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Multiple-Order Tensor Field Theory: Enumeration of unitary invariant observables

T0 review · 2 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A new formula counts every mixed-order tensor interaction.

desk verdict A genuine mixed-order extension of tensor-invariant counting whose Burnside formula is sound, but the one-field-per-color-type identification is a real scope restriction the paper does not flag. read the letter →

arxiv 2505.13236 v1 pith:A3AXQSTJ submitted 2025-05-19 math-ph hep-thmath.MP

Joseph Ben Geloun , Arnauld Solente This is my paper · ORCID
classification math-phhep-thmath.MP MSC 05A1505C3020B0581T32
keywords unitaryinvariantstensorfieldtheorymultiple-ordercontractionscoloredbipartitegraphsBurnside'slemmamatrixmodelsmixed-orderobservablesenumerationofinteractions
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 proposes extending tensor field theory from a single tensor order to collections of tensors of orders $d'\le d$, and asks a basic enumerative question: given prescribed numbers of tensors of each order and color type, how many inequivalent unitary-invariant contractions exist? The answer, Theorem 4, is a Burnside-lemma formula counting orbits of a permutation group action on colored bijections, each orbit being one observable. If the formula is correct, the full menu of mixed-order tensor interactions—vectors, matrices, and higher-order fields contracted together—becomes systematically enumerable, not just the fixed-order case studied before. The authors show their formula reduces to the known fixed-order count, and they compute new integer sequences for model interactions mixing order-3 tensors with matrices and vectors.

What carries the argument

The load-bearing object is the colored bipartite graph $G=(\Lambda,\Gamma,\sigma)$: vertices carry color types $A\subseteq[d]$ (so a vertex of type $A$ is a tensor of order $|A|$), and $\sigma_c$ is a bijection between the $c$-colored vertices of the two sides. The contraction kernel is the product of Kronecker deltas $\delta(i_c^{(v)},j_c^{(\sigma_c(v))})$ over all edges. The machinery is Burnside's lemma applied to the action $\alpha(\Lambda,\Gamma)$, which permutes the vertices on each side by elements of $H(\Lambda)$ and $H(\Gamma)$ that preserve color types; the orbit-stabilizer theorem converts the fixed-point count into a sum over integer partitions of the per-type cardinalities, with symmetry factors $\mathrm{Sym}(\mu)$ and a delta condition on color-wise sums. This machinery turns the geometric question 'how many distinct interactions?' into a finite arithmetic sum that a computer can evaluate.

What would settle it

For the paper's Section 4 configuration (two order-3 tensors on one side; one order-3 tensor, one $\{1,2\}$ matrix, and one $\{3\}$ vector on the other, with colors 1, 2, 3), enumerate all triples $\sigma=(\sigma_1,\sigma_2,\sigma_3)$ modulo the $H(\Lambda)\times H(\Gamma)$ relabelings by brute-force computer search; if the number of inequivalent contractions is not 20, Theorem 4 is wrong. The same search in the fixed-order case should return $Z^3_2=4$.

Watch

Extended reading notes

Core claim

The central discovery is that unitary invariants built from tensor fields of several orders are exactly the orbits of the group action $\alpha(\Lambda,\Gamma)$ of $H(\Lambda)\times H(\Gamma)$ on the set $S=\times_{c=1}^{d}S(c)$ of colored bijections between two compatible colored vertex sets $\Lambda=(V,\phi)$ and $\Gamma=(W,\psi)$. A vertex carries a color type $A\subseteq[d]$ and represents a tensor of order $|A|$; a bijection $\sigma_c$ pairs the half-edges of color $c$ on the two sides, producing the contraction. Theorem 4 states that $Z(\Lambda,\Gamma)$ equals the displayed sum over partitions $\mu_A\vdash n(A)$ and $\nu_A\vdash m(A)$, weighted by symmetry factors and constrained by the delta condition that, for every color $c$, the sums $\sum_{A\in F(c)}\mu_A$ and $\sum_{A\in F(c)}\nu_A$ match. Each orbit corresponds to one inequivalent contraction, invariant under $U(N)^{\otimes|\phi(v)|}$ for every vertex, and the formula reproduces $Z^d_n=\sum_{\mu\vdash n}\mathrm{Sym}(\mu)^{d-2}$ in the fixed-order limit.

