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The character table of the symmetric group $S_n$, of permutations of $n$ objects, is of fundamental interest in theoretical physics, combinatorics as well as computational complexity theory. We investigate the implications of an identity, which has a geometrical interpretation in combinatorial topological field theories, relating the column sum of normalised central characters of $S_n$ to a sum of structure constants of multiplication in the centre of the group algebra of $S_n$. The identity leads to the proof that a combinatorial computation of the column sum belongs to complexity class \shP. The sum of structure constants has an interpretation in terms of the counting of branched covers of the sphere. This allows the identification of a tractable subset of the structure constants related to genus zero covers. We use this subset to prove that the column sum for a conjugacy class labelled by partition $\lambda$ is non-vanishing if and only if the permutations in the conjugacy class are even. This leads to the result that the determination of the vanishing or otherwise of the column sum is in complexity class \pP. The subset gives a positive lower bound on the column sum for any even $ \lambda$. For any disjoint decomposition of $ \lambda$ as $\lambda_1 \sqcup \lambda_2 $ we obtain a lower bound for the column sum at $ \lambda$ in terms of the product of the column sums for $ \lambda_1$ and$\lambda_2$. This can be expressed as a super-additivity property for the logarithms of column sums of normalized characters., Comment: 52 pages + Appendices, 9 Figures

$U(N)^{\otimes r} \otimes O(N)^{\otimes q}$ invariants are constructed by contractions of complex tensors of order $r+q$, also denoted $(r,q)$. These tensors transform under $r$ fundamental representations of the unitary group $U(N)$ and $q$ fundamental representations of the orthogonal group $O(N)$. Therefore, $U(N)^{\otimes r} \otimes O(N)^{\otimes q}$ invariants are tensor model observables endowed with a tensor field of order $(r,q)$. We enumerate these observables using group theoretic formulae, for arbitrary tensor fields of order $(r,q)$. Inspecting lower-order cases reveals that, at order $(1,1)$, the number of invariants corresponds to a number of 2- or 4-ary necklaces that exhibit pattern avoidance, offering insights into enumerative combinatorics. For a general order $(r,q)$, the counting can be interpreted as the partition function of a topological quantum field theory (TQFT) with the symmetric group serving as gauge group. We identify the 2-complex pertaining to the enumeration of the invariants, which in turn defines the TQFT, and establish a correspondence with countings associated with covers of diverse topologies. For $r>1$, the number of invariants matches the number of ($q$-dependent) weighted equivalence classes of branched covers of the 2-sphere with $r$ branched points. At $r=1$, the counting maps to the enumeration of branched covers of the 2-sphere with $q+3$ branched points. The formalism unveils a wide array of novel integer sequences that have not been previously documented. We also provide various codes for running computational experiments., Comment: 49 pages, 9 figures

Machine Learning approaches like clustering methods deal with massive datasets that present an increasing challenge. We devise parallel algorithms to compute the Multi-Slice Clustering (MSC) for 3rd-order tensors. The MSC method is based on spectral analysis of the tensor slices and works independently on each tensor mode. Such features fit well in the parallel paradigm via a distributed memory system. We show that our parallel scheme outperforms sequential computing and allows for the scalability of the MSC method.

Field theories with combinatorial non-local interactions such as tensor invariants are interesting candidates for describing a phase transition from discrete quantum-gravitational to continuum geometry. In the so-called cyclic-melonic potential approximation of a tensorial field theory on the $r$-dimensional torus it was recently shown using functional renormalization group techniques that no such phase transition to a condensate phase with a tentative continuum geometric interpretation is possible. Here, keeping the same approximation, we show how to overcome this limitation amending the theory by local degrees freedom on $\mathbb{R}^d$. We find that the effective $r-1$ dimensions of the torus part dynamically vanish along the renormalization group flow while the $d$ local dimensions persist up to small momentum scales. Consequently, for $d>2$ one can find a phase structure allowing also for phase transitions., Comment: 25+4 pages, 5 figures

We define the computational task of detecting projectors in finite dimensional associative algebras with a combinatorial basis, labelled by representation theory data, using combinatorial central elements in the algebra. In the first example, the projectors belong to the centre of a symmetric group algebra and are labelled by Young diagrams with a fixed number of boxes $n$. We describe a quantum algorithm for the task based on quantum phase estimation (QPE) and obtain estimates of the complexity as a function of $n$. We compare to a classical algorithm related to the projector identification problem by the AdS/CFT correspondence. This gives a concrete proof of concept for classical/quantum comparisons of the complexity of a detection task, based in holographic correspondences. A second example involves projectors labelled by triples of Young diagrams, all having $n$ boxes, with non-vanishing Kronecker coefficient. The task takes as input the projector, and consists of identifying the triple of Young diagrams. In both of the above cases the standard QPE complexities are polynomial in $n$. A third example of quantum projector detection involves projectors labelled by a triple of Young diagrams, with $m,n$ and $m+n$ boxes respectively, such that the associated Littlewood-Richardson coefficient is non-zero. The projector detection task is to identify the triple of Young diagrams associated with the projector which is given as input. This is motivated by a two-matrix model, related via the AdS/CFT correspondence, to systems of strings attached to giant gravitons. The QPE complexity in this case is polynomial in $m$ and $n$., Comment: 38 pages (including Appendices) ; 2 figures

