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REVIEW 3 major objections 4 minor 78 references

$F$-extremization determines certain large-$N$ CFTs

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A universal variational principle fixes the conformal data of all melonic large-N CFTs: extremize the universal part of the sphere free energy of the corresponding generalized free field theory, subject only to the infrared marginality of…

desk verdict The constrained tilde-F extremization principle for melonic CFTs is real and largely proved, but the 'complete classification' wording outruns the proof. read the letter →

arxiv 2412.10499 v1 pith:AMKC47NS submitted 2024-12-13 hep-th math-phmath.MP

Ludo Fraser-Taliente , John Wheater This is my paper · ORCID
classification hep-thmath-phmath.MP
keywords melonicCFTsspherefreeenergyF-maximizationlarge-Nlimit2PIeffectiveactionSYKmodeltensorfieldtheoriesgeneralizedfields
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 every melonic large-$N$ CFT -- including the generalized SYK models, tensor field theories, and vector models -- is fully determined by a single constrained extremization principle. One extremizes $\widetilde{F}$, the universal part of the sphere free energy of a generalized free field theory with the same field content but arbitrary conformal dimensions, subject only to the linear constraints that each melonic interaction is marginal in the infrared. If true, this gives a complete classification of melonic CFTs and extends the $F$- and $a$-maximization methods known from supersymmetric theories to non-supersymmetric CFTs in continuous dimension $d$. The proof runs through the two-particle-irreducible effective action and, equivalently, through Schwinger-Dyson resummation, showing that the running couplings become Lagrange multipliers that enforce the marginality constraints. A sympathetic reader would care because it reduces a zoo of individually solved models to one variational principle and connects extremization of free energy with counting the number of infrared degrees of freedom.

What carries the argument

The load-bearing object is $\widetilde{F}$, the universal part of the sphere free energy, defined by $\widetilde{F} = -\sin(\pi d/2) \log Z_{S^d}$ for a collection of generalized free fields; it is finite in continuous dimension and interpolates between the Weyl anomaly coefficients and the odd-dimensional sphere free energy. The proof passes through the 2PI effective action $\Gamma[\{G_\phi\}]$ evaluated on the sphere with conformal propagators: the exact quantum solution extremizes this functional, and in the melonic limit the interaction vertices collapse into linear constraints on the scaling dimensions, with the renormalized squared couplings acting as Lagrange multipliers. The key technical fact is that the complete melon integral $\widetilde{M}(m_m)$ vanishes linearly in the marginality parameter $m_m = \sum_\phi q^m_\phi \Delta_\phi - d$ near $m_m = 0$, which is why only the infrared marginality constraints survive in the conformal limit.

What would settle it

Compute the full two-point Schwinger-Dyson solution of a multi-field melonic model at a non-integer dimension, say the quartic Yukawa model at $d=2.5$, and check whether every solution lying in the infrared wedge also satisfies $m_m = 0$ for every melon and extremizes $\widetilde{F}$; a solution with $m_m \neq 0$ that still gives a consistent flat-space CFT would refute the classification.

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Extended reading notes

Core claim

The central claim is that, for any melonic QFT in $d$ dimensions, the infrared CFT is specified by constrained extremization of $\widetilde{F}$, defined for a mean field theory with the same field content but arbitrary trial scaling dimensions $\Delta_\phi$. Concretely, with melonic interactions of schematic form $g_m \prod_\phi \phi^{q^m_\phi}$, the physical dimensions extremize $\widetilde{F}(\{\Delta_\phi\}) = \sum_\phi \widetilde{F}_\phi(\Delta_\phi, \rho'_\phi)$ subject to $\sum_\phi q^m_\phi \Delta_\phi - d = 0$ for every interaction that does not run to zero. Equivalently, the melonic CFTs are precisely the conformal mean field theories -- theories whose correlators are sums of products of two-point functions -- with constrained extremal $\widetilde{F}$. The paper establishes this by showing that the conformal slice of the 2PI effective action on the sphere reduces exactly to $\widetilde{F}$ plus linear constraints, and verifies the procedure on the quartic Yukawa, Popović, supersymmetric, and large-$n$ vector model examples.

