameterization, the partition function per site (4.3) can be written as [33], ... (4.8) ... As shown in [33] this partition function exactly coincides with that of the vertex model z∞^{(IRC)}(λ1,λ2,λ3) = z∞^{vert}(λ1,λ2,λ3). (4.13)"

    The exact partition-function results (4.8), the vertex–IRC equality (4.13), and the external-field result (4.15) are reported from [33], a companion paper by the same four authors ('to be published, 2026'), with no derivation in this text. These formulas are the paper's main exact-solvability predictions. They contain no fitted constants — only the parameter η and spherical-triangle data — so there is no reduction of output to input. But the only support provided is a self-citation chain to an unpublished work, making the central results unverifiable from the present manuscript.

1 more flagged steps
  1. other [§2.2, eqs. (2.15)–(2.17); §3.2–§3.3]
    "Note, that in general the constant η does not belong to the set S. Therefore, the 'shifted functions' φ(x±η) in (2.15) and (2.17) should be understood as an analytic continuation of φ(x) from x∈S. ... A solution of the defining relation (2.11)-(2.18) for this case was obtained by Andersen and Kashaev [31] ... A solution of (2.11)-(2.18) for this case was obtained by Woronowicz [32]"

    The simplified vertex weights (2.23), the IRC weights (2.28), the second vertex R-matrix (2.33), and hence all §4 partition-function results require the Fourier self-duality (2.15) with its consistency constraint (2.16). For the Andersen–Kashaev and Woronowicz examples this property is not proved here; it is imported from [31] (co-authored by Kashaev) and [32]. The paper itself states that η∉S in general, so φ(x±η) requires an analytic continuation not specified for these new cases. Thus a key premise of the concrete models is carried by citation and by an unspecified continuation, rather than by a derivation in the text. This is a missing verification, not a definitional identification; if (2.15) or the continuation fails, the simplified weights and partition functions are not well define

full rationale

The paper does not exhibit a fitted parameter called a prediction, nor does any equation reduce to its input by construction. The ZTE (2.22) is a nontrivial six-operator identity, distinct from the defining pentagon identity (2.12), and the partition-function formulas (4.8), (4.13), (4.15) contain no fitted constants — only η and spherical-triangle data. The derivation chain is therefore a genuine mathematical proposal if the deferred proofs hold. The score is elevated to 4 because the paper's central claims are not self-contained: the proof of (2.22) is deferred to the same authors' unpublished [29], the exact partition-function results are deferred to the same authors' unpublished [33], and the self-duality (2.15) underpinning all concrete simplified weights is imported from [31]/[32] with an unspecified analytic continuation for η∉S. These are load-bearing self-citations and omitted verifications, not demonstrations of logical circularity; hence a moderate score rather than a higher one.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The central claims rest on (i) the existence of quantum dilogarithms with properties (2.11)-(2.15) on the three groups (established in prior literature, partly by the same authors), (ii) the unimodularity regime (4.1), and (iii) two unpublished companion papers [29], [33] by the same authors that carry the proofs of the tetrahedron equations and the partition-function calculations. There is no data fitting and no invented physical entity; the free parameters are genuine model parameters. The main ledger signal is the deferral: a large fraction of the paper's asserted content is uncertified within the text.