Enhanced tensor field theories (eTFT) have dominant graphs that differ from the melonic diagrams of conventional tensor field theories. They therefore describe pertinent candidates to escape the so-called branched polymer phase, the universal geometry found for tensor models. For generic order $d$ of the tensor field, we compute the perturbative $\beta$-functions at one-loop of two just-renormalizable quartic eTFT coined by $+$ or $\times$, depending on their vertex weights. The models $+$ has two quartic coupling constants $(\lambda, \lambda_{+})$, and two 2-point couplings(mass, $Z_a$). Meanwhile, the model $\times$ has two quartic coupling constants $(\lambda, \lambda_{\times})$ and three 2-point couplings (mass, $Z_a$, $Z_{2a}$). At all orders, both models have a constant wave function renormalization: $Z=1$ and therefore no anomalous dimension. Despite such peculiar behavior, both models acquire nontrivial radiative corrections for the coupling constants. The RG flow of the model $+$ exhibits a particular asymptotic safety: $\lambda_{+}$ is marginal without corrections thus is a fixed point of arbitrary constant value. All remaining couplings determine relevant directions and get suppressed in the UV. Concerning the model $\times$, $\lambda_{\times}$ is marginal and again a fixed point (arbitrary constant value), $\lambda$, $\mu$ and $Z_a$ are all relevant couplings and flow to 0. Meanwhile $Z_{2a}$ is a marginal coupling and becomes a linear function of the time scale. This model can neither be called asymptotically safe or free., Comment: 48 pages, 20 figures, scaling dimensions corrected, some statements corrected, typos fixed

Several methods for triclustering three-dimensional data require the cluster size or the number of clusters in each dimension to be specified. To address this issue, the Multi-Slice Clustering (MSC) for 3-order tensor finds signal slices that lie in a low dimensional subspace for a rank-one tensor dataset in order to find a cluster based on the threshold similarity. We propose an extension algorithm called MSC-DBSCAN to extract the different clusters of slices that lie in the different subspaces from the data if the dataset is a sum of r rank-one tensor (r > 1). Our algorithm uses the same input as the MSC algorithm and can find the same solution for rank-one tensor data as MSC., Comment: 13 pages, improved version, typos removed, text restructured, same results

We propose a new method of multiway clustering for 3-order tensors via affinity matrix (MCAM). Based on a notion of similarity between the tensor slices and the spread of information of each slice, our model builds an affinity/similarity matrix on which we apply advanced clustering methods. The combination of all clusters of the three modes delivers the desired multiway clustering. Finally, MCAM achieves competitive results compared with other known algorithms on synthetics and real datasets.

Bi-partite ribbon graphs arise in organising the large $N$ expansion of correlators in random matrix models and in the enumeration of observables in random tensor models. There is an algebra $\mathcal{K}(n)$, with basis given by bi-partite ribbon graphs with $n$ edges, which is useful in the applications to matrix and tensor models. The algebra $\mathcal{K}(n)$ is closely related to symmetric group algebras and has a matrix-block decomposition related to Clebsch-Gordan multiplicities, also known as Kronecker coefficients, for symmetric group representations. Quantum mechanical models which use $\mathcal{K}(n)$ as Hilbert spaces can be used to give combinatorial algorithms for computing the Kronecker coefficients., Comment: 13 pages, 1 figure. References updated, typos fixed. We thank the Editors Konstantinos Anagnostopoulos, Peter Schupp, George Zoupanos, for the invitation to contribute to this special volume on "Non-commutativity and physics". arXiv admin note: text overlap with arXiv:2010.04054

Several methods of triclustering of three dimensional data require the specification of the cluster size in each dimension. This introduces a certain degree of arbitrariness. To address this issue, we propose a new method, namely the multi-slice clustering (MSC) for a 3-order tensor data set. We analyse, in each dimension or tensor mode, the spectral decomposition of each tensor slice, i.e. a matrix. Thus, we define a similarity measure between matrix slices up to a threshold (precision) parameter, and from that, identify a cluster. The intersection of all partial clusters provides the desired triclustering. The effectiveness of our algorithm is shown on both synthetic and real-world data sets.