Load-bearing premise

The argument assumes that the only infrared solutions that can be conformally mapped to flat space are the $R$-independent extrema of $\widetilde{F}$; if some $R$-dependent extremum of the conformal slice nevertheless produced a valid flat-space CFT through a different limiting procedure, the proposed classification would miss it.

Editorial extensions

If this is right

  • Every melonic CFT -- SYK-like, tensor, or vector -- can be solved by one constrained extremization instead of a model-by-model Schwinger-Dyson analysis.
  • The procedure extends $F$- and $a$-maximization to non-supersymmetric CFTs in continuous dimension, so supersymmetric extremization results can be imported directly for melonic SCFTs.
  • The classification is complete in the strict large-$N$ limit: the infrared CFT is exactly a mean field theory with constrained extremal $\widetilde{F}$, with all finite-symmetry details reduced to the dimensions of the symmetry representations.
  • The extremization generically produces a discrete set of candidate vacua, including complex scaling dimensions at some values of $d$ and missing solutions at certain integer dimensions, which can be compared against direct model computations.
  • Because $\widetilde{F}$ interpolates between $a$ and $F$, the result supports interpreting the infrared fixed point as extremizing the effective number of infrared degrees of freedom, in the spirit of the generalized $c$, $F$, and $a$ theorems.

Reading between the lines

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

  • If the claim is correct, the same constrained extremization may apply beyond strictly melonic diagrams to any large-$N$ limit dominated by factorization; the test would be whether non-melonic corrections shift the extremum at order $1/N$.
  • The coincidence between melonic and supersymmetric extremization suggests that, in unitary integer dimensions, the physical vacuum might always be a maximum of $\widetilde{F}$; if so, the vacuum selection problem among the discrete infrared solutions reduces to maximizing the count of infrared degrees of freedom.
  • Tuning the free propagator changes the infrared wedge, so the same procedure should reproduce long-range and $\Box^k$ CFTs; a direct check would be to run the extremization for the long-range SYK line and compare with its known solution.
  • Once the extremal dimensions are known, mean field theory fixes all higher-point functions and OPE data, so the newly predicted vacua come with concrete spectral predictions that could be tested by direct diagrammatic computation.
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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

3 major / 4 minor

Summary. The paper claims that the IR conformal data of all 'melonic' large-N CFTs (generalized SYK models, tensor models, and certain vector models) are determined by a constrained extremization of the universal part of the sphere free energy, tilde F, computed for generalized free fields with trial scaling dimensions. The constraints are linear marginality conditions sum_phi q^m_phi Delta_phi = d for each melonic interaction. The derivation is given twice: first via the 2PI effective action on the sphere (Section 5), and then via direct Schwinger-Dyson resummation in the conformal limit (Appendix A). The paper applies the procedure to the quartic Yukawa model, the Popovic model, a supersymmetric component model, and the large-n vector models, where it reproduces the standard 1/n anomalous dimension. It also discusses additional extrema, complex solutions, and gaps at integer dimensions.

Significance. If the central claim is fully established, the paper provides a genuinely unifying principle for a large class of exactly solvable large-N CFTs and extends the logic of F- and a-maximization to non-supersymmetric theories in continuous dimension. The 2PI derivation is structurally sound and the reproduction of the known vector-model anomalous dimension (6.6) is a valuable check. The paper also deserves credit for giving two distinct routes to the result, for tabulating concrete numerical solution spaces, and for being honest about unresolved issues such as missing solutions at integer d. The main weakness is that the completeness direction of the classification is asserted rather than rigorously proved, and the status of spurious extrema of the constrained tilde F problem is not fully clarified.

