A generalization of the Virasoro algebra to arbitrary dimensions

Razvan Gurau

REVIEW

A generalization of the Virasoro algebra to arbitrary dimensions

Razvan Gurau

High Energy Physics - Theory · Submitted 30 May 2011 · 60 references

Pith's verdict: The tree-indexed algebra closure is real and checkable; the 'Virasoro in arbitrary dimensions' framing outruns what the paper actually demonstrates.

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CitationRazvan Gurau. "A generalization of the Virasoro algebra to arbitrary dimensions." arXiv:1105.6072 (2011). https://pith.science/paper/VBONXVE6

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Editor's note

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Worth your time if you work on tensor models or GFT, but read it for Theorem 1, not for the title. The genuine content is the algebra: operators L_T indexed by colored rooted D-ary trees, defined in eq (31), restricted to class functions of the tree couplings, and shown in Theorem 1 to close under commutation with the structure constants being sums over gluing vertices. The proof runs through Lemmas 1-5 and is laid out in enough detail that you can check it on a page of scratch paper. That is a real, self-contained piece of mathematics and it does not depend on the physics derivation that follows. Credit where due: the generic-interaction colored model in Section IV.C is also a sensible construction, and the observation that the melonic-bubble operators are invariant under the tree equivalence relation is a nice touch that makes the class-function restriction natural rather than ad hoc. The paper is also candid about what it does not have — Section V says outright that it is unclear whether the algebra is new or a relabeling of something known, and that closure fails at subleading orders. Those are acknowledged, so I won't hold them against it as hidden problems. Two soft spots. First, the step from the exact SDE (65) to the leading-order constraint (66) rests on the bound in eq (61) for the non-melonic terms, which is argued from the degree expansion and Lemma 7 rather than computed graph by graph. The stress-test flagged this and I agree it's the load-bearing gap, but I'd add that it's the standard rigor level in this literature and the argument that those graphs are non-melonic is convincing on its face. Second, and softer than the reader's write-up suggests: the D=2 reduction to the Virasoro generators L'_m is asserted in one sentence in Section II, not shown. Since that identification is what the title promises, it deserves an explicit computation, even a short appendix. As it stands the honest description is 'a tree-indexed Lie algebra that reduces to Virasoro at D=2', not 'Virasoro in arbitrary dimensions'. Who should read it: anyone in the tensor-model or GFT community, and anyone curious whether the loop-equation machinery survives above two dimensions. It does not yet give critical exponents or a continuum limit, and it doesn't claim to. Send it to referees — the algebra result is worth a careful check, and the D=2 identification should be demanded explicitly.

Referee report

Major objections (2)

[Section II, sentence following Eq. (7)] The title and abstract claim a 'generalization of the Virasoro algebra to arbitrary dimensions', and the only concrete evidence offered for the 'Virasoro' identification is the sentence after Eq. (7): 'If one specializes the operators we define below for D=2 (and takes into account the cyclicity of the trace), one obtains the operators L'_m.' This is asserted, not shown. Please give the explicit D=2 computation, including how the factor 2N in the linear term of L'_m and the 1/N^2 in the quadratic term arise from the tree-indexed operators of Eq. (31), and what role the cyclicity of the trace plays. If the reduction holds only up to a relabeling or rescaling of the generators, that should be stated, since it bears directly on the title claim.
[Section IV.D, Eq. (61) and the paragraph following Eq. (60)] The central physical claim, Eq. (66), follows from the exact SDE (65) only if the non-melonic terms collected in '⟨...⟩' are suppressed relative to N^D Z. The paper bounds them by Eq. (61), (1/Z)⟨...⟩ ≤ N^{D-2/(D-1)!}, but the contributing graph classes are never enumerated: the text identifies them verbally (a color-D line joining the root melon to a distinct melon; a melonic bubble joined to a non-melonic bubble) and asserts they cannot be melonic. Since this is the load-bearing step for the whole constraint analysis, please either enumerate the contributing classes and show ω(G) ≥ 1 for each, or cite the specific result in [22-24] that establishes the bound for these correlation functions. The argument that the reconnected line cannot be a tree line of a melonic graph (end of Section IV.B) should be made explicit rather than left to the reader.

