We introduce a linearized version of group field theory. It can be viewed either as theory over the additive vector space or an asymptotic expansion any around unit element. prove exact power-counting theorems for graph such models. For colored models power counting amplitude is further computed in terms homology graph. An theorem also established particular class graphs nonlinearized models, which satisfy planarity condition. Examples and connections with previous results are discussed.

Using standard field theoretical techniques, we survey pure Yang–Mills theory on the noncommutative torus, including Feynman rules and BRS symmetry. Although in general free of any infrared singularity, is ultraviolet divergent. Because an invariant regularization scheme, this turns out to be renormalizable detailed computation one-loop counterterms given, leading asymptotically theory. Besides, it that nonplanar diagrams are overall convergent when θ irrational.

We investigate the relationship between universal topological polynomials for graphs in mathematics and parametric representation of Feynman amplitudes quantum field theory. In this first article we consider translation invariant theories with usual heat-kernel-based propagator. show how Symanzik theory are particular multivariate versions Tutte polynomial, new noncommutative special Bollobás–Riordan polynomials.

We investigate the group field theory formulation of Engle-Pereira-Rovelli-Livine/Freidel-Krasnov (EPRL/FK spin-foam models. These models aim at a dynamical, i.e., nontopological 4D quantum gravity. introduce saddle point method for general amplitudes and compare it with existing results, in particular, second order correction to EPRL/FK propagator.

We consider a simple modification of the amplitude defining dynamics loop quantum gravity, corresponding to introduction cosmological constant, and possibly related $SL(2,\mathbb{C}{)}_{q}$ extension theory recently considered by Fairbairn-Meusburger Han. show that, in context spinfoam cosmology, this yields de Sitter solution.

We study the effect of non-Gaussian average over random couplings in a complex version celebrated Sachdev-Ye-Kitaev (SYK) model. Using Polchinski-like equation and tensor Gaussian universality, we show that this averaging leads to modification variance distribution at leading order $N$. then derive form effective action all orders. An explicit computation case quartic perturbation is performed for both SYK model mentioned above generalization proposed D. Gross V. Rosenhaus [J. High Energy...

Group field theories are particular quantum defined on D copies of a group which reproduce spin foam amplitudes space-time dimension D. In these lecture notes, we present the general construction theories, merging ideas from tensor models and loop gravity.This is organized as follows.In first section, basic aspects theory matrix models.The second section devoted to in last examine properties formulation BF EPRL model.We conclude with few possible research topics, like continuum limit based...

We define a new topological polynomial extending the Bollobas-Riordan one, which obeys four-term reduction relation of deletion/contraction type and has natural behavior under partial duality. This allows to write down completely explicit combinatorial evaluation polynomials, occurring in parametric representation non-commutative Grosse-Wulkenhaar quantum field theory. An solution for commutative theories based on Mehler kernel is also provided.

We define in this paper several Hopf algebras describing the combinatorics of so-called multi-scale renormalization quantum field theory. After a brief recall main mathematical features renormalization, we assigned graphs, that are graphs with appropriate decorations for framework. then on these and Gallavotti-Nicolò trees, particular class trees encoding supplementary informations graphs. Several morphisms between combinatorial Connes-Kreimer algebra given. Finally, scale dependent...

The generating function of the cumulants in random matrix models, as well themselves, can be expanded asymptotic (divergent) series indexed by maps. While at fixed genus sums over maps converge, genera do not. In this paper we obtain alternative expansions both for and that cure problem. We provide explicit convergent cumulants, remainders their perturbative expansion (in size maps) topological maps). show any cumulant is an analytic inside a cardioid domain complex plane prove Borel summable origin.

In this paper, we review some general formulations of exact renormalisation group equations and loop for tensor models tensorial field theories.We illustrate the use these in derivation leading order expectation values observables models.Furthermore, to establish a suitable scaling dimension interactions Abelian theories with closure constraint.We also present analogues models.

We prove by explicit calculation that Feynman graphs in noncommutative Yang-Mills theory made of repeated insertions into itself arbitrarily many one-loop ghost propagator corrections are renormalizable local counterterms. This provides a strong support for the renormalizability conjecture model.

A new path integral representation of Lorentzian Engle–Pereira–Rovelli–Livine spinfoam model is derived by employing the theory unitary . The taken as a starting point semiclassical analysis. relation between and classical simplicial geometry studied via large-spin asymptotic expansion amplitude with all spins uniformly large. More precisely, in regime, there an equivalence critical configuration (with certain nondegeneracy assumption) geometry. Such allows us to classify configurations...

In S. Giombi, I. Klebanov, F. Popov, Prakash and G. Tarnopolsky, {\it Phys. Rev.} {\bf D} 98 (2018) 10, 105005, a prismatic tensor model was introduced. We study here the diagrammatics double scaling limit of this model, using intermediate field method. explicitly exhibit next-to-leading order Feynman graphs in $1/N$ expansion. Using appropriate combinatorial tools, we further general term expansion compute $2-$point function limit.

We introduce non-linear $\sigma$-models in the framework of noncommutative geometry with special emphasis on models defined torus. choose as target spaces two point space and circle illustrate some characteristic features corresponding $\sigma$-models. In particular we construct a $\sigma$-model instanton topological charge equal to 1. also define investigate properties analogue Wess-Zumino-Witten model.

Abstract We derive an exact renormalization group equation in the context of (colored) field theories. This describes variation effective action as some modes fields are integrated out. From a combinatorial point view, can be expressed using boundary triangulation and corresponding identifies its simplexes, image In theory, terms parametrized by spin networks, while theory Feynman graphs correspond to foams. provides formulation theories that only involves graphs.