major comments (3)
  1. [Section 5.4, Eqs. (5.23)-(5.24) and Appendix A] The completeness direction of the classification is asserted rather than proved. The text states that R-dependent extrema of tilde F 'do not satisfy (5.15)', but it does not demonstrate this, and it immediately discards 'other solutions' to (5.23a). Moreover, (5.23a) also admits solutions with tilde M(m_m)=0 for positive integer m_m or with g_m=0, and these are not analysed as separate cases. Since the abstract claims a 'complete classification', the proof should supply the missing converse: for a nonzero melon, a conformal propagator solves the flat-space Schwinger-Dyson equation (5.15) only when m_m=0, and then the coupling is fixed by (A.17c). Appendix A.3 already contains the dimensional-analysis argument that achieves this, but the main text does not connect it to the R-dependence discussion in Section 5.4. Please either promote that argument into the main proof or qualify the 'complete classification' claim.
  2. [Sections 4.1, 5.4 and 7, Eqs. (4.2), (5.25), (A.17c)] The variational problem treats the Lagrange multipliers g'_m as unconstrained numbers, but for a physical melonic theory they are proportional to positive squared couplings in the conventions of (5.26). Equation (A.17c) then imposes a sign condition on the derivatives d tilde F_phi/dDelta_phi that is never checked. If the word 'precisely' in the abstract is intended as an iff statement, the paper should either prove that every IR-wedge extremum with m_m=0 is realizable with physical couplings, or explicitly state that the classification is one-way: every melonic CFT arises this way, but not every extremum of tilde F necessarily corresponds to a physical QFT vacuum. This matters because several solutions displayed in Figures 6-8 are complex or lie outside the IR wedge, and no criterion is given for selecting the physical ones.
  3. [Section 5.3, Eq. (5.18)] The derivation of the extremization equations is obscured by an apparent power-of-R inconsistency. Equation (5.18) contains (2R)^{2m_m}, while the subsequent extremization equations (5.23a,b) and the expansion (5.24) use (2R)^{-2m_m}. With the definition of tilde M in (5.21), the extremization equations do not follow from (5.18) as written. The sign of the exponent in (5.18) should be corrected, or the definition of tilde M adjusted, so that the reader can verify the variational steps explicitly.
minor comments (4)
  1. [References] Reference [59] is cited as 'Kutsakov' in Section 4; the correct spelling is 'Kutasov'.
  2. [Abstract and Section 3.2] The abstract says 'the knownF and a-maximization procedures'; an article and spacing are missing ('the known F- and a-maximization procedures').
  3. [Section 7.3, Eq. (7.6)] The sentence defining the quartic Yukawa model as 'hλprismatic' appears garbled; the notation should be defined or removed.
  4. [Section 5.2.1, Eq. (5.11)] The role of N versus M^q in the tensor-model normalization is explained in the text, but it would help to state explicitly in (5.13) that the sum over fields counts each dynamical field once and that N is the total number of colors entering the large-N counting.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the tilde F-extremization claim is derived from the 2PI effective action and Schwinger-Dyson equations, not assumed; the few self-citations are illustrative only.

full rationale

The central claim—that melonic CFTs are precisely the conformal mean field theories with constrained extremal tilde F—is proved in Section 5 from the 2PI effective action, with an independent diagrammatic derivation in Appendix A starting from the two-point Schwinger-Dyson equations. The SDE derivation postulates only the conformal ansatz for two-point functions and then derives both the dimensional constraint sum_phi q^m_phi Delta_phi = d and the extremal equations (A.17); these are consequences of the SDE holding in the conformal scaling limit, not inputs. The R-independence criterion in Section 5.4 is a physical consistency condition for a CFT on the sphere, not a hidden assumption of the answer: R-dependent extrema of the conformal slice cannot satisfy the full SDE at large R, and the appendix shows that the mm=0 solutions do satisfy the SDE. The Lagrange multipliers in (5.25) are identified with renormalized coupling constants via (5.26), an identification that follows from comparing the 2PI variations with the SDE, so no fitted parameter is renamed as a prediction. The self-citations to the authors' companion paper [27] concern example models (the quartic Yukawa model and component superfield vacua) and are not load-bearing for the theorem; the generalized free field free-energy input is quoted from [51], whose authors do not overlap with the present paper. The procedure reproduces known independent results—for example, the standard large-n O(n) anomalous dimension in (6.6) and the SYK solution—which provide external consistency checks. The claimed completeness is supported by the appendix's direct derivation from the SDE; the caveat in Section 8 item 4 about missing solutions at certain integer d is a limitation of solution existence for particular models, not a circular step. No step in the derivation chain reduces to its own input by construction.

