Correlators in two rainbow tensor and complex multi-matrix models

Bei Kang; Ke Wu; Lu-Yao Wang; Wei-Zhong Zhao

REVIEW 3 major objections 4 minor 38 references

Correlators in two rainbow tensor and complex multi-matrix models

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read This paper constructs two rank-3 rainbow tensor models whose partition functions are exponentials of W-operators, and shows that every Gaussian correlator of gauge-invariant operators equals a topology-weighted sum over triples of colored…

desk verdict The Dessin/Wick correlator formulas are solid and new, but the advertised W-representation is asserted rather than derived—fixable, referee-worthy. read the letter →

arxiv 2505.08132 v1 pith:NPR2VL4G submitted 2025-05-13 hep-th math-phmath.MP

Bei Kang , Lu-Yao Wang , Ke Wu , Wei-Zhong Zhao This is my paper · ORCID

classification hep-thmath-phmath.MP

keywords tensormodelsrainbowrank-3tensorsgauge-invariantoperatorsW-representationcoloredDessinsd'enfantsHurwitznumberscomplexmulti-matrix

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 claims that two families of rank-3 rainbow tensor models—theories with several complex tensors $T^j_{i_1i_2i_3}$ transforming under $U(N_1)\otimes U(N_2)\otimes U(N_3)$—are exactly solvable. In each family the full Gaussian partition function with arbitrary couplings to connected gauge-invariant operators is generated by an exponential of a cut-and-join operator, $Z = \exp(\widehat W)\cdot 1$, and every correlator admits a second, graphical formula: a sum over triples of colored Dessins weighted only by Euler characteristics and automorphism group orders. The bridge between the two formulas is a one-to-one correspondence between connected operators and colored bipartite maps. A sympathetic reader would care because this turns correlator computations in rainbow tensor models into finite combinatorial and topological bookkeeping, and because the same W-representation machinery then yields complex multi-matrix models as degradations.

What carries the argument

The load-bearing objects are the W-operator $\widehat W$, built from cut-and-join coefficients $(\Delta_j)_{\bar\mu}$ and $(\Lambda_j)$ that encode contractions of tensor indices, and colored Dessins—bipartite graphs embedded in an oriented surface whose edges carry colors indexing the $\sharp L^2$ tensor-conjugation pairs. The W-operator acts on coupling constants and produces the partition function from the constant 1; the grading operator $\widehat D$ satisfies $[\widehat D,\widehat W]=\widehat W$, so $\widehat W$ raises the level by one and powers of $\widehat W$ give the correlators level by level. On the Dessin side, the one-to-one correspondence between connected gauge-invariant operators and colored bipartite maps turns each Gaussian expectation into a pasting of three Dessins, with each pasting weighted by $N_i^{\chi(D_i)}$ factors and divided by automorphism group sizes. The paper claims the two mechanisms give identical correlator formulas.