Load-bearing premise

The enumeration treats all tensors with the same order and same set of colors as one and the same field; if a model contains two distinct fields of identical color type, that identification breaks down and the orbit count no longer lists the model's interactions.

Editorial extensions

If this is right

  • For any prescribed list of tensor orders and color types, every distinct unitary-invariant interaction is one term in the sum, so the formula gives the full interaction inventory of a mixed-order tensor model.
  • Setting all tensors to a single full order $d$ returns the known count $Z^d_n=\sum_{\mu\vdash n}\mathrm{Sym}(\mu)^{d-2}$; the multiple-order formula is a genuine extension, not a separate bookkeeping device.
  • Mixed models coupling order-3 tensors with matrices and vectors acquire explicit countable interaction sets; the paper's worked example yields 20 inequivalent contractions, and the resulting integer sequences had not been catalogued before.
  • The provided implementation lets one scan 'tensor theory space' and select candidate interactions, such as the displayed $M^4\phi^4$-type action, before any renormalization analysis.

Reading between the lines

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

  • If a model contains two physically distinct fields with the same color type—say two different matrix fields both of type $\{1,2\}$—the identification rule (34)-(35) fails, so Theorem 4 would undercount the joint interactions; an extension would need extra color labels for field species.
  • The formula's factorized delta conditions look like gluing conditions, which suggests the known correspondence between fixed-order tensor invariants and branched covers of a sphere may extend to mixed orders; the authors explicitly leave that topological interpretation open.
  • A testable extension is to attach weights by $n(A)$ and form generating functions from the sum; the large-$n$ growth of the new sequences should then be governed by the partition with the largest symmetry factor, mirroring known asymptotics for fixed-order invariants.
  • Because the counting is purely combinatorial, the same sum enumerates all edge-colored bipartite graphs with prescribed vertex color types, so the formula can be used outside tensor field theory as a graph enumerator.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper proposes a framework for multiple-order tensor field theory in which fields of different orders (encoded as vertex color types) are contracted to form unitary invariants. It defines multiple-order tensor contractions as colored bipartite graphs, proves invariance under the unitary group and under a permutation group action, and derives via Burnside's lemma and the orbit-stabilizer theorem a closed formula (Theorem 4) for the number of inequivalent contractions. The known fixed-order formula Z_d^n = sum Sym(mu)^(d-2) is recovered as a corollary. The paper also gives a worked example with rank-3 tensors coupled to matrices and vectors, a hand-check of the resulting 20 orbits, tables of new integer sequences, and Python code implementing the counting formula.

Significance. If the issues identified below are resolved, the paper would be a useful and novel contribution. The enumeration formula is derived self-contained, with no fitted parameters, and the fixed-order formula appears as a limiting corollary rather than as an input, so the recovery is an external consistency check rather than circular reasoning. The provisioning of Python code, the nontrivial worked example, and the explicit recovery of known results are clear strengths. The main caveats are that the formalism only counts one field per color type and that the unitary-invariance proof as printed needs a correction in the transformation of the conjugate family.