Minor objections (10)

[Eq. (5)] The second trace factor is written Tr[(MM†)^k]^{n-k}; it should be Tr[(MM†)^{n-k}].
[Eq. (6) and the following sentence] The sum in L'_m is written Σ_{k=1}^{n-1} with index n, but the operator is L'_m; the summation index should be tied to m.
[Section IV.D, sentence following Eq. (60)] The text states 'the scaling of Tr_T[T̄,T] is N^{D-1}', whereas Eq. (54) gives ⟨Tr_B⟩ ~ N^{1+2/(D-2)!ω(B)} (i.e. N for melonic B). Please reconcile this bookkeeping, since the bound (61) depends on it.
[Section V and Section IV.C] 'we expects' should be 'we expect', and 'an uni que' should be 'a unique'.
[PACS numbers] 05.40-a should presumably be 05.40.-a.
[Whole manuscript] The version I received has broken spacing throughout ('dimen sions', 'T hey', 'l eading', etc.); this may be an extraction artifact, but please check the final typeset version.
[Eq. (32)] The bracket is the antisymmetrization of the grafting product (T2,T1) ↦ Σ_{V∈T2} T2⋆_V T1, which is the defining operation of a pre-Lie (Vinberg) algebra. Making this explicit would give a concrete lead on the open question in Section V about whether the algebra is new or a relabeling of a known structure.
[Section V] The text states that Eq. (67) 'completely defines the free energy at leading order'; a system of first-order PDEs determines F_∞ only up to integration constants or boundary conditions, so this should be softened or the relevant boundary data specified.
[Eq. (31) and Eq. (67)] The notation t_T and t_{[T]} is used somewhat interchangeably; please state once and for all that the operators act on class functions and use one notation consistently.
[Figures 1-3] The vertex labels are hard to read in the version I received; please ensure the final figures are legible at print size.

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The paper

The paper claims

In two dimensions the loop equations of a random matrix model are the positive half of the Virasoro algebra, which is the bridge from matrix models to conformal field theory and continuum 2D gravity. Gurau asks whether an analogous bridge,

The reading

In two dimensions the loop equations of a random matrix model are not a mere list of identities: they are the positive half of the Virasoro algebra, and that is the bridge from matrix models to conformal field theory and to continuum 2D gravity. Gurau asks whether anything comparable survives when the matrix is replaced by a rank-$D$ tensor, the bookkeeping device for random triangulations in $D$ dimensions. His answer is yes at leading order in the large-$N$ expansion. After promoting the tensor model to one with an independent coupling for every melonic interaction, the leading Schwinger–Dyson equations become a family of differential operators indexed by colored rooted $D$-ary trees, and those operators close under commutation. If the construction holds, the partition function of a tensor model in any dimension obeys an infinite set of algebraic constraints standing in the same relation to the model as the Virasoro constraints do in $D=2$.

What carries the argument

The load-bearing object is the grafting operation $star_V$ on colored rooted $D$-ary trees: cut the branches at a vertex $V$ of one tree, insert the second tree there, and reglue the branches at the maximal vertices of the appropriate color. This operation is what makes the algebra work, because it is associative in the right sense ($(T star_V T_1) star_V T_2 = T star_V (T_1 star_{( )} T_2)$ and the two orders of grafting at distinct vertices commute), and because the equivalence $T sim T'$ generated by moving a branch of color $i$ past the maximal vertex of color $i$ leaves the tensor-network operator $Tr_T[bar T,T]$ invariant. The operator $L_{T_1}$ combines a derivative term with one derivative per non-empty branch of the root of $T_1$ together with a sum over all trees $T$ and all vertices $V in T$ weighted by $t_T$ and differentiated with respect to $t_{T star_V T_1}$. The proof of closure is a bookkeeping exercise in which the cross terms cancel by the associativity lemmas and the surviving terms reassemble into the stated commutator; the final cancellation uses that the test function depends only on class variables, which is an explicit restriction the author imposes on the domain of the operators.

