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Joseph Ben Geloun

Publications and source records attributed to Joseph Ben Geloun.

At least 37 records · Page 2Linked to original sources
arXiv2020

On-the-fly Optimization of Parallel Computation of Symbolic Symplectic Invariants

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.

cs.DC↗
arXiv2018

Minimizers of the dynamical Boulatov model

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.

gr-qc↗
arXiv2018

Functional Renormalization Group analysis of rank 3 tensorial group field theory: The full quartic invariant truncation

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.

hep-th↗
arXiv2017

A Renormalizable SYK-type Tensor Field Theory

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.

hep-th↗
arXiv2017

Renormalizable Enhanced Tensor Field Theory: The quartic melonic case

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 $ϕ: (U(1)^{D})^{d} \to \mathbb{C}$ and with interactions of the form $p^{2a}ϕ^4$ 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.

hep-th↗
arXiv2017

Tensor Models, Kronecker coefficients and Permutation Centralizer Algebras

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.

hep-th↗
arXiv2017

Embedding Half-Edge Graphs in Punctured Surfaces

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.

math.CO↗
arXiv2016

Nontrivial UV behavior of rank-4 tensor field models for quantum gravity

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.

gr-qc↗
arXiv2016

Renormalizable Tensor Field Theories

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.

hep-th↗
arXiv2016

Functional Renormalisation Group analysis of Tensorial Group Field Theories on $\mathbb{R}^d$

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.

hep-th↗
arXiv2015

On an ordering-dependent generalization of Tutte polynomial

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.

math-ph↗
arXiv2015

Functional Renormalisation Group analysis of a Tensorial Group Field Theory on $\mathbb{R}^3$

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.

hep-th↗
arXiv2015

A power counting theorem for a $p^{2a}ϕ^4$ tensorial group field theory

We introduce a tensorial group field theory endowed with weighted interaction terms of the form $p^{2a} ϕ^4$. The model can be seen as a field theory over $d=3,4$ copies of $U(1)$ where formal powers of Laplacian operators, namely $Δ^{a}$, $a>0$, act on tensorial $ϕ^4$-interactions producing, after Fourier transform, $p^{2a}ϕ^4$ 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 $ϕ^4$ tensorial models become, in some cases, the dominant ones in the $p^{2a}ϕ^4$ models. Finally, we explore sufficient conditions on the parameter $a$ yielding a list of potentially super-renormalizable $p^{2a}ϕ^4$ models.

hep-th↗
arXiv2015

Functional Renormalisation Group Approach for Tensorial Group Field Theory: a Rank-3 Model

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.

hep-th↗
arXiv2014

Recipe theorems for polynomial invariants on ribbon graphs with half-edges

We provide recipe theorems for the Bollobàs 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 $\mathcal{R}$.

math.CO↗
arXiv2014

Parametric Representation of Rank d Tensorial Group Field Theory: Abelian Models with Kinetic Term $\sum_{s}|p_s| + μ$

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 $ϕ^{2n}$ 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.

hep-th↗
arXiv2013

On the finite amplitudes for open graphs in Abelian dynamical colored Boulatov-Ooguri models

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.

hep-th↗