major comments (2)
  1. [Sec. 3.2, Definition 9 and Sec. 3.3.1, Theorem 2, Eqs. (45), (50)] As printed, the contraction I(G;(T(v)),(R(w))) is formed from two families of covariant tensors, and Theorem 2 states invariance under the fundamental action of U(N) on both families. This cannot be correct for complex unitary invariance: if both T_i and R_i transform by the same unitary U, the contraction T_i R_i transforms by U^T U, which is not the identity. The R-family must consist of conjugate fields transforming by \bar U. Accordingly, Eq. (45) and the unitarity step after Eq. (50) should use \bar U (with \bar U_{ba} U_{bc} = \delta_{ac}), not U for both factors. With the current formulas the proof is valid only if U^T U = I, i.e. for orthogonal rather than unitary transformations. The authors should either state explicitly that R(w) are conjugate tensors and correct the transformation rules, or clarify the intended convention; this is load-bearing because it is the proof that the enumerated objects are unitary invariants.
  2. [Sec. 3.2, Eqs. (34)-(35), and Sec. 3.4, Theorem 4] The identification rule (Eqs. (34)-(35)) declares all tensors attached to vertices of the same color type to be equal and indistinguishable. This is not merely a harmless convention: it restricts the scope of Theorem 4 to models with at most one field per color type. For example, with two distinct matrix fields A and B of type {1,2} on each side, Theorem 4 gives Z(Lambda,Gamma)=2, whereas direct enumeration of contractions with distinguishable A,B gives 4 inequivalent contractions. The paper's title and abstract promise enumeration of observables for multiple-order TFT without this qualification. The authors should either restrict the claim explicitly in the abstract and in the statement of Theorem 4, or extend the formalism to labeled fields. The internal derivation of Theorem 4 is sound for the single-field-per-type setting, but the advertised scope is broader than what is proven.
minor comments (6)
  1. [Sec. 2.1, Eq. (3)] In Definition 1, the transformation rule for the conjugate tensor \bar T is written with the same matrix Lambda as for T; it should involve \bar Lambda. This is likely the source of the U/\bar U ambiguity in Theorem 2.
  2. [Sec. 3.3.2, Theorem 3, Eq. (69)] The proof ends with the equality I(Lambda,Gamma,eta.sigma.pi^{-1};...) = I(...), while the statement of Theorem 3 is invariance under eta.sigma.pi. Since pi ranges over the group H(Lambda), the two formulations are equivalent, but the mismatch should be pointed out to avoid confusion.
  3. [Sec. 4.2] The combinatorial construction of the 16 remaining graphs is described informally, and the claim that the 4+4+8 graphs are pairwise non-isomorphic is not fully shown. This is acceptable as a validation example because the 20-orbit count is independently produced by the implementation of Theorem 4, but a sentence explaining why the cases are disjoint would strengthen the presentation.
  4. [Sec. 4.2] There is a typo: 'Height configurations' in Figure 7's caption should read 'Eight configurations'.
  5. [Abstract and Sec. B.2] The abstract states that the paper unveils integer sequences 'not documented elsewhere'; the evidence in Section B.2 is only that the first few values were not found in the OEIS. The wording should be softened to 'apparently new' or the OEIS search parameters should be reported.
  6. [Appendix C] The displayed Python code omits the required import statements for sympy and itertools, and the function name 'couting_orbits' is a typo for 'counting_orbits'. These are minor because the code is illustrative, but they should be fixed for reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; Theorem 4 is a self-contained Burnside orbit count with the fixed-order formula recovered only as a limiting corollary.

full rationale

The paper's central enumeration, Theorem 4, is derived directly from Burnside's lemma applied to the explicitly defined group action α(Λ,Γ) on the set of colored bijections. The orbit count Z(Λ,Γ) in Eq. (70) is computed from the cardinality functions n and m and the symmetry factors of integer partitions; no fitted parameter, no empirically inferred constant, and no target observable is used as an input. The identification rule in Eqs. (34)-(35), declaring equal-type tensors indistinguishable, is a modeling convention that defines the equivalence relation whose orbits are counted. It restricts the scope of the theorem to one field per color type, but it is not circular: the theorem does not presuppose the number of orbits it claims to compute. The fixed-order formula Z_d^n = Σ_{μ⊢n} Sym(μ)^{d-2} appears in Corollary 1 and Eq. (87) as a limiting case derived from Theorem 4, not as a premise of the proof, so recovering it is an external consistency check rather than a circular reduction. Self-citations to [37] and related work provide context and review, but the proof of Theorem 4 is self-contained and does not import its conclusion from those references. No step in the derivation chain reduces to its own inputs, and there is no fitted quantity renamed as a prediction. The paper's limitation about distinguishable same-type fields is a scope caveat, not a circularity.

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

The derivation relies only on standard group theory plus the explicitly stated identification of equal-type tensors. No free parameters are fitted, and no new physical entities are introduced.

assumptions (5)
  • standard math Burnside's lemma (Appendix Proposition A.2)
    Used in the proof of Theorem 4 to average fixed points of the action of H(Lambda)xH(Gamma) on the set of colored bijections.
  • standard math Orbit-stabilizer theorem (Appendix Proposition A.1)
    Used to express the size of a conjugacy class in terms of the symmetry factor Sym(mu).
  • standard math Symmetric group ambivalence and symmetry factors of conjugacy classes
    Used when replacing the conjugacy class of pi_c by that of pi_c^{-1} and when writing |C(mu)| = n!/Sym(mu).
  • domain assumption All tensors with the same color type are the same field (Section 3.2, equations (34)-(35))
    Defines the equivalence used to quotient contractions by vertex relabeling. It restricts the enumeration to one field per color type and is the main scope limitation.
  • domain assumption Finite-dimensional complex vector space E of dimension N for all fields
    The unitary invariance argument uses finite unitary matrices U(N); the paper does not treat infinite-dimensional or functional subtleties.