What would settle it

Compute a level-three Gaussian correlator in the two-tensor model (4.18) by direct Wick contraction for small $N_1,N_2,N_3$ and compare with the colored-Dessin sum (4.21), equivalently with the coefficient read from $\exp(\widehat W'_I)\cdot 1$. The cut-and-join coefficients needed at level three are not given in closed form, so any mismatch—or any non-polynomial dependence on $N_i$ in the intermediate coefficients—would break the claimed W-representation; agreement at several small $N_i$ values would support it.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the rainbow tensor model (4.1) with measure $\exp(-\sum_{j=1}^{\sharp L}\mathrm{Tr}\,T_j\bar T_j)$ and couplings to connected operators, and its cyclic variant (4.25) with measure $\exp(-\sum_{j=1}^{\sharp L-1}\mathrm{Tr}\,T_j\bar T_{j+1}-\mathrm{Tr}\,T_{\sharp L}\bar T_1)$, are both W-representable: $Z_I=\exp(\widehat W_I)\cdot 1$ and $Z_{II}=\exp(\widehat W_{II})\cdot 1$. From this representation the correlators are extracted as coefficients in powers of $\widehat W$, and equivalently, through the operator--Dessin correspondence, as the sum $$\langle\!\langle R_{\tilde\$\sigma$}^{(\vec a,\vec b)}\rangle\!\rangle = \frac{1}{N_1}\sum_{D_1,D_2,D_3\in \mathcal D_{\rm col}} \frac{$N_1^{{\frac12\chi(D_1)}}$$N_2^{{\chi(D_2)-\frac12\chi(D_1)}}$$N_3^{{\chi(D_3)-\frac12\chi(D_1)}}$}{|\mathrm{Aut}(D_1)||\mathrm{Aut}(D_2)||\mathrm{Aut}(D_3)|},$$ with the prefactor $1/N_1$ replaced by $1/N_2$ in the cyclic model (4.31). The same computation also yields the count of independent operators at level $l$ in terms of Hurwitz numbers, relates disconnected to connected operators through the plethystic logarithm, and produces complex multi-matrix models (5.1) and (5.2) as $N_1=1$ degradations of the two tensor models.

Load-bearing premise

The W-representation derivation assumes that the cut-and-join coefficients $(\Delta_j)$ and $(\Lambda_j)$ are well-defined polynomials in $N_1,N_2,N_3$ at every level and that the algebra of connected gauge-invariant operators closes under cut and join; the paper states this structure but does not prove it or write the coefficients in general.

Editorial extensions

If this is right

Gaussian correlators in the rank-3 rainbow models depend only on the Euler characteristics and automorphism groups of three colored Dessins, so at each level the whole set of correlators is fixed by a finite topological bookkeeping.
The W-representation gives an algebraic generation of the same correlators: expanding $\exp(\widehat W)\cdot 1$ in the coupling constants and reading off coefficients replaces Gaussian integration by differentiation.
Specializing to one tensor reproduces the Aristotelian rainbow model, and further restricting couplings yields the red-rainbow model whose correlators reduce to sums over Young diagrams involving $GL(N)$ representation dimensions and symmetric-group characters.
Setting $N_1=1$ and rescaling couplings degrades the tensor models into the complex multi-matrix models (5.1) and (5.2), whose correlators are the corresponding limits of the tensor correlators.
The number of independent operators at level $l$ is expressed through Hurwitz numbers, and disconnected operators are recovered from connected ones by a plethystic exponential, completing the enumeration at every level.

Reading between the lines

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

Because the Dessin weights are powers of $N_i$ through Euler characteristics, the formulas are ready-made for a large-$N$ genus expansion, even though the paper does not analyze that limit.
The W-operator construction may extend to rainbow tensors of rank $r>3$; if the algebra of connected operators closes there, the colored-Dessin bookkeeping would have to be replaced by higher-dimensional maps, which is a testable next step.
The multi-matrix degradations, especially the multi-trace family (5.8), are natural candidates for exact character expansions analogous to the red-rainbow formula; checking for such expansions would extend the paper's results beyond what it states.
The two independent formulas—W-operator and Dessin sum—can be cross-checked at a level where the cut coefficients are not written out, such as level three in the two-tensor model, providing a self-consistency test of the whole construction.

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,

The paper extends the W-representation / Dessin program from one- and two-tensor rank-3 rainbow models to arbitrary numbers of tensors. The genuinely new pieces are the Hurwitz-number counting formula (2.12), the colored-Dessin description, and the explicit Wick-theorem correlator sums (4.17) and (4.31). The two-tensor example (4.22) matches direct computation, and the #L=1 limit reduces to the known Aristotelian/red rainbow results. That part is worth taking seriously.