free parameters (4)
  • b (modular parameter of the Faddeev / Andersen–Kashaev dilogarithms, q = e^{iπb²})
    Genuine model/deformation parameter, not fitted. Enters the dilogarithm (3.2) and, via η = i(b + b^{-1})/2, appears in the partition function (4.8) only as an overall coefficient; regime restricted to Im b = 0 or |b| = 1 in §4.
  • η_0 (free parameter of the Woronowicz dilogarithm, q = −e^{iη_0})
    Free parameter of the Woronowicz example (§3.3), not fitted. No partition-function result is given for this case, so it does not affect the central numerical claims.
  • λ_1, λ_2, λ_3 (spectral parameters)
    Arbitrary parameters of the commuting transfer-matrix families (§2.3). They are physical parameters of the models, not fit constants; the partition function depends on them through the spherical-triangle data β_j via (4.6)-(4.7).
  • ρ, ρ_1, ρ_W (normalization factors)
    Scalar factors fixed by (2.24), (2.25), (2.30). Gauge choices chosen by hand so that vertex and IRC partition functions coincide; they carry no physical free-energy content.
assumptions (7)
  • domain assumption Quantum dilogarithm satisfies inversion (2.11) and the pentagon identity (2.12).
    Defining properties of φ (§2.2). Existence asserted for the three examples: proved for the Faddeev case in [40], quoted from [31] and [32] for the Andersen–Kashaev and Woronowicz cases. The claimed proof of the tetrahedron equation 'is only based on' these two properties (§2.3).
  • ad hoc to paper Fourier self-duality (2.15) with consistency constraint γ²φ(0)⁶G(η) = 1 (2.16).
    Additional imposed property, not needed for the general R-matrix (2.19) but required for the simplified weights (2.23), the IRC weights (2.28), the R-matrix (2.33), and the partition-function analysis of §4. The paper states (2.16) arises 'as a consistency condition' but does not prove it for the examples.
  • domain assumption Analytic continuation of φ(x ± η) for η ∉ S.
    Explicitly assumed in §2.2: 'the constant η does not belong to the set S... should be understood as an analytic continuation'. For the Faddeev case this is classical (product formula (3.2)); for the other examples it is imported by citation.
  • domain assumption Unimodularity regime Im b = 0 or |b| = 1, Re b > 0 (eq. (4.1)).
    Restriction imposed in §4 so that |φ(x)| = 1 for real x; all claimed partition-function results are derived only in this regime, although the abstract states the partition function 'can be exactly calculated' without this qualifier.
  • ad hoc to paper The R-matrix (2.19)/(2.23) satisfies the vertex ZTE (2.22) and the W-weights (2.28) satisfy the IRC ZTE (2.31).
    Load-bearing assertion presented as 'We claim...' in §2.3, with proof deferred to the unpublished companion [29]. Also relied upon for the commutativity of transfer matrices and hence the definition of integrability.
  • ad hoc to paper Exact partition-function identities (4.8), (4.13), (4.15) and the vertex↔IRC equivalence.
    Presented as results with details 'as shown in [33]' (§4.1, §4.2, §4.3), [33] being an unpublished companion 'to be published, 2026'. The text does not contain the derivation.
  • standard math Self-dual LCA group harmonic analysis: existence of F, f(x,y), G(x) satisfying (2.7)-(2.9).
    Standard Fourier/Gaussian-exponent structure on Pontryagin self-dual locally compact Abelian groups, cited to Weil [34] (Sect. 2.2).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum Dilogarithms and New Integrable Lattice Models in Three Dimensions." pith.science (2026). https://pith.science/paper/375F6WOS

@misc{pith2026251223338,
  author       = {Pith},
  title        = {Pith review of: Quantum Dilogarithms and New Integrable Lattice Models in Three Dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/375F6WOS}},
  note         = {Machine review of arXiv:2512.23338}
}
read the original abstract

In this paper we introduce a new class of integrable 3D lattice models, possessing continuous families of commuting layer-to-layer transfer matrices. Algebraically, this commutativity is based on a very special construction of local Boltzmann weights in terms of quantum dilogarithms satisfying the inversion and pentagon identities. We give three examples of such quantum dilogarithms, leading to integrable 3D lattice models. The partition function per site in these models can be exactly calculated in the limit of an infinite lattice by using the functional relations, symmetry and factorization properties of the transfer matrix. The results of such calculations for 3D models associated with the Faddeev modular quantum dilogarithm are briefly presented.

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

  1. Solving the tetrahedron equation by Teichm\"uller TQFT

    math-ph 2026-02 conditional novelty 6.0 of 10

    Boltzmann weights built from Teichmüller TQFT on shaped triangulations with line defects exactly solve the bicolored tetrahedron equations.

Reference graph

Works this paper leans on

47 extracted references · 3 linked inside Pith · cited by 1 Pith paper

  1. [29]

    V., Kashaev, R

    Bazhanov, V. V., Kashaev, R. M., Mangazeev, V. V., and Sergeev, S. M. Quantum dilogarithms and new integrable lattice models in three dimensions. II. Zamolodchikov tetrahedron equation. to be published, 2026

    2026

  2. [33]

    V., Kashaev, R

    Bazhanov, V. V., Kashaev, R. M., Mangazeev, V. V., and Sergeev, S. M. Quantum dilogarithms and new integrable lattice models in three dimensions. III. Partition function. to be published, 2026

    2026

  3. [31]

    Andersen, J. E. and Kashaev, R. Complex Quantum Chern-Simons. arXiv:1409.1208, 2014

    arXiv 2014

  4. [32]

    Woronowicz, S. L. Operator equalities related to the quantumE(2) group. Comm. Math. Phys.144(1992) 417–428

    1992

  5. [1]

    Zamolodchikov, A. B. Tetrahedra equations and integrable systems in three-dimensional space. Soviet Phys. JETP52(1980) 325–336