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Pith. "Pith review of Multiple-Order Tensor Field Theory: Enumeration of unitary invariant observables." pith.science (2026). https://pith.science/paper/A3AXQSTJ

@misc{pith2026250513236,
  author       = {Pith},
  title        = {Pith review of: Multiple-Order Tensor Field Theory: Enumeration of unitary invariant observables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A3AXQSTJ}},
  note         = {Machine review of arXiv:2505.13236}
}
abstract

In Tensor Field Theory (TFT), observables are defined through tensor field contractions that produce unitary invariants for complex-valued tensor fields. Traditionally, these observables are constructed using tensor fields of a fixed order $d$. Here, we propose an extended theoretical framework for TFT that incorporates tensor fields of varying orders $d'$, satisfying $d' \leq d$. We then establish a comprehensive group-theoretic formalism that enables the systematic enumeration of these complex TFT observables. This approach encompasses existing counting methods and therefore recovers known results in specific limiting cases. Additionally, we provide computational tools to facilitate the enumeration of these invariants, unveiling novel integer sequences that have not been documented elsewhere.

Figures

Figures reproduced from arXiv: 2505.13236 by the authors.

Figure 1
Figure 1. Graph associated with the vector contraction [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Graph associated with the matrix contraction [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Illustration of TabcTdefTabfTdec (left), MabMcdMefMadMebMcf (middle) and TabcdTabcd (right). Below, we define the gauge group action that describes the invariance of unitary con￾tractions according to point (ii) of proposition 1. Definition 3 (Group action). Let d, n ∈ N ∗ and H1 = H2 = Sn. We define the group action α d n on (Sn) ×d by: α d n :  H1 × H2 × (Sn) ×d → (Sn) ×d (γ, ρ, σ) 7→ γσρ = (γσ1ρ, γσ2ρ, · · · , γ… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Illustration of an edge-colored bipartite graph. [PITH_FULL_IMAGE:figures/full_fig_p027_4.png]
Figure 5
Figure 5. Figure 5: Four graphs representing multiple-order contractions of tensors issued from [PITH_FULL_IMAGE:figures/full_fig_p028_5.png]
Figure 6
Figure 6. Figure 6: Four graphs are obtained from figure 4 by gluing the red vector and matrix field [PITH_FULL_IMAGE:figures/full_fig_p028_6.png]
Figure 7
Figure 7. Figure 7: Height configurations obtained from figure 4 by gluing the two vectors to one or two tensors. 5 Concluding remarks The definition of multiple-order tensor contraction is provided for any d ′ ≤ d, where d is a positive integer, demonstrating that these tensor contractio…
Figure 8
Figure 8. Figure 8: A possible multiple-order contraction generalizing the Grosse-Wulkenhaar trace [PITH_FULL_IMAGE:figures/full_fig_p030_8.png]

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Reference graph

Works this paper leans on

52 extracted references · 30 canonical work pages

  1. [1]

    Di Francesco, P

    P. Di Francesco, P. H. Ginsparg, and J. Zinn-Justin, 2-D Gravity and random ma- trices, Phys. Rept. 254, 1–133 (1995), arXiv:hep-th/9306153

    arXiv 1995

  2. [2]

    Scaling limits of random planar maps with large faces

    J.-F. Le Gall and G. Miermont, in XVIth International Congress on Mathematical Physics (World Scientific, 2010) arXiv:0907.3262 [math.PR]

    work page Pith review arXiv 2010

  3. [3]

    Ambjorn, B

    J. Ambjorn, B. Durhuus, and T. Jonsson, Three-dimensional simplicial quantum gravity and generalized matrix models , Mod. Phys. Lett. A 6, 1133–1146 (1991)

    work page 1991

  4. [4]