The counting of the dimension of the space of $U(N) \times U(N) \times U(N)$ polynomial invariants of a complex $3$-index tensor as a function of degree $n$ is known in terms of a sum of squares of Kronecker coefficients. For $n \le N$, the formula can be expressed in terms of a sum of symmetry factors of partitions of $n$ denoted $Z_3(n)$. We derive the large $n$ all-orders asymptotic formula for $ Z_3(n)$ making contact with high order results previously obtained numerically. The derivation relies on the dominance in the sum, of partitions with many parts of length $1$. The dominance of other small parts in restricted partition sums leads to related asymptotic results. The result for the $3$-index tensor observables gives the large $n$ asymptotic expansion for the counting of bipartite ribbon graphs with $n$ edges, and for the dimension of the associated Kronecker permutation centralizer algebra. We explain how the different terms in the asymptotics are associated with probability distributions over ribbon graphs. The large $n$ dominance of small parts also leads to conjectured formulae for the asymptotics of invariants for general $d$-index tensors. The coefficients of $ 1/n$ in these expansions involve Stirling numbers of the second kind along with restricted partition sums., Comment: 44 pages

We define solvable quantum mechanical systems on a Hilbert space spanned by bipartite ribbon graphs with a fixed number of edges. The Hilbert space is also an associative algebra, where the product is derived from permutation group products. The existence and structure of this Hilbert space algebra has a number of consequences. The algebra product, which can be expressed in terms of integer ribbon graph reconnection coefficients, is used to define solvable Hamiltonians with eigenvalues expressed in terms of normalized characters of symmetric group elements and degeneracies given in terms of Kronecker coefficients, which are tensor product multiplicities of symmetric group representations. The square of the Kronecker coefficient for a triple of Young diagrams is shown to be equal to the dimension of a sub-lattice in the lattice of ribbon graphs. This leads to an answer to the long-standing question of a combinatoric interpretation of the Kronecker coefficients. As an avenue to explore quantum supremacy and its implications for computational complexity theory, we outline experiments to detect non-vanishing Kronecker coefficients for hypothetical quantum realizations/simulations of these quantum systems. The correspondence between ribbon graphs and Belyi maps leads to an interpretation of these quantum mechanical systems in terms of quantum membrane world-volumes interpolating between string geometries., Comment: Version 1 - 49 pages; 12 pages of Appendices; 1 figure; revision V2 - added Lemma 1 which simplifies the proof of the main theorems; revision V3 - published version

Real or complex tensor model observables, the backbone of the tensor theory space, are classical (unitary, orthogonal, symplectic) Lie group invariants. These observables represent as colored graphs, and that representation gives an handle to study their combinatorial, topological and algebraic properties. We give here an overview of the symmetric group-theoretic formulation of the enumeration of unitary and orthogonal invariant observables which turns out to bear a rich structure. From their counting formulae, one finds a correspondence with topological field theory on 2-cellular complexes that brings other interpretations of the same countings. Furthermore, tensor model observables span an algebra that turns out to be semi-simple. Dealing with complex tensors, we discuss the representation theoretic base of the algebra making explicit its Wedderburn-Artin decomposition. The real case is more subtle as a base of its Wedderburn-Artin decomposition is yet unknown., Comment: Contribution to the Corfu Summer Institute 2019, compiling 1907.04668, 1708.03524 and ,1307.6490, 25 pages, 8 figs

Group invariants are used in high energy physics to define quantum field theory interactions. In this paper, we are presenting the parallel algebraic computation of special invariants called symplectic and even focusing on one particular invariant that finds recent interest in physics. Our results will export to other invariants. The cost of performing basic computations on the multivariate polynomials involved evolves during the computation, as the polynomials get larger or with an increasing number of terms. However, in some cases, they stay small. Traditionally, high-performance software is optimized by running it on a smaller data set in order to use profiling information to set some tuning parameters. Since the (communication and computation) costs evolve during the computation, the first iterations of the computation might not be representative of the rest of the computation and this approach cannot be applied in this case. To cope with this evolution, we are presenting an approach to get performance data and tune the algorithm during the execution.