The soft spot is exactly what the stress-test flags: the W-representation is not actually derived. The coefficients (Δ_j) and (Λ_j) that define Ŵ_I in (4.8) are left as abstract polynomials; the derivation of (4.6) from the Ward identity is a one-line assertion; and the grading/commutator identities (4.9), (4.13), which are needed for Z=exp(Ŵ)·1, are asserted without proof. So equations (4.14) and (4.25), and the expression (4.16), are not yet verified. This matters because the abstract advertises W-representations as a main tool. It is not a fatal flaw for the whole paper, since the Dessin-based correlator formula (4.17) is derived directly from Wick's theorem and passes checks, so it stands independently. But the W-representation claims need either explicit computation of the coefficients or a reference to a derivation that does the job; otherwise they should be relabeled as conjectural.

I also found the counting formula (2.12) plausible but only lightly checked; it reduces to known cases but no nontrivial #L>1 example is given. The citation pattern is fine—the self-citations are to the prior papers in the same program—and there is no sign of fitting to a target.

Who is this for? People working on tensor model combinatorics and W-representations will find the Dessin correlator formula useful. It deserves a serious referee: the central claim is checkable and the gaps are the kind a revision can fix. I would not desk-reject. Send it to a referee who knows the Dessin/W-representation literature, and ask them to verify or disprove (4.9)-(4.13) and to demand the coefficients in (4.4)-(4.5) or a precise statement of what remains open.

Referee Report

3 major / 4 minor

Summary. The paper constructs two families of rank-3 rainbow tensor models with an arbitrary number ♯L of complex tensors transforming under U(N1)⊗U(N2)⊗U(N3). Section 2 counts gauge-invariant operators at each level, expressing the count through Hurwitz numbers. Section 3 identifies connected operators with colored Dessins. Section 4 defines two tensor models, model I with Gaussian weight exp(−Σ Tr T_j \bar T_j) and model II with cyclic weight exp(−Σ Tr T_j \bar T_{j+1}), and claims W-representations Z = exp(\hat W)·1 for both. Two compact correlator formulas are presented for each model: one from the W-operator expansion and one from a Wick-theorem sum over colored Dessins. Section 5 degrades the models to complex multi-matrix models. The paper claims the W-representation construction as a derivation but leaves the cut-and-join coefficients and closure/grading properties unproved.

Significance. The Dessin-based correlator formula (4.17) and its model-II analogue (4.31) are explicit, parameter-free sums obtained from Wick's theorem; they pass consistency checks in the two-tensor example (4.22) and reduce to known single-tensor character formulas (4.24). The counting formula (2.12) is explicit and reproduces the known ♯L=1 case. These are genuine contributions. The W-representation claims (4.14) and (4.25), however, are advertised as an independent derivation of correlators but are not substantiated, so the paper currently does not deliver a complete proof of one of its two central tools.

major comments (3)

[4.1, Eqs. (4.3)–(4.14)] The step from the Ward identity (4.3) to (4.6) is asserted as 'not difficult,' but the cut-and-join coefficients (Δ_j) and (Λ_j) in (4.4)–(4.5) are never computed, and the paper does not show that the connected operators close under these operations, that the commutator (4.9) holds, or that the grading relations (4.13) are satisfied. Without these checks the W-representation Z_I = exp(\hat W_I)·1 and the correlator formula (4.16) derived from it are unverified. This is load-bearing because the W-representation is one of the two advertised methods in the abstract and Section 4.
[4.2, Eqs. (4.20) and (4.22)] The explicit two-tensor correlators are stated to follow from calculating \hat W'_I, but \hat W'_I in (4.20) is expressed entirely through the unspecified coefficients (Δ_1), (Δ_2), (Λ_1), (Λ_2). The table (4.22) therefore cannot be checked from the W-representation method. Only one entry (R(1,1,1,1)_{((12),(12),(12))}) is verified by the Dessin method; the remaining seven entries should be either derived explicitly from (4.17) or accompanied by the computed coefficients.
[4.3, Eqs. (4.27)–(4.29)] The same gap occurs for model II. The operators (\tilde Δ_j) and (\tilde Λ_j) are introduced in (4.28)–(4.29) but not computed; the W-operator \hat W_{II} in (4.27) and \hat W'_{II} in (4.34) are only templates. The asserted W-representation (4.25), the grading conditions, and the correlator formula (4.30) are therefore not established for the general multi-tensor model.