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

The central claim rests primarily on the melonic dominance and large-N factorization assumptions that define the theory class, plus the conformal slice reduction. The R-independence criterion is the main ad hoc element. No free parameters are fitted; all quantities are solved from the extremization.

assumptions (6)
  • domain assumption Only melonic diagrams contribute to the leading large-N 2PI effective action
    Defines the class of melonic QFTs; used in Section 5.2.2 to truncate Gamma_2 to the nm complete melon integrals.
  • domain assumption Large-N factorization: the leading correlators are Gaussian, so the CFT is a mean field theory
    Used throughout (Section 5.3) to identify tilde F as a sum of generalized free field free energies and to drop N-subleading terms.
  • domain assumption The IR fixed point is conformal and its two-point functions take the unique conformal form (5.16)
    Section 5.3: without this, the functional extremization cannot be reduced to a function extremization over scaling dimensions Delta_Phi.
  • domain assumption Free propagator term C^{-1} drops out because Delta_Phi > Delta^free_Phi (IR wedge)
    Section 5.2 and 5.4; fields at or below free dimension define long-range models which are excluded.
  • ad hoc to paper R-independent extrema of the conformal slice are exactly the full extrema of the 2PI action
    Section 5.4: solutions with R dependence are asserted to be inconsistent and discarded; this is a new selection criterion specific to this paper.
  • domain assumption No symmetry breaking and no accidental IR symmetries
    Assumed in Sections 5.2 and 3.4.2; if broken, the classification changes.

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Pith. "Pith review of $F$-extremization determines certain large-$N$ CFTs." pith.science (2026). https://pith.science/paper/AMKC47NS

@misc{pith2026241210499,
  author       = {Pith},
  title        = {Pith review of: $F$-extremization determines certain large-$N$ CFTs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AMKC47NS}},
  note         = {Machine review of arXiv:2412.10499}
}
abstract

We show that the conformal data of a range of large-$N$ CFTs, the melonic CFTs, are specified by constrained extremization of the universal part of the sphere free energy $F=-\log Z_{S^d}$, called $\tilde{F}$. This family includes the generalized SYK models, the vector models (O$(N)$, Gross-Neveu, etc.), and the tensor field theories. The known $F$ and $a$-maximization procedures in SCFTs are therefore extended to these non-supersymmetric CFTs in continuous $d$. We establish our result using the two-particle irreducible (2PI) effective action, and, equivalently, by Feynman diagram resummation. $\tilde{F}$ interpolates in continuous dimension between the known $C$-functions, so we interpret this result as an extremization of the number of IR degrees of freedom, in the spirit of the generalized $c,F,a$-theorems. The outcome is a complete classification of the melonic CFTs: they are the conformal mean field theories which extremize the universal part of the sphere free energy, subject to an IR marginality condition on the interaction Lagrangian.

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Reference graph

Works this paper leans on

78 extracted references · 26 canonical work pages

  1. [1]

    Giombi and I

    S. Giombi and I. R. Klebanov,Interpolating betweena and F, arXiv:1409.1937. 1, 1, 3.2, 3.4, 3.4, 2

    arXiv 1937

  2. [2]

    S. S. Pufu,The F-Theorem and F-Maximization, arXiv:1608.02960. 1, 3.2, 3.3.1, 3.4.2, 5.3

    arXiv

  3. [3]

    Melonic CFTs

    D. Benedetti,Melonic CFTs, arXiv:2004.08616. 1, 2

    work page Pith review arXiv 2004

  4. [4]