    1980

  6. [2]

    Zamolodchikov, A. B. Tetrahedron equations and the relativistic S matrix of straight strings in (2+1)-dimensions. Commun. Math. Phys.79(1981) 489–505

    1981

  7. [3]

    Bazhanov, V. V. and Stroganov, Y. G. On commutativity conditions for transfer matrices on multidimensional lattice. Theor. Math. Phys.52(1982) 685–691

    1982

  8. [4]

    J.Exactly Solved Models in Statistical Mechanics

    Baxter, R. J.Exactly Solved Models in Statistical Mechanics. Academic, London, 1982

    1982

Show all 47 references
  1. [5]

    Baxter, R. J. On Zamolodchikov’s solution of the tetrahedron equations. Commun. Math. Phys.88(1983) 185–205

    1983

  2. [6]

    Bazhanov, V. V. and Stroganov, Y. G. Free Fermions on Three-dimensional Lattice and Tetrahedron Equations. Nucl. Phys. B230(1984) 435–454

    1984

  3. [7]

    Baxter, R. J. The Yang-Baxter Equations and the Zamolodchikov Model. Physica18D(1986) 321–247

    1986

  4. [8]

    Maillet, J. M. and Nijhoff, F. Integrability for multidimensional lattice models. Phys. Lett. B224(1989) 389

    1989

  5. [9]

    Bazhanov, V. V. and Baxter, R. J. New solvable lattice models in three-dimensions. J. Statist. Phys.69(1992) 453–585

    1992

  6. [10]

    Bazhanov, V. V. and Baxter, R. J. Star triangle relation for a three-dimensional model. J. Statist. Phys.71(1993) 839–864. 16

    1993

  7. [11]

    M., Mangazeev, V

    Kashaev, R. M., Mangazeev, V. V., and Stroganov, Y. G. Spatial symmetry, local integrability and tetrahedron equations in the Baxter-Bazhanov model. Int. J. Mod. Phys. A8(1993) 587

    1993

  8. [12]

    Korepanov, I. G. Tetrahedral Zamolodchikov algebras corresponding to Baxter’sL-operators. Comm. Math. Phys.154(1993) 85–97

    1993

  9. [13]

    Inversion and symmetry relations for a three-dimensional solvable model

    Bazhanov, V. Inversion and symmetry relations for a three-dimensional solvable model. Int. J. Mod. Phys. B07(1993) 3501–3515

    1993

  10. [14]

    Labeling schemes for tetrahedron equations and dualities between them

    Hietarinta, J. Labeling schemes for tetrahedron equations and dualities between them. J. Phys. A27(1994) 5727–5748

    1994

  11. [15]

    Kapranov, M. M. and Voevodsky, V. A. 2-categories and Zamolodchikov tetrahedra equations. Proceedings of Symposia in Pure Mathematics56(1994) 177–259

    1994

  12. [16]

    M., Mangazeev, V

    Sergeev, S. M., Mangazeev, V. V., and Stroganov, Y. G. The vertex formulation of the Bazhanov-Baxter model. J. Stat. Phys.82(1996) 31–50

    1996

  13. [17]

    Kashaev, R. M. On discrete three-dimensional equations associated with the local Yang-Baxter relation. Lett. Math. Phys.38(1996) 389–397

    1996

  14. [18]

    Baxter, R. J. and Bazhanov, V. V. Two-layer Zamolodchikov model. InXIIth International Congress of Mathematical Physics (ICMP ’97) (Brisbane), pages 15–23. Int. Press, Cambridge, MA, 1999

    1999

  15. [19]

    Sergeev, S. M. Solutions of the functional tetrahedron equation connected with the local Yang- Baxter equation for the ferro-electric condition. Lett. Math. Phys.45(1998) 113–119

    1998

  16. [20]

    Kashaev, R. M. and Sergeev, S. M. On pentagon, ten-term, and tetrahedron relations. Comm. Math. Phys.195(1998) 309–319

    1998

  17. [21]

    Kashaev, R. M. and Volkov, A. Y. From the tetrahedron equation to universalR-matrices. In L. D. Faddeev’s Seminar on Mathematical Physics, volume 201 ofAmer. Math. Soc. Transl. Ser. 2, pages 79–89. Amer. Math. Soc., Providence, RI, 2000

    2000

  18. [22]

    Bazhanov, V. V. and Sergeev, S. M. Zamolodchikov’s tetrahedron equation and hidden struc- ture of quantum groups. J. Phys.A39(2006) 3295–3310