minor comments (4)

[Eq. (4.17)] The symbol N1 is used both for the size of U(N_1) and for the normalization constant in (4.17); please use separate notations (e.g., \mathcal N_1) to avoid ambiguity.
[Eqs. (5.6)–(5.7)] The equality between the large-N1 limit of the normalized sum over σ1 and the evaluation at N1=1 is nontrivial; a one-sentence explanation (e.g., leading-power reduction) would help readers.
[Through the text] There are numerous typographical issues in the displayed equations, including missing braces in (2.8) and inconsistent placement of sums in (4.8); a careful proofreading pass is needed.
[Section 3] The 'one-to-one correspondence' between connected operators and colored Dessins is asserted by construction; since this bijection is central to the graphical correlator formula, it would be helpful to state the inverse map or cite the precise result from [7,8].

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Dessin correlator formulas are Wick-theorem evaluations, and the unproved W-representation identities are rigor gaps rather than circular borrowings.

full rationale

The paper's advertised new correlator expressions, (4.17) and (4.31), are obtained by a direct Gaussian evaluation: the text states "we have used the Wick theorem in the second line" and then sums over permutations gamma and tau1,tau2,tau3. The final sum over colored Dessins D1,D2,D3 is a bijective relabeling of those Wick contractions via the operator/Dessin correspondence of Section 3, not a fitted or independently assumed result. No parameter is tuned to correlator data. The W-representation claims (4.14) and (4.25) are weaker: (4.6) is asserted as "not difficult to obtain" from the Ward identity, and (4.9) and (4.13) are stated without proof, with the cut-and-join coefficients (Delta_j) and (Lambda_j) left as unspecified polynomials. That is an incompleteness or correctness risk, not circularity, because W_I is not defined in terms of the correlators it is supposed to predict; it is a differential operator whose coefficients are supposed to come from the cut/join algebra. The paper's citations of the authors' earlier W-representation papers [28,30] are background for previously known one-tensor and two-tensor cases; the multi-tensor extension is not obtained merely by citing them, and the Dessin-based correlator computation is independent of those citations. Accordingly, no step in the derivation chain reduces by construction to its own input.

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

The paper's central formulas rely on standard combinatorial facts (Burnside's lemma, Wick's theorem, Hurwitz number identities), on prior Dessin correspondence, and on an unproven algebraic closure and W-representation step. No free parameters are fitted to data.

assumptions (5)

standard math The identity connecting double sums to Hurwitz numbers (Eq. 2.9 from Ref [11]) and its extensions (2.10)-(2.11).
Used in Section 2 to express the count of independent operators in terms of Hurwitz numbers.
standard math Gaussian complex tensor integration is governed by Wick's theorem with propagator δ_{aa'}δ_{bb'}δ_{cc'}δ_j^k.
Basis of the correlator computations in Section 4, particularly Eq. (4.17).
domain assumption The general operator/Dessin correspondence of Refs [7,8,36] extends to the colored multi-tensor setting and is a bijection.
Section 3 establishes the correspondence by example at level 2 but does not give a full proof; the correlator formula's graphical interpretation rests on it.
ad hoc to paper The cut-and-join coefficients (Δ_j) and (Λ_j) in (4.4)-(4.5) are polynomials in N_i with integer coefficients, and the algebra generated by connected operators closes under these operations.
Section 4.1 asserts this without computation; it is the key input for the W-representation (4.14).
domain assumption The differential identities (4.6), (4.9), and (4.13) hold, so Z_I = exp(Ŵ_I)·1 reproduces the partition function.
Stated as 'it is not difficult to obtain' in Section 4.1; not fully derived in the text.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Correlators in two rainbow tensor and complex multi-matrix models." pith.science (2026). https://pith.science/paper/NPR2VL4G