    Sasakura, Tensor model for gravity and orientability of manifold, Mod

    N. Sasakura, Tensor model for gravity and orientability of manifold, Mod. Phys. Lett. A 6, 2613–2624 (1991)

    work page 1991

  5. [5]

    Gross, Tensor models and simplicial quantum gravity in > 2-D, Nucl

    M. Gross, Tensor models and simplicial quantum gravity in > 2-D, Nucl. Phys. B Proc. Suppl. 25, 144–149 (1992)

    work page 1992

  6. [6]

    D. V. Boulatov, A Model of three-dimensional lattice gravity , Mod. Phys. Lett. A 7, 1629–1646 (1992), arXiv:hep-th/9202074

    arXiv 1992

  7. [7]

    Freidel, Group field theory: An overview , International Journal of Theoretical Physics 44, 1769–1783 (2005)

    L. Freidel, Group field theory: An overview , International Journal of Theoretical Physics 44, 1769–1783 (2005)

    work page 2005

  8. [8]

    Oriti, The group field theory approach to quantum gravity (2007), arXiv:gr- qc/0607032 [gr-qc]

    D. Oriti, The group field theory approach to quantum gravity (2007), arXiv:gr- qc/0607032 [gr-qc]

    work page arXiv 2007

Show all 52 references
  1. [9]

    Marchetti, D

    L. Marchetti, D. Oriti, A. G. A. Pithis, and J. Th¨ urigen, Mean-Field Phase Tran- sitions in Tensorial Group Field Theory Quantum Gravity , Phys. Rev. Lett. 130, 141501 (2023), arXiv:2211.12768 [gr-qc]

    2023 arXiv

  2. [10]

    Carrozza, Tensor models and group field theories: combinatorics, large N and renormalization (2024), arXiv:2404.07834 [math-ph]

    S. Carrozza, Tensor models and group field theories: combinatorics, large N and renormalization (2024), arXiv:2404.07834 [math-ph] . 40

    2024 arXiv

  3. [11]

    Carrozza, Tensorial Methods and Renormalization in Group Field Theories , Springer Theses (Springer International Publishing, 2016)

    S. Carrozza, Tensorial Methods and Renormalization in Group Field Theories , Springer Theses (Springer International Publishing, 2016)

    2016

  4. [12]

    Rivasseau, Quantum Gravity and Renormalization: The Tensor Track , AIP Conf

    V. Rivasseau, Quantum Gravity and Renormalization: The Tensor Track , AIP Conf. Proc. 1444, 18–29 (2012), arXiv:1112.5104 [hep-th]

    2012 arXiv

  5. [13]

    Gurau, Random Tensors (Oxford University Press, 2017)

    R. Gurau, Random Tensors (Oxford University Press, 2017)

    2017

  6. [14]

    Bonzom, R

    V. Bonzom, R. Gurau, A. Riello, and V. Rivasseau, Critical behavior of colored tensor models in the large n limit , Nuclear Physics B 853, 174–195 (2011)

    2011

  7. [15]

    Gurau and J

    R. Gurau and J. P. Ryan, Melons are branched polymers , Annales Henri Poincar´ e 15, 2085–2131 (2013)

    2013

  8. [16]

    Ben Geloun and V

    J. Ben Geloun and V. Rivasseau, A Renormalizable 4-Dimensional Tensor Field Theory, Commun. Math. Phys. 318, 69–109 (2013), arXiv:1111.4997 [hep-th]

    2013 arXiv

  9. [17]

    Ben Geloun, Renormalizable Models in Rank d≥ 2 Tensorial Group Field Theory, Commun

    J. Ben Geloun, Renormalizable Models in Rank d≥ 2 Tensorial Group Field Theory, Commun. Math. Phys. 332, 117–188 (2014), arXiv:1306.1201 [hep-th]

    2014 arXiv

  10. [18]

    Eichhorn and T

    A. Eichhorn and T. A. Koslowski, Continuum limit in matrix models for quantum gravity from the Functional Renormalization Group, Phys. Rev. D 88, 084016 (2013), 1309.1690

    2013 arXiv

  11. [19]