    D. J. Gross and V. Rosenhaus,A Generalization of Sachdev-Ye-Kitaev, arXiv:1610.01569. 1, 2, 5.2.1

    arXiv

  5. [5]

    Witten,An SYK-Like Model Without Disorder, arXiv:1610.09758

    E. Witten,An SYK-Like Model Without Disorder, arXiv:1610.09758. 1, 2

    arXiv

  6. [6]

    Giombi, I

    S. Giombi, I. R. Klebanov, and G. Tarnopolsky,Bosonic tensor models at largeN and small ϵ, arXiv:1707.03866. 1, 2

    arXiv

  7. [7]

    Zinn-Justin,Quantum Field Theory and Critical Phenomena

    J. Zinn-Justin,Quantum Field Theory and Critical Phenomena. Oxford University Press, 5 ed. 1, 6.1

    work page

  8. [8]

    Maldacena and D

    J. Maldacena and D. Stanford,Comments on the Sachdev-Ye-Kitaev model, arXiv:1604.07818. 1, 2, 5.2.1

    arXiv

Show all 78 references
  1. [9]

    Rosenhaus,An introduction to the SYK model, arXiv:1807.03334

    V. Rosenhaus,An introduction to the SYK model, arXiv:1807.03334. 2

    arXiv

  2. [10]

    Murugan, D

    J. Murugan, D. Stanford, and E. Witten,More on Supersymmetric and 2d Analogs of the SYK Model, arXiv:1706.05362. 7.3

    arXiv

  3. [11]

    J. Liu, E. Perlmutter, V. Rosenhaus, and D. Simmons-Duffin,d-dimensional SYK, AdS Loops, and 6j Symbols, arXiv:1808.00612

    arXiv

  4. [12]

    Bulycheva, I

    K. Bulycheva, I. R. Klebanov, A. Milekhin, and G. Tarnopolsky,Spectra of Operators in Large N Tensor Models, arXiv:1707.09347

    arXiv

  5. [13]

    Yoon,SYK Models and SYK-like Tensor Models with Global Symmetry, arXiv:1707.01740

    J. Yoon,SYK Models and SYK-like Tensor Models with Global Symmetry, arXiv:1707.01740

    arXiv

  6. [14]

    Bulycheva,A note on the SYK model with complex fermions, arXiv:1706.07411

    K. Bulycheva,A note on the SYK model with complex fermions, arXiv:1706.07411

    arXiv

  7. [15]

    Y. Gu, A. Kitaev, S. Sachdev, and G. Tarnopolsky,Notes on the complex Sachdev-Ye-Kitaev model, arXiv:1910.14099

    1910 arXiv

  8. [16]

    Turiaci and H

    G. Turiaci and H. Verlinde,Towards a 2d QFT Analog of the SYK Model, arXiv:1701.00528

    arXiv

  9. [17]

    Marcus and S

    E. Marcus and S. Vandoren,A new class of SYK-like models with maximal chaos, arXiv:1808.01190

    arXiv

  10. [18]

    W. Fu, D. Gaiotto, J. Maldacena, and S. Sachdev,Supersymmetric SYK models, arXiv:1610.08917

    arXiv

  11. [19]

    Berkooz, P

    M. Berkooz, P. Narayan, M. Rozali, and J. Simón,Comments on the Random Thirring Model, arXiv:1702.05105

    arXiv

  12. [20]

    Chang, S

    C.-M. Chang, S. Colin-Ellerin, C. Peng, and M. Rangamani,A 3d disordered superconformal fixed point, arXiv:2108.00027

    arXiv

  13. [21]

    Chang and X

    C.-M. Chang and X. Shen,Disordered N = (2, 2) Supersymmetric Field Theories, arXiv:2307.08742

    arXiv

  14. [22]

    Biggs, J

    A. Biggs, J. Maldacena, and V. Narovlansky,A supersymmetric SYK model with a curious low energy behavior, arXiv:2309.08818. 1, 4 – 40 –

    arXiv

  15. [23]