@misc{pith2026250508132,
  author       = {Pith},
  title        = {Pith review of: Correlators in two rainbow tensor and complex multi-matrix models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NPR2VL4G}},
  note         = {Machine review of arXiv:2505.08132}
}

abstract

We construct two rainbow tensor models with multi-tensors of rank-$3$ and present their $W$-representations. We give the formula of counting number of independent gauge-invariant operators in terms of Hurwitz numbers and establish a one-to-one correspondence between connected operators and colored Dessins. By means of the colored Dessins and $W$-representations, respectively, we derive two compact expressions of correlators for each of rainbow tensor models. Furthermore, two complex multi-matrix models from the degradations of the constructed rainbow tensor models are also discussed.

Figures

Figures reproduced from arXiv: 2505.08132 by the authors.

**Figure 2.** Correspondence between R˜ (1,1,1,1),(12) (id,id,id) and the Dessin. Since the only choice of the contraction γ is the permutation (12), the Dessin D1 corresponding to the correlator ⟨⟨R˜ (1,1,1,1),(12) (id,id,id) ⟩⟩I,♯L=2 is determined by (id, (12), (12)) (see [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** The Dessin D1 of ⟨⟨R˜ (1,1,1,1),(12) (id,id,id) ⟩⟩I,♯L=2 . Let us delete white vertices in the Dessin assigned to the operator R˜ (1,1,1,1),(12) (id,id,id) and black vertices in the Dessin D1 corresponding to the correlator ⟨⟨R˜ (1,1,1,1),(12) (id,id,id) ⟩⟩I,♯L=2 , and glue the remaining vertices with the valence kept. It gives the Dessin D2 corresponding to the correlator ⟨⟨R˜ (1,1,1,1),(12) (id,id,id) ⟩⟩I,♯L=2 (se… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: The Dessin D2 of ⟨⟨R˜ (1,1,1,1),(12) (id,id,id) ⟩⟩I,♯L=2 . Let us delete black vertices in the Dessin assigned to the operator R˜ (1,1,1,1),(12) (id,id,id) and convert the remaining white vertices to black vertices. Then we delete black vertices in the Dessin D1 corre…

**Figure 5.** Figure 5: The Dessin D3 of ⟨⟨R˜ (1,1,1,1),(12) (id,id,id) ⟩⟩I,♯L=2 . It is obvious that χ(D1) = χ(D2) = χ(D3) = 2 from Figs. 3-5 and N1 = 1. Thus from the second line of (4.21), we reach the result of the last line of (4.22). When ♯L = 1, the rainbow model (4.1) becomes the Aris…

**Figure 6.** Figure 6: Dessins corresponding to R (a1,0,··· ,0,b1,0,··· ,0) σ˜ . The Dessins D1 assigned to correlators ⟨⟨R(a1,0,··· ,0,b1,0,··· ,0) σ˜ ⟩⟩I,♯L=1 are obtained by taking any permutation in S a1 to label the edges and going around the white vertices. The Dessins D2 and D3 are t…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

38 extracted references · 18 canonical work pages

[1]

Ben Geloun and S

J. Ben Geloun and S. Ramgoolam, Counting tensor model observables and branched covers of the 2-sphere, Ann. Inst. Henri Poincar´e Comb. Phys. Interact. 1 (2014) 77 [arXiv:1307.6490]

arXiv 2014
[2]

Gauge Invariants, Correlators and Holography in Bosonic and Fermionic Tensor Models

R. de Mello Koch, D. Gossman and L. Tribelhorn, Gauge invariants, correlators and holography in bosonic and fermionic tensor models, JHEP 09 (2017) 011 [arXiv:1707.01455]

work page Pith review arXiv 2017
[3]

Ben Geloun and S

J. Ben Geloun and S. Ramgoolam, Tensor models, Kronecker coefficients and permutation central- izer algebras, JHEP 11 (2017) 092 [arXiv:1708.03524]

arXiv 2017
[4]