    2006

  19. [23]

    V., Mangazeev, V

    Bazhanov, V. V., Mangazeev, V. V., and Sergeev, S. M. Quantum geometry of 3-dimensional lattices. J. Stat. Mech (2008) P07004

    2008

  20. [24]

    Sergeev, S. M. Super-tetrahedra and super-algebras. J. Math. Phys.50(2009) 083519

    2009

  21. [25]

    Tetrahedron equation and quantum cluster algebras

    Inoue, R., Kuniba, A., and Terashima, Y. Tetrahedron equation and quantum cluster algebras. J. Phys. A57(2024) 085202

    2024

  22. [26]

    Tetrahedron Equation and Quantum R Matrices for modular double ofU q(D(2) n+1),U q(A(2) 2n ) andU q(C (1) n )

    Kuniba, A., Okado, M., and Sergeev, S. Tetrahedron Equation and Quantum R Matrices for modular double ofU q(D(2) n+1),U q(A(2) 2n ) andU q(C (1) n ). arXiv:1409.1986, 2014

    1986 arXiv

  23. [27]

    Sergeev, S. M. Quantum integrable models in discrete 2+1 dimensional space-time: auxil- iary linear problem on a lattice, zero curvature representation, isospectral deformation of the Zamolodchikov-Bazhanov-Baxter model. Particles and Nuclei35(2004) 1051–1111

    2004

  24. [28]

    Faddeev, L. D. and Kashaev, R. M. Quantum Dilogarithm. Mod. Phys. Lett.A9(1994) 427–434. 17

    1994

  25. [30]

    Faddeev, L. D. Current - like variables in massive and massless integrable models. InInterna- tional School of Physics ’Enrico Fermi’: 127th Course: Quantum Groups and Their Physical Applications, pages 117–136, 1994

    1994

  26. [34]

    Weil, A.L’int´ egration dans les groupes topologiques et ses applications. Actual. Sci. Ind., no

  27. [35]

    The Yang-Baxter relation and gauge invariance

    Kashaev, R. The Yang-Baxter relation and gauge invariance. J. Phys. A49(2016) 164001

    2016

  28. [36]

    Sergeev, S. M. Geometry of quadrilateral nets: Second Hamiltonian form. Journal of Geometry and Physics59(2009) 1150–1154

    2009

  29. [37]

    Sergeev, S. M. Quantum 2 + 1 evolution model. J. Phys. A: Math. Gen.32(1999) 5693–5714

    1999

  30. [38]

    Baxter, R. J. Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisen- berg chain. II. Equivalence to a generalized Ice-type lattice model. Ann. Phys.76(1973) 25–47

    1973

  31. [39]

    and Kashaev, R

    Garoufalidis, S. and Kashaev, R. Quantum dilogarithms over local fields and invariants of 3-manifolds. arXiv:2306.01331, 2023

    2023 arXiv

  32. [40]

    D., Kashaev, R

    Faddeev, L. D., Kashaev, R. M., and Volkov, A. Y. Strongly coupled quantum discrete Liouville theory. I. Algebraic approach and duality. Commun. Math. Phys.219(2001) 199–219

    2001

  33. [41]

    Kashaev, R. M. and Nakanishi, T. Classical and Quantum Dilogarithm Identities. SIGMA7 (2011) 102

    2011

  34. [42]

    3-Manifolds and 3d Indices

    Dimofte, T., Gaiotto, D., and Gukov, S. 3-Manifolds and 3d Indices. Adv. Theor. Math. Phys. 17(2013) 975–1076

    2013

  35. [43]

    V., Mangazeev, V

    Bazhanov, V. V., Mangazeev, V. V., and Sergeev, S. M. Quantum geometry of 3-dimensional lattices. J. Stat. Mech.0807(2008) P07004

    2008

  36. [44]

    Bobenko, A. I. Discrete conformal maps and surfaces. London Math. Soc. Lecture Note Ser. 255(1999) 97–108

    1999

  37. [45]

    and Santini, P

    Doliwa, A. and Santini, P. M. Multidimensional quadrilateral lattices are integrable. Phys. Lett. A233(1997) 365–372

    1997

  38. [46]

    Konopelchenko, B. G. and Schief, W. K. Three-dimensional integrable lattices in Euclidean spaces: conjugacy and orthogonality. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.454 (1998) 3075–3104. 18

    1998

  39. [869]

    [This book has been republished by the author at Princeton, N

    Hermann et Cie., Paris, 1940. [This book has been republished by the author at Princeton, N. J., 1941.]

    1940

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.