Extended reading notes

Core claim

The paper does three things in sequence. First it writes down a colored tensor model with generic interaction: instead of a single coupling, every melonic bubble — equivalently every colored rooted $D$-ary tree $T$ — gets its own coupling $t_T$, mirroring the step from a one-coupling matrix model to $V(MM^dagger)=sum_j t_j (MM^dagger)^j$. Second, it derives the Schwinger–Dyson equations of this model at leading order in $1/N$ and shows they collapse to the statement $L_{T_1} Z = 0$ for every tree $T_1$, where $L_{T_1}$ is a differential operator built from the branches of $T_1$ and from all ways of grafting $T_1$ onto another tree at a vertex. Third, it proves (Theorem 1) that these operators, restricted to functions of the class variables $t_{[T]}$, obey $[L_{T_2},L_{T_1}] = sum_{V in T_2} L_{T_2 star_V T_1} - sum_{V in T_1} L_{T_1 star_V T_2}$ — a Lie algebra whose index set is trees rather than integers. Translating the constraints into the large-$N$ free energy gives equation (67), a closed differential equation for $F_infty(t_{[T]})$; the operator $L_{()}$ is identified with the generator of dilations.

Load-bearing premise

The claim that the tree-indexed constraints actually hold for the partition function rests on the scaling estimate in equation (61): any graph containing a non-melonic insertion, or a line joining two distinct melons, is asserted to scale at most as $N^{D-2/(D-1)!}$ and therefore to drop out at leading order. That bound is argued from the degree expansion and Lemma 7 rather than checked graph by graph, and the placeholder terms in equation (65) are never enumerated.

Editorial extensions

If this is right

Equation (67) gives, in any dimension, a closed differential equation for the leading-order free energy $F_infty(t_{[T]})$ — the author states it completely defines that free energy, which makes it a better starting point than the perturbative series for locating multicritical points.
Because the model now carries one coupling per melonic bubble, the multiple interaction terms that matrix models needed to produce multicritical points become available in $D ge 3$.
The dilation generator is one of the algebra's elements ($L_{()}$), so the continuum theory the constraints encode is expected to be scale invariant.
The tree-indexed algebra supplies a concrete candidate symmetry of the large-$N$ tensor model, in the same structural role that the Virasoro constraints play for 2D gravity.

Reading between the lines

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

Structurally the bracket looks like a many-body version of the Virasoro bracket: where $[L_m,L_n]=(m-n)L_{m+n}$ has a coefficient counting index differences, here the structure constants count the vertices at which one tree can be grafted into another. Whether a central extension exists — which the paper explicitly leaves open — is the sharp test of whether this is genuinely Virasoro-like or a dif
The author notes that closure holds at leading order but fails at all orders, and that the full SDEs would require tensor-network operators for non-melonic bubbles. That suggests the leading-order algebra is the analogue of the single-trace loop algebra, and the full algebra would need the analogue of multi-trace operators.
If equation (67) does determine $F_infty$ uniquely, the natural next experiment is to solve it at low order in $D=3$ and compare the critical exponents against the known one-coupling melonic results; agreement would be evidence that the tree couplings are the right coordinates for the multicritical landscape.

Original abstract

Colored tensor models generalize matrix models in higher dimensions. They admit a 1/N expansion dominated by spherical topologies and exhibit a critical behavior strongly reminiscent of matrix models. In this paper we generalize the colored tensor models to colored models with generic interaction, derive the Schwinger Dyson equations in the large N limit and analyze the associated algebra of constraints satisfied at leading order by the partition function. We show that the constraints form a Lie algebra (indexed by trees) yielding a generalization of the Virasoro algebra in arbitrary dimensions.

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.65 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Colored tensor models generalize matrix models in higher dimensions. They admit a 1/N expansion dominated by spherical topologies and exhibit a critical behavior strongly reminiscent of matrix models. In this paper we generalize the colored tensor models to colored models with generic interaction, derive the Schwinger Dyson equations in the large N limit and analyze the associated algebra of constraints satisfied at leading order by the partition function. We show that the constraints form a Lie algebra (indexed by trees) yielding a generalization of the Virasoro algebra in arbitrary dimensions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Colored tensor models generalize matrix models in higher dimensions.	background 1.00	0.05	0.02	asserted 1.00
s1	They admit a 1/N expansion dominated by spherical topologies and exhibit a critical behavior strongly reminiscent of matrix models.	background 1.00	0.14	0.03	asserted 1.00
s2	In this paper we generalize the colored tensor models to colored models with generic interaction, derive the Schwinger Dyson equations in the large N limit and analyze the associated algebra of constraints satisfied at leading order by the partition function.	contribution 0.99	0.97	0.10	asserted 0.93
s3	We show that the constraints form a Lie algebra (indexed by trees) yielding a generalization of the Virasoro algebra in arbitrary dimensions.	result 0.97	0.95	0.90	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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sources read

arXiv listing abstract v1

not examined

full text
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relations to other papers