On the counting of $O(N)$ tensor invariants

R. Avohou, J. Ben Geloun and N. Dub, On the counting of O(N) tensor invariants, Adv. Theor. Math. Phys. 24 (2020) 4 [arXiv:1907.04668]

work page Pith review arXiv 2020
[5]

Ben Geloun, On the counting tensor model observables as U(N) and O(N) classical invariants, PoS CORFU2019 175 (2020) [arXiv:2005.01773]

J. Ben Geloun, On the counting tensor model observables as U(N) and O(N) classical invariants, PoS CORFU2019 175 (2020) [arXiv:2005.01773]

arXiv 2020
[6]

Avohou, J

R. Avohou, J. Ben Geloun and R. Toriumi, Counting U(N) N r N O(N) N q invariants and tensor model observables, Eur. Phys. J. C 84 (2024) 839 [arXiv:2404.16404]

arXiv 2024
[7]

Correspondence between Feynman diagrams and operators in quantum field theory that emerges from tensor model

N. Amburg, H. Itoyama, A. Mironov, A. Morozov, D. Vasiliev and R. Yoshioka, Correspondence between Feynman diagrams and operators in quantum field theory that emerges from tensor model, Eur. Phys. J. C 80 (2020) 471 [arXiv:1911.10574]. 18

work page Pith review arXiv 2020
[8]

Review on the Operator/Feynman diagram/Dessins d'enfant Correspondence in Tensor Model

H. Itoyama, A. Mironov, A. Morozov and R. Yoshioka, Review on the operator /Feynman dia- gram/dessins d’enfants, Int. J. Mod. Phys. A 37 (2022) 03 [arXiv:2109.05245]

work page Pith review arXiv 2022

Show all 38 references

[9]

Itoyama, A

H. Itoyama, A. Mironov and A. Morozov, Ward identities and combinatorics of rainbow tensor models, JHEP 06 (2017) 115 [arXiv:1704.08648]

2017 arXiv
[10]

Itoyama, A

H. Itoyama, A. Mironov and A. Morozov, Rainbow tensor model with enhanced symmetry and extreme melonic dominance, Phys. Lett. B 771 (2017) 180 [arXiv:1703.04983]

2017 arXiv
[11]

Itoyama, A

H. Itoyama, A. Mironov and A. Morozov, Cut and join operator ring in tensor models, Nucl. Phys. B 932 (2018) 52 [arXiv:1710.10027]

2018 arXiv
[12]

Itoyama, A

H. Itoyama, A. Mironov and A. Morozov, Tensorial generalization of characters, JHEP 12 (2019) 127 [arXiv:1909.06921]

2019 arXiv
[13]

Morozov and Sh

A. Morozov and Sh. Shakirov, Generation of matrix models by ˆW-operators, JHEP 04 (2009) 064 [arXiv:0902.2627]

2009 arXiv
[14]

Alexandrov, Cut-and-Join operator representation for Kontsewich-Witten tau-function, Mod

A. Alexandrov, Cut-and-Join operator representation for Kontsewich-Witten tau-function, Mod. Phys. Lett. A 26 (2011) 2193 [arXiv:1009.4887]

2011 arXiv
[15]

Alexandrov, Cut-and-join description of generalized Brezin-Gross-Witten model, Adv

A. Alexandrov, Cut-and-join description of generalized Brezin-Gross-Witten model, Adv. Theor. Math. Phys. 22 (2018) 1347 [arXiv:1608.01627]

2018 arXiv
[16]

Cassia, R

L. Cassia, R. Lodin and M. Zabzine, On matrix models and their q-deformations, JHEP 10 (2020) 126 [arXiv:2007.10354]

2020 arXiv
[17]

Mironov, V

A. Mironov, V . Mishnyakov and A. Morozov, Non-AbelianW-representation for GKM, Phys. Lett. B 823 (2021) 136721 [arXiv:2107.02210]