$O(N)$ invariants are the observables of real tensor models. We use regular colored graphs to represent these invariants, the valence of the vertices of the graphs relates to the tensor rank. We enumerate $O(N)$ invariants as $d$-regular graphs, using permutation group techniques. We also list their generating functions and give (software) algorithms computing their number at an arbitrary rank and an arbitrary number of vertices. As an interesting property, we reveal that the algebraic structure which organizes these invariants differs from that of the unitary invariants. The underlying topological field theory formulation of the rank $d$ counting shows that it corresponds to counting of coverings of the $d-1$ cylinders sharing the same boundary circle and with $d$ defects. At fixed rank and fixed number of vertices, an associative semi-simple algebra with dimension the number of invariants naturally emerges from the formulation. Using the representation theory of the symmetric group, we enlighten a few crucial facts: the enumeration of $O(N)$ invariants gives a sum of constrained Kronecker coefficients; there is a representation theoretic orthogonal base of the algebra that reflects its dimension; normal ordered 2-pt correlators of the Gaussian models evaluate using permutation group language, and further, via representation theory, these functions provide other representation theoretic orthogonal bases of the algebra., Comment: 44 pages, 6 figures; typos fixed, references updated

We study the Euler-Lagrange equation of the dynamical Boulatov model which is a simplicial model for 3d Euclidean quantum gravity augmented by a Laplace-Beltrami operator. We provide all its solutions on the space of left and right invariant functions that render the interaction of the model an equilateral tetrahedron. Surprisingly, for a non-linear equation of motion, the solution space forms a vector space. This space distinguishes three classes of solutions: saddle points, global and local minima of the action. Our analysis shows that there exists one parameter region of coupling constants for which the action admits degenerate global minima., Comment: closest to published version

In this paper we consider the complete momentum-independent quartic order truncation for the effective average action of a real Abelian rank 3 tensorial group field theory. This complete truncation includes non-melonic as well as double-trace interactions. In the usual functional renormalization group perspective, the inclusion of more operators that belong to the underlying theory space corresponds to an improvement of the truncation of the effective average action. We show that the inclusion of non-melonic and double-trace operators in the truncation brings subtleties. In particular, we discuss the assignment of scaling dimensions to the non-melonic sector and how the inclusion of double-trace operators considerably changes the results for critical exponents when they are not included. We argue that this is not a particular problem of the present model by comparing the results with a pure tensor model. We discuss how these issues should be investigated in future work., Comment: 12 pages, 1 figure

In this paper we introduce a simple field theoretic version of the Carrozza-Tanasa-Klebanov-Tarnopolsky (CTKT) "uncolored" holographic tensor model. It gives a more familiar interpretation to the previously abstract modes of the SYK or CTKT models in terms of momenta. We choose for the tensor propagator the usual Fermionic propagator of condensed matter, with a spherical Fermi surface, but keep the CTKT interactions. Hence our field theory can also be considered as an ordinary condensed matter model with a non-local and non-rotational invariant interaction. Using a multiscale analysis we prove that this field theory is just renormalizable to all orders of perturbation theory in the ultraviolet regime., Comment: 24 pages, 2 figures

Amplitudes of ordinary tensor models are dominated at large $N$ by the so-called melonic graph amplitudes. Enhanced tensor models extend tensor models with special scalings of their interactions which allow, in the same limit, that the sub-dominant amplitudes to be "enhanced", that is to be as dominant as the melonic ones. These models were introduced to explore new large $N$ limits and to probe different phases for tensor models. Tensor field theory is the quantum field theoretic counterpart of tensor models and enhanced tensor field theory enlarges this theory space to accommodate enhanced tensor interactions. We undertake the multi-scale renormalization analysis for two types of enhanced quartic melonic theories with rank $d$ tensor fields $\phi: (U(1)^{D})^{d} \to \mathbb{C}$ and with interactions of the form $p^{2a}\phi^4$ reminiscent of derivative couplings expressed in momentum space. Scrutinizing the degree of divergence of both theories, we identify generic conditions for their renormalizability at all orders of perturbation. For a first type of theory, we identify a 2-parameter space of just-renormalizable models for generic $(d,D)$. These models have dominant non-melonic four-point functions. Finally, by specifying the parameters, we detail the renormalization analysis of a second type of model. Lying in between just- and super-renormalizability, that model is more exotic: all four-point amplitudes are convergent, however it exhibits an infinite family of divergent two-point amplitudes., Comment: 52 pages, 12 figures

It is known that graphs cellularly embedded into surfaces are equivalent to ribbon graphs. In this work, we generalize this statement to broader classes of graphs and surfaces. Half-edge graphs extend abstract graphs and are useful in quantum field theory in physics. On the other hand, ribbon graphs with half-edges generalize ribbon graphs and appear in a different type of field theory emanating from matrix models. We then give a sense of embeddings of half-edge graphs in punctured surfaces and determine (minimal/maximal) conditions for an equivalence between these embeddings and half-edge ribbon graphs. Given some assumptions on the embedding, the geometric dual of a cellularly embedded half-edge graph is also identified., Comment: 20 pages, 15 figures