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P abstract

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S sections
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C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 2.** FIG. 2. Gluing of two trees at a vertex

**Figure 3.** FIG. 3. Two equivalent trees

**Figure 7.** FIG. 7. A melon graph and its associated colored GN rooted tree.

Figures reproduced from arXiv:1105.6072 by the authors.

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A Renormalizable 4-Dimensional Tensor Field Theory
hep-th 2011-11 conditional novelty 8.0 of 10

A rank-4 tensor field theory on U(1)^4 with a standard bosonic propagator and melonic phi^6-type interactions is proved renormalizable to all orders by multiscale analysis and a power-counting theorem.
Universality for Random Tensors
math.PR 2011-11 accept novelty 8.0 of 10

Random tensors of rank D at least 3 with trace-invariant, properly uniformly bounded cumulants converge in distribution to a Gaussian tensor model in the large N limit.
Double Scaling in Tensor Models with a Quartic Interaction
hep-th 2013-07 conditional novelty 7.0 of 10

For a quartically interacting rank-D tensor model with D<6, the subleading 1/N contributions are dominated by 'cherry tree' graphs, which resum to a stable double scaling limit with upper critical dimension six.
The 1/N Expansion of Tensor Models Beyond Perturbation Theory
math-ph 2013-04 accept novelty 7.0 of 10

The 1/N expansion of quartic tensor models is established beyond perturbation theory, with explicit series and bounded rest terms, and the models become Gaussian in the large-N limit.
Renormalization of an SU(2) Tensorial Group Field Theory in Three Dimensions
hep-th 2013-03 conditional novelty 7.0 of 10

The SU(2) tensorial group field theory in three dimensions with tensor-invariant interactions up to sixth order is just renormalizable, with all divergences melonic and absorbable into a finite set of counterterms.

Show 18 more citing papers

Renormalization of Tensorial Group Field Theories: Abelian U(1) Models in Four Dimensions
hep-th 2012-07 conditional novelty 7.0 of 10

Four-dimensional Abelian tensorial group field theories with gauge invariance conditions are proven super-renormalizable for any polynomial interaction, with divergences confined to melonic tadpoles removable by a new...
Functional Renormalisation Group Approach for Tensorial Group Field Theory: a Rank-3 Model
hep-th 2014-11 conditional novelty 6.0 of 10

The first functional renormalisation group treatment of tensorial group field theory yields explicit beta functions for a rank-3 model, a non-autonomous flow traced to the group-manifold radius, and Gaussian plus non-...
Discrete Renormalization Group for SU(2) Tensorial Group Field Theory
hep-th 2014-07 conditional novelty 6.0 of 10

Within a truncated four-coupling continuous flow, the rank-3 SU(2) tensorial group field theory is not asymptotically free for positive marginal couplings, contrary to prior expectation.
Tensorial methods and renormalization in Group Field Theories
hep-th 2013-10 conditional novelty 6.0 of 10

Group Field Theories with tensorial interactions and a closure constraint admit a consistent multiscale renormalization analysis, and a 3d SU(2) model is shown just-renormalizable at the perturbative level.
Renormalizable Models in Rank $d\geq 2$ Tensorial Group Field Theory
hep-th 2013-06 accept novelty 6.0 of 10

Generalizing multi-scale power counting to a one-parameter family of tensor field theories, the paper classifies which are renormalizable and finds the rank d at least 3 ones are asymptotically free in the UV.
The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders
hep-th 2012-03 conditional novelty 6.0 of 10

The Schwinger-Dyson equations of random tensor models close a Lie algebra of constraints at all orders in 1/N, with the leading-order tree algebra sitting inside it as a subalgebra.
Random tensor models in the large N limit: Uncoloring the colored tensor models
hep-th 2012-02 accept novelty 6.0 of 10