2021 arXiv
[18]

Mironov, A

A. Mironov, A. Morozov, Superintegrability summary, Phys. Lett. B 835 (2022) 137573 [arXiv:2201.12917]

2022 arXiv
[19]

Mironov, A

A. Mironov, A. Morozov and Z. Zakirova, New insights into superintegrability from unitary matrix models, Phys. Lett. B 831 (2022) 137178 [arXiv:2203.03869]

2022 arXiv
[20]

Wang, C.H

R. Wang, C.H. Zhang, F.H. Zhang and W.Z. Zhao, CFT approach to constraint operators for ( β- deformed) hermitian one-matrix models, Nucl. Phys. B 985 (2022) 115989 [arXiv:2203.14578]

2022 arXiv
[21]

Mishnyakov and A

V . Mishnyakov and A. Oreshina, Superintegrability in β-deformed Gaussian Hermitian matrix model from W-operators, Eur. Phys. J. C 82 (2022) 548 [arXiv:2203.15675]

2022 arXiv
[22]

R. Wang, F. Liu, C.H. Zhang and W.Z. Zhao, Superintegrability for (β-deformed) partition function hierarchies with W-representations, Eur. Phys. J. C 82 (2022) 902 [arXiv:2206.13038]. 19

2022 arXiv
[23]

Bawane, P

A. Bawane, P. Karimi and P. Sułkowski, Proving superintegrability inβ-deformed eigenvalue mod- els, SciPost Phys. 13 (2022) 069 [arXiv:2206.14763]

2022 arXiv
[24]

Alexandrov, On W-operators and superintegrability for dessins d’enfant, Eur

A. Alexandrov, On W-operators and superintegrability for dessins d’enfant, Eur. Phys. J. C 83 (2023) 147 [arXiv:2212.10952]

2023 arXiv
[25]

Mironov, V

A. Mironov, V . Mishnyakov, A. Morozov, A. Popolitov, R. Wang and W.Z. Zhao, Interpolating matrix models for WLZZ series, Eur. Phys. J. C 83 (2023) 377 [arXiv:2301.04107]

2023 arXiv
[26]

F. Liu, R. Wang, J. Yang and W.Z. Zhao, Calogero-Sutherland-type quantum systems, generalized hypergeometric functions and superintegrability for integral chain, arXiv:2502.18921

arXiv
[27]

Itoyama, A

H. Itoyama, A. Mironov and A. Morozov, Complete solution to Gaussian tensor model and its integrable properties, Phys. Lett. B 802 (2020) 135237 [arXiv:1910.03261]

2020 arXiv
[28]

Kang, L.Y

B. Kang, L.Y . Wang, K. Wu, J. Yang and W.Z. Zhao, W-representation of rainbow tensor model, JHEP 05 (2021) 228 [arXiv:2104.01332]

2021 arXiv
[29]

L.Y . Wang, R. Wang, K. Wu and W.Z. Zhao, W-representations of the fermionic matrix and Aris- totelian tensor models, Nucl. Phys. B 973 (2021) 115612 [arXiv:2110.14269]

2021 arXiv
[30]

Kang, L.Y

B. Kang, L.Y . Wang, K. Wu and W.Z. Zhao, A two-tensor model with order-three, Eur. Phys. J. C 84 (2024) 239 [arXiv:2301.06046]

2024 arXiv
[31]

Orlov and D

A. Orlov and D. M. Shcherbin, Hypergeometric solutions of soliton equations, Theor. Math. Phys. 128 (2001) 906

2001
[32]

Okounkov and R

A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz numbers, and completed cycles, Ann. Math. 163 (2006) 517 [arXiv:math/0204305]

2006 arXiv
[33]

Orlov, Hypergeometric functions as infinite-soliton tau functions, Theor

A. Orlov, Hypergeometric functions as infinite-soliton tau functions, Theor. Math. Phys.146 (2006) 183 [arXiv:nlin/0305001]