We show that the counting of observables and correlators for a 3-index tensor model are organized by the structure of a family of permutation centralizer algebras. These algebras are shown to be semi-simple and their Wedderburn-Artin decompositions into matrix blocks are given in terms of Clebsch-Gordan coefficients of symmetric groups. The matrix basis for the algebras also gives an orthogonal basis for the tensor observables which diagonalizes the Gaussian two-point functions. The centres of the algebras are associated with correlators which are expressible in terms of Kronecker coefficients (Clebsch-Gordan multiplicities of symmetric groups). The color-exchange symmetry present in the Gaussian model, as well as a large class of interacting models, is used to refine the description of the permutation centralizer algebras. This discussion is extended to a general number of colors $d$: it is used to prove the integrality of an infinite family of number sequences related to color-symmetrizations of colored graphs, and expressible in terms of symmetric group representation theory data. Generalizing a connection between matrix models and Belyi maps, correlators in Gaussian tensor models are interpreted in terms of covers of singular 2-complexes. There is an intriguing difference, between matrix and higher rank tensor models, in the computational complexity of superficially comparable correlators of observables parametrized by Young diagrams., Comment: 81 pages; 5 figures; 5 tables; references updated and added

We investigate the universality classes of rank-4 colored bipartite U(1) tensor field models near the Gaussian fixed point with the functional renormalization group. In a truncation that contains all power counting relevant and marginal operators, we find a one-dimensional UV attractor that is connected with the Gaussian fixed point. Hence this is first evidence that the model could be asymptotically safe due to a mechanism similar to the one found in the Grosse-Wulkenhaar model, whose UV behavior near the Gaussian fixed point is also described by one-dimensional attractor that contains the Gaussian fixed point. However, the cancellation mechanism that is responsible for the simultaneous vanishing of the beta functions is new to tensor models, i.e. it does not occur in vector or matrix models., Comment: Latex, 4 pages + references

Extending tensor models at the field theoretical level, tensor field theories are nonlocal quantum field theories with Feynman graphs identified with simplicial complexes. They become relevant for addressing quantum topology and geometry in any dimension and therefore form an interesting class of models for studying quantum gravity. We review the class of perturbatively renormalizable tensor field theories and some of their features., Comment: Based on the talk Tensor Models and Renormalization given at the International Congress on Mathematical Physics, ICMP2015, 27th July - 1st August, 2015, Santiago de Chile, Chile

Rank-d Tensorial Group Field Theories are quantum field theories defined on a group manifold $G^{\times d}$, which represent a non-local generalization of standard QFT, and a candidate formalism for quantum gravity, since, when endowed with appropriate data, they can be interpreted as defining a field theoretic description of the fundamental building blocks of quantum spacetime. Their renormalisation analysis is crucial both for establishing their consistency as quantum field theories, and for studying the emergence of continuum spacetime and geometry from them. In this paper, we study the renormalisation group flow of two simple classes of TGFTs, defined for the group $G=\mathbb{R}$ for arbitrary rank, both without and with gauge invariance conditions, by means of functional renormalisation group techniques. The issue of IR divergences is tackled by the definition of a proper thermodynamic limit for TGFTs. We map the phase diagram of such models, in a simple truncation, and identify both UV and IR fixed points of the RG flow. Encouragingly, for all the models we study, we find evidence for the existence of a phase transition of condensation type., Comment: 43 pages, 9 figures

A generalization of Tutte polynomial involved in the evaluation of the moments of the integrated geometric Brownian in the Ito formalism is discussed. The new combinatorial invariant depends on the order in which the sequence of contraction-deletions have been performed on the graph. Thus, this work provides a motivation for studying an order-dependent Tutte polynomial in the context of stochastic differential equations. We show that in the limit of the control parameters encoding the ordering going to zero, the multivariate Tutte-Fortuin-Kasteleyn polynomial is recovered., Comment: 17 pages, 2 figures

We study a model of Tensorial Group Field Theory (TGFT) on $\mathbb{R}^3$ from the point of view of the Functional Renormalisation Group. This is the first attempt to apply a renormalisation procedure to a TGFT model defined over a non-compact group manifold. IR divergences (with respect to the metric on $\mathbb{R}$) coming from the non-compactness of the group are regularised via compactification, and a thermodynamic limit is then taken. We identify then IR and UV fixed points of the RG flow and find strong hints of a phase transition of the TGFT system from a symmetric to a broken or condensate phase in the IR., Comment: 7 pages, 2 Figures