The 1/N expansion, melonic dominance, and continuum limit of colored tensor models extend to the most general single complex tensor model, with Virasoro constraints and multicritical exponents gamma_m = 1 - 1/m.
3D Tensor Field Theory: Renormalization and One-loop $\beta$-functions
hep-th 2011-12 accept novelty 6.0 of 10

The rank-3 tensor model over U(1)^3 is renormalizable at all orders, and its symmetric-coupling version is asymptotically free in the UV at one loop.
Functional Renormalisation Group analysis of Tensorial Group Field Theories on $\mathbb{R}^d$
hep-th 2016-01 conditional novelty 5.0 of 10

Truncated FRG flows of rank-d TGFTs on R, with and without gauge invariance, show asymptotic freedom in the UV and non-Gaussian IR fixed points consistent with a condensation-type phase transition.
Counting Tensor Model Observables and Branched Covers of the 2-Sphere
hep-th 2013-07 accept novelty 5.0 of 10

Counting gauge-invariant contractions of rank-d tensor fields equals counting degree-n branched covers of the 2-sphere with d branch points, with generating functions and code provided for the enumeration.
Melons are branched polymers
math-ph 2013-02 conditional novelty 5.0 of 10

Melonic graphs, the leading-order graphs of tensor models, have Hausdorff dimension 2 and spectral dimension 4/3, matching branched polymers.
Two and four-loop $\beta$-functions of rank 4 renormalizable tensor field theories
hep-th 2012-05 conditional novelty 5.0 of 10

The rank-4 tensor field model of arXiv:1111.4997 is asymptotically free in both phi-6 sectors, safe in the phi-4(1) sector, and has a Landau ghost in the anomalous phi-4(2) sector.
Quantum Gravity and Renormalization: The Tensor Track
hep-th 2011-12 conditional novelty 4.0 of 10

The paper proposes quantizing gravity as a renormalizable quantum field theory of large random tensors, with a phase transition (geometrogenesis) converting a pre-geometric tensor phase into ordinary spacetime.
Invitation to Random Tensors
hep-th 2016-09 accept novelty 2.0 of 10

Random tensors extend random matrix models to higher dimensions, and a 1/N expansion indexed by a new invariant called the degree turned a stalled program into a working theory of random D-dimensional spaces.
Flowing in Group Field Theory Space: a Review
gr-qc 2016-03 accept novelty 2.0 of 10

Tensor-invariant bubble interactions supply group field theories with enough locality to renormalize, yielding a classified table of power-counting renormalizable models and hints of Wilson-Fisher-like fixed points.
Group Field Theory and Loop Quantum Gravity
gr-qc 2014-08 accept novelty 2.0 of 10

Group field theory reformulates loop quantum gravity as a quantum field theory of spin network vertices, giving spin foam models a generating functional and a route to continuum and cosmological limits.
The Spin Foam Approach to Quantum Gravity
gr-qc 2012-05 accept novelty 2.0 of 10

The review presents the spin foam quantization of gravity as a constrained BF path integral, focusing on the EPRL and Freidel-Krasnov models and on 2+1 gravity as a solvable guide.
Colored Tensor Models - a Review
hep-th 2011-09 accept novelty 2.0 of 10

Colored tensor models provide a controlled large-N framework for random geometry in D at least 3, with melonic graphs dominating the leading order and exhibiting a critical continuum limit.

How later papers used it

67 later papers cite it

Evidence: substantial. 53 of the citing papers share no author with it, and 6 of those carry a confident label from Pith's citation classifier.

Pith read 23 of the 111 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

3 later papers by other authors are listed below: 2 built on it and 1 used its method or tool.

Builds on it 2 of 3 papers with this label

Stated a lemma that is a direct generalization of Lemma 7 in this paper.

A New Large N Expansion for General Matrix-Tensor Models (2017) · 0.97 · paraphrased: the licence does not allow quoting
Stated a lemma that is a direct generalization of Lemma 7 in this paper.

New Limits for Large $N$ Matrix and Tensor Models: Large $D$, Melons and Applications (2019) · 0.97 · paraphrased: the licence does not allow quoting

Uses its method or tool 1 paper

Used the same procedure of this paper, which mainly integrates the partition function with respect to all colors save one.