2006 arXiv
[34]

I. P. Goulden and D. M. Jackson, The KP hierarchy, branched covers, and triangulations, Adv. Math. 219 (2008) 932 [arXiv:0803.3980]

2008 arXiv
[35]

Ambjørn and L

J. Ambjørn and L. Chekhov, The matrix model for dessins d’enfants, Ann. Inst. H. Poincare D Comb. Phys. Interact. 1 (2014) 337 [arXiv:1404.4240]

2014 arXiv
[36]

de Mello Koch and S

R. de Mello Koch and S. Ramgoolam, From matrix models and quantum fields to Hurwitz space and the absolute Galois group, arXiv:1002.1634

arXiv
[37]

L.Y . Wang, V . Mishnyakov, A. Popolitov, F. Liu and R. Wang, W-representations for multi-character partition functions and theirβ-deformations, Phys. Lett. B 851 (2024) 138570 [arXiv:2301.12763]

2024 arXiv
[38]

Wang, Y .S

L.Y . Wang, Y .S. Zhu, S.K. Yao and B. Kang, LargeN limit of complex multi-matrix model, Nucl. Phys. B 998 (2024) 116425 [arXiv:2312.08761]. 20

2024 arXiv

Pith tools

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

[1] [1]

Ben Geloun and S

J. Ben Geloun and S. Ramgoolam, Counting tensor model observables and branched covers of the 2-sphere, Ann. Inst. Henri Poincar´e Comb. Phys. Interact. 1 (2014) 77 [arXiv:1307.6490]

arXiv 2014

[2] [2]

Gauge Invariants, Correlators and Holography in Bosonic and Fermionic Tensor Models

R. de Mello Koch, D. Gossman and L. Tribelhorn, Gauge invariants, correlators and holography in bosonic and fermionic tensor models, JHEP 09 (2017) 011 [arXiv:1707.01455]

work page Pith review arXiv 2017

[3] [3]

Ben Geloun and S

J. Ben Geloun and S. Ramgoolam, Tensor models, Kronecker coefficients and permutation central- izer algebras, JHEP 11 (2017) 092 [arXiv:1708.03524]

arXiv 2017

[4] [4]

On the counting of $O(N)$ tensor invariants

R. Avohou, J. Ben Geloun and N. Dub, On the counting of O(N) tensor invariants, Adv. Theor. Math. Phys. 24 (2020) 4 [arXiv:1907.04668]

work page Pith review arXiv 2020

[5] [5]

Ben Geloun, On the counting tensor model observables as U(N) and O(N) classical invariants, PoS CORFU2019 175 (2020) [arXiv:2005.01773]

J. Ben Geloun, On the counting tensor model observables as U(N) and O(N) classical invariants, PoS CORFU2019 175 (2020) [arXiv:2005.01773]

arXiv 2020

[6] [6]

Avohou, J

R. Avohou, J. Ben Geloun and R. Toriumi, Counting U(N) N r N O(N) N q invariants and tensor model observables, Eur. Phys. J. C 84 (2024) 839 [arXiv:2404.16404]

arXiv 2024

[7] [7]

Correspondence between Feynman diagrams and operators in quantum field theory that emerges from tensor model

N. Amburg, H. Itoyama, A. Mironov, A. Morozov, D. Vasiliev and R. Yoshioka, Correspondence between Feynman diagrams and operators in quantum field theory that emerges from tensor model, Eur. Phys. J. C 80 (2020) 471 [arXiv:1911.10574]. 18

work page Pith review arXiv 2020

[8] [8]

Review on the Operator/Feynman diagram/Dessins d'enfant Correspondence in Tensor Model

H. Itoyama, A. Mironov, A. Morozov and R. Yoshioka, Review on the operator /Feynman dia- gram/dessins d’enfants, Int. J. Mod. Phys. A 37 (2022) 03 [arXiv:2109.05245]

work page Pith review arXiv 2022