We introduce a tensorial group field theory endowed with weighted interaction terms of the form $p^{2a} \phi^4$. The model can be seen as a field theory over $d=3,4$ copies of $U(1)$ where formal powers of Laplacian operators, namely $\Delta^{a}$, $a>0$, act on tensorial $\phi^4$-interactions producing, after Fourier transform, $p^{2a}\phi^4$ interactions. Using multi-scale analysis, we provide a power counting theorem for this type of models. A new quantity depending on the incidence matrix between vertices and faces of Feynman graphs is invoked in the degree of divergence of amplitudes. As a result, generally, the divergence degree is enhanced compared to the divergence degree of models without weighted vertices. The subleading terms in the partition function of the $\phi^4$ tensorial models become, in some cases, the dominant ones in the $p^{2a}\phi^4$ models. Finally, we explore sufficient conditions on the parameter $a$ yielding a list of potentially super-renormalizable $p^{2a}\phi^4$ models., Comment: 15 pages, 7 figures

We set up the Functional Renormalisation Group formalism for Tensorial Group Field Theory in full generality. We then apply it to a rank-3 model over U(1) x U(1) x U(1), endowed with a linear kinetic term and nonlocal interactions. The system of FRG equations turns out to be non-autonomous in the RG flow parameter. This feature is explained by the existence of a hidden scale, the radius of the group manifold. We investigate in detail the opposite regimes of large cut-off (UV) and small cut-off (IR) of the FRG equations, where the system becomes autonomous, and we find, in both case, Gaussian and non-Gaussian fixed points. We derive and interpret the critical exponents and flow diagrams associated with these fixed points, and discuss how the UV and IR regimes are matched at finite N. Finally, we discuss the evidence for a phase transition from a symmetric phase to a broken or condensed phase, from an RG perspective, finding that this seems to exist only in the approximate regime of very large radius of the group manifold, as to be expected for systems on compact manifolds., Comment: 27 pages, 14 figures; v2: some clarifications added, minor changes, matches published version

We provide recipe theorems for the Bollob\`as and Riordan polynomial $\mathcal{R}$ defined on classes of ribbon graphs with half-edges introduced in arXiv:1310.3708[math.GT]. We also define a generalized transition polynomial $Q$ on this new category of ribbon graphs and establish a relationship between $Q$ and $\mathcal{R}$., Comment: 24 pages, 14 figures

We consider the parametric representation of the amplitudes of Abelian models in the so-called framework of rank $d$ Tensorial Group Field Theory. These models are called Abelian because their fields live on $U(1)^D$. We concentrate on the case when these models are endowed with particular kinetic terms involving a linear power in momenta. New dimensional regularization and renormalization schemes are introduced for particular models in this class: a rank 3 tensor model, an infinite tower of matrix models $\phi^{2n}$ over $U(1)$, and a matrix model over $U(1)^2$. For all divergent amplitudes, we identify a domain of meromorphicity in a strip determined by the real part of the group dimension $D$. From this point, the ordinary subtraction program is applied and leads to convergent and analytic renormalized integrals. Furthermore, we identify and study in depth the Symanzik polynomials provided by the parametric amplitudes of generic rank $d$ Abelian models. We find that these polynomials do not satisfy the ordinary Tutte's rules (contraction/deletion). By scrutinizing the "face"-structure of these polynomials, we find a generalized polynomial which turns out to be stable only under contraction., Comment: 69 pages, 35 figures

In this paper, we analyze the Bollob\'as and Riordan polynomial $\mathcal{R}$ for ribbon graphs with half-ribbons introduced in [Combinatorics, Probability and Computing 31, 507-549, 2022]. We prove the universality property of a multivariate version of $\mathcal{R}$ whereas $\mathcal{R}$ itself turns out to be universal for a subclass of ribbon graphs with half-ribbons. We also show that $\mathcal{R}$ can be defined on some equivalence classes of ribbon graphs involving half-ribbons moves and that the new polynomial is universal on these classes., Comment: 21 pages, 15 figures

In the work [Int. J. Theor. Phys. 50, 2819 (2011)], it has been proved that the radiative corrections of the 2-point function in the SU(2) Boulatov tensor model generates a relevant (in the Renormalization Group sense) contribution of the form of a Laplacian. Such a term which was missing in the initial Boulatov model action should be added in that action before discussing the renormalization analysis of this model. In this work, by linearizing the group manifold, we prove that the amplitudes associated with Feynman graphs with external legs of the colored Boulatov model over U(1)^3 endowed with a Laplacian dynamics are all convergent. We conjecture that the same feature happens for the corresponding Boulatov model over SU(2). Higher rank models are also discussed., Comment: 10 pages, 4 Figures