    Choudhury, A

    S. Choudhury, A. Dey, I. Halder, L. Janagal, S. Minwalla, and R. Poojary,Notes on Melonic O(N )q−1 Tensor Models, arXiv:1707.09352. 1, 1

    arXiv

  16. [24]

    I. R. Klebanov, F. Popov, and G. Tarnopolsky,TASI Lectures on LargeN Tensor Models, arXiv:1808.09434

    arXiv

  17. [25]

    Gurau,Notes on Tensor Models and Tensor Field Theories, arXiv:1907.03531

    R. Gurau,Notes on Tensor Models and Tensor Field Theories, arXiv:1907.03531. 1, 2, 2

    1907 arXiv

  18. [26]

    S. S. Gubser, C. Jepsen, Z. Ji, and B. Trundy,Higher melonic theories, arXiv:1806.04800. 4

    arXiv

  19. [27]

    Fraser-Taliente and J

    L. Fraser-Taliente and J. Wheater,Melonic limits of the quartic Yukawa model and general features of melonic CFTs, arXiv:2410.09152. 2, 2, 4.1, 4.1, 5.2.2, 7.1, 7.3, 7.3, 5

    arXiv

  20. [28]

    Prakash and R

    S. Prakash and R. Sinha,A Complex Fermionic Tensor Model ind Dimensions, arXiv:1710.09357. 5.2.2

    arXiv

  21. [29]

    I. R. Klebanov, P. N. Pallegar, and F. K. Popov,Majorana Fermion Quantum Mechanics for Higher Rank Tensors, arXiv:1905.06264

    1905 arXiv

  22. [30]

    Benedetti, N

    D. Benedetti, N. Delporte, S. Harribey, and R. Sinha,Sextic tensor field theories in rank3 and 5, arXiv:1912.06641. 2, 2, A.2

    1912 arXiv

  23. [31]

    Chang, S

    C.-M. Chang, S. Colin-Ellerin, and M. Rangamani,On Melonic Supertensor Models, arXiv:1806.09903. 1, 2, 7.3

    arXiv

  24. [32]

    Chang, S

    C.-M. Chang, S. Colin-Ellerin, C. Peng, and M. Rangamani,Disordered vector models: From higher spins to incipient strings, arXiv:2112.09157. 1, 6.1, 6.1

    arXiv

  25. [33]

    Benedetti and N

    D. Benedetti and N. Delporte,Remarks on a melonic field theory with cubic interaction, arXiv:2012.12238. 1, 2, 9

    2012 arXiv

  26. [34]

    L. Fei, S. Giombi, I. R. Klebanov, and G. Tarnopolsky,Generalized F-Theorem and theϵ Expansion, arXiv:1507.01960. 1, 1, 2

    arXiv

  27. [35]

    A. B. Zamolodchikov,Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, . 1, 3.2

  28. [36]

    J. L. Cardy,Is There a c Theorem in Four-Dimensions?, . 1, 3.1, 5.3

  29. [37]

    Benedetti and R

    D. Benedetti and R. Gurau,2PI effective action for the SYK model and tensor field theories, arXiv:1802.05500. 1, 5, 5.8, 5.2, 5.2.1, 8, 1

    arXiv

  30. [38]

    Benedetti,The Melonic Large-N Limit in Quantum Field Theory,

    D. Benedetti,The Melonic Large-N Limit in Quantum Field Theory, . 2, 2

  31. [39]

    Kitaev and S

    A. Kitaev and S. J. Suh,The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, arXiv:1711.08467. 2, 5.2.1

    arXiv

  32. [40]

    Carrozza and V

    S. Carrozza and V. Pozsgay,SYK-like tensor quantum mechanics withSp(N ) symmetry, arXiv:1809.07753. 2

    arXiv

  33. [41]

    Harribey,Sextic tensor model in rank3 at next-to-leading order, arXiv:2109.08034

    S. Harribey,Sextic tensor model in rank3 at next-to-leading order, arXiv:2109.08034. 2

    arXiv

  34. [42]