    Eichhorn and T

    A. Eichhorn and T. A. Koslowski, Towards phase transitions between discrete and continuum quantum spacetime from the renormalization group , Phys. Rev. D 90, 104039 (2014), 1408.4127

    2014 arXiv

  12. [20]

    Benedetti, J

    D. Benedetti, J. Ben Geloun, and D. Oriti, Functional Renormalisation Group Ap- proach for Tensorial Group Field Theory: a Rank-3 Model , JHEP 03, 084, 1411.3180

    arXiv

  13. [21]

    Ben Geloun, R

    J. Ben Geloun, R. Martini, and D. Oriti, Functional Renormalization Group analysis of a Tensorial Group Field Theory on R3, EPL 112, 31001 (2015), 1508.01855

    2015 arXiv

  14. [22]

    Ben Geloun, R

    J. Ben Geloun, R. Martini, and D. Oriti, Functional renormalization group analysis of tensorial group field theories on Rd, Phys. Rev. D 94, 024017 (2016), 1601.08211

    2016 arXiv

  15. [23]

    Krajewski and R

    T. Krajewski and R. Toriumi, Exact Renormalisation Group Equations and Loop Equations for Tensor Models , SIGMA 12, 068 (2016), 1603.00172

    2016 arXiv

  16. [25]

    Eichhorn, T

    A. Eichhorn, T. Koslowski, J. Lumma, and A. D. Pereira, Towards background inde- pendent quantum gravity with tensor models , Class. Quant. Grav. 36, 155007 (2019), arXiv:1811.00814 [gr-qc]

    2019 arXiv

  17. [27]

    Eichhorn, J

    A. Eichhorn, J. Lumma, A. D. Pereira, and A. Sikandar, Universal critical behavior in tensor models for four-dimensional quantum gravity , JHEP 02, 110, 1912.05314

    1912 arXiv

  18. [28]

    Lahoche and D

    V. Lahoche and D. Ousmane Samary, Large-d behavior of the Feynman amplitudes for a just-renormalizable tensorial group field theory , arXiv , arXiv:1911.08601 (2019), 1911.08601

    2019 arXiv

  19. [29]

    Lahoche, D

    V. Lahoche, D. Ousmane Samary, and A. D. Pereira, Renormalization group flow of coupled tensorial group field theories: Towards the Ising model on random lattices , Phys. Rev. D 101, 064014 (2020), 1911.05173

    2020 arXiv

  20. [30]

    Lahoche and D

    V. Lahoche and D. Ousmane Samary, Revisited functional renormalization group approach for random matrices in the large-N limit, Phys. Rev. D 101, 106015 (2020), 1909.03327

    2020 arXiv

  21. [31]

    Lahoche and D

    V. Lahoche and D. Ousmane Samary, Pedagogical comments about nonperturbative Ward-constrained melonic renormalization group flow , Phys. Rev. D 101, 024001 (2020), 2001.00934

    2020 arXiv

  22. [32]

    Bonzom, T

    V. Bonzom, T. Delepouve, and V. Rivasseau, Enhancing non-melonic triangulations: A tensor model mixing melonic and planar maps , Nucl. Phys. B 895, 161–191 (2015), arXiv:1502.01365 [math-ph]

    2015 arXiv

  23. [33]

    Ben Geloun and R

    J. Ben Geloun and R. Toriumi, Renormalizable enhanced tensor field theory: The quartic melonic case, Journal of Mathematical Physics59, 10.1063/1.5022438 (2018)

    2018 doi

  24. [34]

    Ben Geloun and R

    J. Ben Geloun and R. Toriumi, One-loop beta-functions of quartic enhanced tensor field theories, J. Phys. A 57, 015401 (2024), arXiv:2303.09829 [hep-th]

    2024 arXiv

  25. [35]

    R. C. Read, The enumeration of locally restricted graphs (i) , Journal of The London Mathematical Society-second Series , 417–436 (1959)

    1959

  26. [36]

    de Mello Koch and S

    R. de Mello Koch and S. Ramgoolam, Strings from Feynman Graph counting : with- out large N , Phys. Rev. D 85, 026007 (2012), arXiv:1110.4858 [hep-th]

    2012 arXiv

  27. [37]