Lattice gauge theories of permutation groups with a simple topological action (henceforth permutation-TFTs) have recently found several applications in the combinatorics of quantum field theories (QFTs). They have been used to solve counting problems of Feynman graphs in QFTs and ribbon graphs of large $N$, often revealing inter-relations between different counting problems. In another recent development, tensor theories generalizing matrix theories have been actively developed as models of random geometry in three or more dimensions. Here, we apply permutation-TFT methods to count gauge invariants for tensor models (colored as well as non-colored), exhibiting a relationship with counting problems of branched covers of the 2-sphere, where the rank $d$ of the tensor gets related to a number of branch points. We give explicit generating functions for the relevant counting and describe algorithms for the enumeration of the invariants. As well as the classic count of Hurwitz equivalence classes of branched covers with fixed branch points, collecting these under an equivalence of permuting the branch points is relevant to the color-symmetrized tensor invariant counting. We also apply the permutation-TFT methods to obtain some formulae for correlators of the tensor model invariants., Comment: 56 pages, 14 Figures

We investigate the finite and large $N$ behaviors of independent-value O(N)-invariant matrix models. These are models defined with matrix-type fields and with no gradient term in their action. They are generically nonrenormalizable but can be handled by nonperturbative techniques. We find that the functional of any O(N) matrix trace invariant may be expressed in terms of an O(N)-invariant measure. Based on this result, we prove that, in the limit that all interaction coupling constants go to zero, any interacting theory is continuously connected to a pseudo-free theory. This theory differs radically from the familiar free theory consisting in putting the coupling constants to zero in the initial action. The proof is given for generic finite-size matrix models, whereas, in the limiting case $N\rightarrow\infty$, we succeed in showing this behavior for restricted types of actions using a particular scaling of the parameters., Comment: 19 pages

Classes of renormalizable models in the Tensorial Group Field Theory framework are investigated. The rank $d$ tensor fields are defined over $d$ copies of a group manifold $G_D=U(1)^D$ or $G_D= SU(2)^D$ with no symmetry and no gauge invariance assumed on the fields. In particular, we explore the space of renormalizable models endowed with a kinetic term corresponding to a sum of momenta of the form $p^{2a}$, $a\in ]0,1]$. This study is tailored for models equipped with Laplacian dynamics on $G_D$ (case $a=1$) but also for more exotic nonlocal models in quantum topology (case $0

The Bollob\'as-Riordan polynomial [Math. Ann. 323, 81 (2002)] is a universal polynomial invariant for ribbon graphs. We find an extension of this polynomial for a particular family of combinatorial objects, called rank 3 weakly-colored stranded graphs. Stranded graphs arise in the study of tensor models for quantum gravity in physics, and generalize graphs and ribbon graphs. We present a seven-variable polynomial invariant of these graphs, which obeys a contraction/deletion recursion relation similar to that of the Tutte and Bollob\'as-Riordan polynomials. However, it is defined on a much broader class of objects, and furthermore captures properties that are not encoded by the Tutte or Bollob\'as-Riordan polynomials., Comment: 41 pages, 32 figures; improved version (published), some statements corrected

The Bollobas-Riordan polynomial [Math. Ann. 323, 81 (2002)] extends the Tutte polynomial and its contraction/deletion rule for ordinary graphs to ribbon graphs. Given a ribbon graph $\cG$, the related polynomial should be computable from the knowledge of the terminal forms of $\cG$ namely specific induced graphs for which the contraction/deletion procedure becomes more involved. We consider some classes of terminal forms as rosette ribbon graphs with $N\ge 1$ petals and solve their associate Bollobas-Riordan polynomial. This work therefore enlarges the list of terminal forms for ribbon graphs for which the Bollobas-Riordan polynomial could be directly deduced., Comment: 18 pages, 8 figures

Enhanced quantization is an improved program for overcoming difficulties which may arise during an ordinary canonical quantization procedure. We review here how this program applies for a particle on circle., Comment: 6 pages; Contribution to the XXIXth International Colloquium on Group-Theoretical Methods in Physics, Nankai, China, August 20-26, 2012

Recently, a rank four tensor group field theory has been proved renormalizable. We provide here the key points on the renormalizability of this model and its UV asymptotic freedom., Comment: 7 pages, 3 Figures; Contribution to the XXIXth International Colloquium on Group-Theoretical Methods in Physics, Nankai, China, August 20-26, 2012

This note fills a gap in the article with title above [1]. We provide the proof of Equation (82) of Lemma 5 in [1] and thereby complete its power counting analysis with a more precise next-to-leading-order estimate., Comment: 10 pages, 5 figures, tiny changes in Lemma 4