    Giombi, I

    S. Giombi, I. R. Klebanov, F. Popov, S. Prakash, and G. Tarnopolsky,Prismatic Large N Models for Bosonic Tensors, arXiv:1808.04344. 2, 5

    arXiv

  35. [43]

    Prakash,Spectrum of a Gross-Neveu Yukawa model with flavor disorder ind = 3, arXiv:2207.13983

    S. Prakash,Spectrum of a Gross-Neveu Yukawa model with flavor disorder ind = 3, arXiv:2207.13983. 2, 4.1

    arXiv

  36. [44]

    Gurau and H

    R. Gurau and H. Keppler,Duality of Orthogonal and Symplectic Random Tensor Models, arXiv:2207.01993. 2 – 41 –

    arXiv

  37. [45]

    F. K. Popov,Supersymmetric Tensor Model at LargeN and Small ϵ, arXiv:1907.02440. 2, 7.3, 6

    1907 arXiv

  38. [46]

    Lettera and A

    D. Lettera and A. Vichi,A large-N tensor model with four supercharges, arXiv:2012.11600. 2, 7.3, 6

    2012 arXiv

  39. [47]

    D. J. Amit and D. V. I. Roginsky,Exactly soluble limit ofϕ3 field theory with internal Potts symmetry, . 2

  40. [48]

    Nador, D

    V. Nador, D. Oriti, X. Pang, A. Tanasa, and Y.-L. Wang,Generalized Amit-Roginsky model from perturbations of 3D quantum gravity, arXiv:2307.14211. 2

    arXiv

  41. [49]

    D. J. Gross and V. Rosenhaus,A line of CFTs: From generalized free fields to SYK, arXiv:1706.07015. 2, A.2

    arXiv

  42. [50]

    Benedetti, R

    D. Benedetti, R. Gurau, S. Harribey, and K. Suzuki,Long-range multi-scalar models at three loops, arXiv:2007.04603

    2007 arXiv

  43. [51]

    Benedetti, R

    D. Benedetti, R. Gurau, S. Harribey, and D. Lettera,The F-theorem in the melonic limit, arXiv:2111.11792. 3.2, 3.3, 3.11, 5.3, 5.4, 6.2, 5, 1

    arXiv

  44. [52]

    Shen,Long range SYK model and boundary SYK model, arXiv:2308.12598

    X.-Y. Shen,Long range SYK model and boundary SYK model, arXiv:2308.12598. 2, 6.1, 3, A.2

    arXiv

  45. [53]

    Berges, R

    J. Berges, R. Gurau, and T. Preis,Asymptotic freedom in a strongly interacting scalar quantum field theory in four Euclidean dimensions, arXiv:2301.09514. 2

    arXiv

  46. [54]

    Komargodski and A

    Z. Komargodski and A. Schwimmer,On Renormalization Group Flows in Four Dimensions, arXiv:1107.3987. 3.2

    arXiv

  47. [55]

    Karateev, P

    D. Karateev, P. Kravchuk, and D. Simmons-Duffin,Harmonic Analysis and Mean Field Theory, arXiv:1809.05111. 3.3, 5.3, A.1

    arXiv

  48. [56]

    Sun,AdS one-loop partition functions from bulk and edge characters, arXiv:2010.15826

    Z. Sun,AdS one-loop partition functions from bulk and edge characters, arXiv:2010.15826. 3.11, 6.2, 3

    2010 arXiv

  49. [57]

    Harribey,Renormalization in tensor field theory and the melonic fixed point, arXiv:2207.05520

    S. Harribey,Renormalization in tensor field theory and the melonic fixed point, arXiv:2207.05520. 3.11

    arXiv

  50. [58]

    J. A. Minahan,Localizing gauge theories onSd, arXiv:1512.06924. 3.4

    arXiv

  51. [59]

    a-theorem

    D. Kutasov,New Results on the "a-theorem" in Four Dimensional Supersymmetric Field Theory, hep-th/0312098. 4

    arXiv

  52. [60]