3D Tensor Field Theory: Renormalization and One-loop $\beta$-functions (2012) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

66 arXiv papers cite this in 110 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 79 sentences without a confident label

Builds on it 3 papers

The algebra of constraints identified in [ref] and completed in this paper at all order constitutes the first step towards applying tensor models to the study of conformal field theories and quantum gravity in arbitrary dimensions.

The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders · Introduction · 0.83
Finally, we shall need the following crucial lemma, which is a direct generalization of Lemma 7 in [ref] .

A New Large N Expansion for General Matrix-Tensor Models · Degree · 0.97
Finally, we shall need the following crucial lemma, which is a direct generalization of Lemma 7 in [ref] .

New Limits for Large $N$ Matrix and Tensor Models: Large $D$, Melons and Applications · Jackets and degree · 0.97

Uses its method or tool 1 paper

As in [ref] , one attributes a different coupling constant to different tensor network operators and simply rewrites (omitting the color index $0$) S^, 0 = ∑_ℬ λ_ℬ/ Sym(ℬ) Tr_ℬ [φ̅φ].

3D Tensor Field Theory: Renormalization and One-loop $\beta$-functions · Rank 3 tensor model over U(1) · 0.97

Compares with it 1 paper

Their derivation is independent from existent Schwinger-Dyson equations for tensor models (e.g. those obtained by Gurău [ref] or Krajewski and Toriumi [ref] ), which, crucially, differ from our SDE in that those SDE reported before are algebraic.

Correlation functions of $\mathrm{U}(N)$-tensor models and their Schwinger-Dyson equations · tocIntroductiontocIntroduction · 1.00

Cites it as support 2 papers

More precisely, integrating out all coloured tensors except for one in a simplicial coloured tensor model gives precisely a model containing a single coloured tensor with bubble interactions, each weighted by the same coupling [ref] .

On 3-Dimensional Quantum Gravity and Quasi-Local Holography in Spin Foam Models and Group Field Theory · Why do we use a Coloured Model? · 0.97
Importantly, our results show that melonic interactions drive the critical behavior of models characterized by different types of interactions (including melonic ones), which is again consistent with the literature [ref] .

Phase transitions in tensorial group field theories: Landau-Ginzburg analysis of models with both local and non-local degrees of freedom · Discussion and conclusion · 0.96

Cites it as background 20 papers

Even more important perhaps, actions for uncolored random tensor theories were developed [ref] .

A Renormalizable 4-Dimensional Tensor Field Theory · Introduction · 0.81
A detailed presentation of these equations can be found in [ref] and a focus on the melonic sub–algebra in [ref] .

The double scaling limit of random tensor models · The Schwinger–Dyson equations · 0.97
Most of the lemmas we present in subsections [ref] and [ref] can be found in [ref] .

Universality for Random Tensors · Random Tensors · 0.93

What it cites · 41 sentences without a confident label

Cites as support 1 paper

It can be shown [ref] that, for $D≥3$, a leading order two point graph must possess a $D$-bubble with exactly two vertices.

Critical behavior of colored tensor models in the large N limit · Leading order graphs · 0.98

Cites as background 27 papers

The independent identically distributed (i.i.d.) colored tensor model in dimension $D$ [ref] is defined by the partition function [equation] $∑_ n$ denotes the sum over all indices $n_ij$ from $1$ to $N$.

Colored Group Field Theory · The i.i.d. colored tensor models with one coupling · 0.89
The independent identically distributed (i.i.d.) colored tensor model in dimension $D$ [ref] is defined by the partition function [equation] $∑_ n$ denotes the sum over all indices $n_ij$ from $1$ to $N$.

Topological Graph Polynomials in Colored Group Field Theory · The i.i.d. colored tensor models with one coupling · 0.89
The independent identically distributed (i.i.d.) colored tensor model in dimension $D$ [ref] is defined by the partition function [equation] $∑_ n$ denotes the sum over all indices $n_ij$ from $1$ to $N$.

The complete 1/N expansion of colored tensor models in arbitrary dimension · The i.i.d. colored tensor models with one coupling · 0.89

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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