We identify new families of renormalizable of tensor models from anterior renormalizable tensor models via a mapping capable of reducing or increasing the rank of the theory without having an effect on the renormalizability property. Mainly, a version of the rank 3 tensor model as defined in [arXiv:1201.0176 [hep-th]], the Grosse-Wulkenhaar model in 4D and 2D generate three different classes of renormalizable models. The proof of the renormalizability is fully performed for the first reduced model. The same procedure can be applied for the remaining cases. Interestingly, we find that, due to the peculiar behavior of anisotropic wave function renormalizations, the rank 3 tensor model reduced to a matrix model generates a simple super-renormalizable vector model., Comment: 22 pages, 7 figures; substantial expansion, more results

We identify the analogues of KMS state in topos theory. Topos KMS states can be viewed as classes of truth objects associated with a measure \mu^\rho (in one-to-one correspondence with an original KMS state \rho) which satisfies topos condition analogues to the ordinary KMS conditions. Topos KMS states can be defined both externally or internally according to whether the automorphism group of geometric morphisms acts externally on the topos structure, or the automorphism group is an object in the topos itself, in which case, it acts internally. We illustrate the formalism by a simple example on the Hilbert space $\Cl^3$.

We apply the quantization scheme introduced in [arXiv:1204.2870] to a particle on a circle. We find that the quantum action functional restricted to appropriate coherent states can be expressed as the classical action plus $\hbar$-corrections. This result extends the examples presented in the cited paper., Comment: 7 pages

A recent rank 4 tensor field model generating 4D simplicial manifolds has been proved to be renormalizable at all orders of perturbation theory [arXiv:1111.4997 [hep-th]]. The model is built out of $\phi^6$ ($\phi^6_{(1/2)}$), $\phi^4$ ($\phi^4_{(1)}$) interactions and an anomalous term ($\phi^4_{(2)}$). The $\beta$-functions of this model are evaluated at two and four loops. We find that the model is asymptotically free in the UV for both the main $\phi^6_{(1/2)}$ interactions whereas it is safe in the $\phi^4_{(1)}$ sector. The remaining anomalous term turns out to possess a Landau ghost., Comment: 31 pages, 31 figures; improved version

We prove that the rank 3 analogue of the tensor model defined in [arXiv:1111.4997 [hep-th]] is renormalizable at all orders of perturbation. The proof is given in the momentum space. The one-loop $\gamma$- and $\beta$-functions of the model are also determined. We find that the model with a unique coupling constant for all interactions and a unique wave function renormalization is asymptotically free in the UV., Comment: 42 pages, 14 figures; improved version, some statements corrected, more comments

The problem of building coherent states from non-normalizable fiducial states is considered. We propose a way of constructing such coherent states by regularizing the divergence of the fiducial state norm. Then, we successfully apply the formalism to particular cases involving systems with a continuous spectrum: coherent states for the free particle and for the inverted oscillator $(p^2 - x^2)$ are explicitly provided. Similar ideas can be used for other systems having non-normalizable fiducial states., Comment: 17 pages, typos corrected, references added

We prove that an integrated version of the Gurau colored tensor model supplemented with the usual Bosonic propagator on $U(1)^4$ is renormalizable to all orders in perturbation theory. The model is of the type expected for quantization of space-time in 4D Euclidean gravity and is the first example of a renormalizable model of this kind. Its vertex and propagator are four-stranded like in 4D group field theories, but without gauge averaging on the strands. Surprisingly perhaps, the model is of the $\phi^6$ rather than of the $\phi^4$ type, since two different $\phi^6$-type interactions are log-divergent, i.e. marginal in the renormalization group sense. The renormalization proof relies on a multiscale analysis. It identifies all divergent graphs through a power counting theorem. These divergent graphs have internal and external structure of a particular kind called melonic. Melonic graphs dominate the 1/N expansion of colored tensor models and generalize the planar ribbon graphs of matrix models. A new locality principle is established for this category of graphs which allows to renormalize their divergences through counterterms of the form of the bare Lagrangian interactions. The model also has an unexpected anomalous log-divergent $(\int \phi^2)^2$ term, which can be interpreted as the generation of a scalar matter field out of pure gravity., Comment: 44 pages, 11 figures, typos corrected, figures added, improved version

The ordinary formalism for classical field theory is applied to dynamical group field theories. Focusing first on a local group field theory over one copy of SU(2) and, then, on more involved nonlocal theories (colored and non colored) defined over a tensor product of the same group, we address the issue of translation and dilatation symmetries and the corresponding Noether theorem. The energy momentum tensor and dilatation current are derived and their properties identified for each case.

Ward-Takahashi identities of the colored Boulatov model are derived using a generic unitary field transformation. In a specific instance, this generic transformation turns out to be a symmetry of the interaction so that particular classes of reduced Ward-Takahashi identities for that symmetry are consequently identified and interpreted., Comment: typos corrected, more comments