    Ben Geloun and S

    J. Ben Geloun and S. Ramgoolam, Counting tensor model observables and branched covers of the 2-sphere , Ann. Inst. H. Poincare D Comb. Phys. Interact. 1, 77–138 (2014), arXiv:1307.6490 [hep-th] . 42

    2014 arXiv

  28. [38]

    R. C. Avohou, J. Ben Geloun, and N. Dub, On the counting ofo(n) tensor invariants, Advances in Theoretical and Mathematical Physics 24, 821–878 (2020)

    2020

  29. [39]

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

    2024 arXiv

  30. [40]

    Ben Geloun and S

    J. Ben Geloun and S. Ramgoolam, Tensor Models, Kronecker coefficients and Per- mutation Centralizer Algebras, JHEP 11, 092, arXiv:1708.03524 [hep-th]

    arXiv

  31. [41]

    Ben Geloun and S

    J. Ben Geloun and S. Ramgoolam, Quantum mechanics of bipartite ribbon graphs: Integrality, Lattices and Kronecker coefficients , Alg. Comb. 6, 547–594 (2023), arXiv:2010.04054 [hep-th]

    2023 arXiv

  32. [42]

    Ben Geloun, On the counting tensor model observables asU(N) andO(N) classical invariants, PoS CORFU2019, 175 (2020), arXiv:2005.01773 [hep-th]

    J. Ben Geloun, On the counting tensor model observables asU(N) andO(N) classical invariants, PoS CORFU2019, 175 (2020), arXiv:2005.01773 [hep-th]

    2020 arXiv

  33. [43]

    Ben Geloun and S

    J. Ben Geloun and S. Ramgoolam, All-orders asymptotics of tensor model ob- servables from symmetries of restricted partitions , J. Phys. A 55, 435203 (2022), arXiv:2106.01470 [hep-th]

    2022 arXiv

  34. [44]

    Dijkgraaf and E

    R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Com- munications in Mathematical Physics 129, 393–429 (1990)

    1990

  35. [45]

    Barcelo and A

    H. Barcelo and A. Ram, Combinatorial representation theory , arXiv Mathematics e-prints (1997), arXiv:math/9707221 [math.RT]

    1997 arXiv

  36. [46]

    F. D. Murnaghan, The analysis of the kronecker product of irreducible representations of the symmetric group , American Journal of Mathematics 60, 761 (1938)

    1938

  37. [47]

    Padellaro, S

    A. Padellaro, S. Ramgoolam, and R.-K. Seong, Row and column detection complex- ities of character tables (2025), arXiv:2503.02543 [hep-th]

    2025

  38. [48]

    de Mello Koch, Y.-H

    R. de Mello Koch, Y.-H. He, G. Kemp, and S. Ramgoolam, Integrality, duality and finiteness in combinatoric topological strings , JHEP 01, 071, arXiv:2106.05598 [hep- th]

    arXiv

  39. [49]

    Ramgoolam and E

    S. Ramgoolam and E. Sharpe, Combinatoric topological string theories and group theory algorithms, JHEP 10, 147, arXiv:2204.02266 [hep-th]

    arXiv

  40. [50]

    Padellaro, R

    A. Padellaro, R. Radhakrishnan, and S. Ramgoolam, Row–column duality and combi- natorial topological strings, J. Phys. A 57, 065202 (2024), arXiv:2304.10217 [hep-th]

    2024 arXiv

  41. [51]

    Lang, Algebra, 3rd ed

    S. Lang, Algebra, 3rd ed. (Springer, New York, 2002). 43

    2002

  42. [52]

    Oriti, L

    D. Oriti, L. Sindoni, and E. Wilson-Ewing, Bouncing cosmologies from quantum gravity condensates, Class. Quant. Grav. 34, 04LT01 (2017), arXiv:1602.08271 [gr- qc]

    2017 arXiv

  43. [53]

    Grosse and R

    H. Grosse and R. Wulkenhaar, Renormalisation of ?4-theory on noncommutative ?4 in the matrix base , Communications in Mathematical Physics 256, 305–374 (2005)

    2005

  44. [54]

    Ben Geloun and T

    J. Ben Geloun and T. A. Koslowski, Nontrivial uv behavior of rank-4 tensor field models for quantum gravity (2016), arXiv:1606.04044 [gr-qc] . 44

    2016 arXiv

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