    Barnes, K

    E. Barnes, K. Intriligator, B. Wecht, and J. Wright,Evidence for the Strongest Version of the 4d a-Theorem, via a-Maximization Along RG Flows, hep-th/0408156

    arXiv

  53. [61]

    Amariti and M

    A. Amariti and M. Siani,F-maximization along the RG flows: A Proposal, arXiv:1105.3979. 4

    arXiv

  54. [62]

    Making Sense of the Legendre Transform

    R. K. P. Zia, E. F. Redish, and S. R. McKay, “Making Sense of the Legendre Transform.” 5.1

  55. [63]

    A. N. Vasil’ev, Y. M. Pis’mak, and Y. R. Khonkonen,1/n expansion: Calculation of the exponents η and ν in the order1/n2 for arbitrary number of dimensions, . 6.1

  56. [64]

    Goykhman and M

    M. Goykhman and M. Smolkin,Vector model in various dimensions, arXiv:1911.08298. 6.1

    1911 arXiv

  57. [65]

    Bekaert and M

    X. Bekaert and M. Grigoriev,Higher order singletons, partially massless fields and their boundary values in the ambient approach, arXiv:1305.0162. 6.2, 3 – 42 –

    arXiv

  58. [66]

    Brust and K

    C. Brust and K. Hinterbichler,Partially Massless Higher-Spin Theory, arXiv:1610.08510. 6.2, 3

    arXiv

  59. [67]

    Zinn-Justin,Four fermion interaction near four-dimensions,

    J. Zinn-Justin,Four fermion interaction near four-dimensions, . 6.2, 6.2

  60. [68]

    Goykhman and R

    M. Goykhman and R. Sinha,CFT data in the Gross-Neveu model, arXiv:2011.07768. 6.2

    2011 arXiv

  61. [69]

    J. A. Gracey,Large N critical exponents for the chiral Heisenberg Gross-Neveu universality class, arXiv:1801.01320. 6.2

    arXiv

  62. [70]

    J. A. Gracey,Critical exponent η at O(1/N3) in the chiral XY model using the largeN conformal bootstrap, arXiv:2101.03385. 6.2

    arXiv

  63. [71]

    Tarnopolsky,On Large N Expansion of the Sphere Free Energy, arXiv:1609.09113

    G. Tarnopolsky,On Large N Expansion of the Sphere Free Energy, arXiv:1609.09113. 6.2

    arXiv

  64. [72]

    D. S. Popović,Anomalous Dimensions in the( ¯ψ ¯ϕ)ψϕ Model Field Theory in 1/N Expansion, . 7.2

  65. [73]

    Hogervorst, S

    M. Hogervorst, S. Rychkov, and B. C. van Rees,Unitarity violation at the Wilson-Fisher fixed point in 4-epsilon dimensions, arXiv:1512.00013. 1

    arXiv

  66. [74]

    Ji and M

    Y. Ji and M. Kelly,Unitarity violation in non-integer dimensional Gross-Neveu-Yukawa model, arXiv:1802.03222. 1

    arXiv

  67. [75]

    Schaub,A Walk ThroughSpin(1, d+ 1), arXiv:2405.01659

    V. Schaub,A Walk ThroughSpin(1, d+ 1), arXiv:2405.01659. 4

    arXiv

  68. [76]

    Benedetti,Instability of complex CFTs with operators in the principal series, arXiv:2103.01813

    D. Benedetti,Instability of complex CFTs with operators in the principal series, arXiv:2103.01813. 2

    arXiv

  69. [77]

    J. Kim, I. R. Klebanov, G. Tarnopolsky, and W. Zhao,Symmetry Breaking in Coupled SYK or Tensor Models, arXiv:1902.02287. 2

    1902 arXiv

  70. [78]

    Gadde,In search of conformal theories, arXiv:1702.07362

    A. Gadde,In search of conformal theories, arXiv:1702.07362. 4 – 43 –

    arXiv

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Reviewed August 11, 2026 · model on record in the stance chip above.