Searcharxiv⌕ Search

arXiv subjects

Thomas Krajewski

Publications and source records attributed to Thomas Krajewski.

30 records · Page 2Linked to original sources
arXiv2010

Linearized Group Field Theory and Power Counting Theorems

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

hep-th↗
arXiv2010

Topological graph polynomials and quantum field theory, Part II: Mehler kernel theories

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

math-ph↗
arXiv2008

Wilsonian renormalization, differential equations and Hopf algebras

In this paper, we present an algebraic formalism inspired by Butcher's B-series in numerical analysis and the Connes-Kreimer approach to perturbative renormalization. We first define power series of non linear operators and propose several applications, among which the perturbative solution of a fixed point equation using the non linear geometric series. Then, following Polchinski, we show how perturbative renormalization works for a non linear perturbation of a linear differential equation that governs the flow of effective actions. Then, we define a general Hopf algebra of Feynman diagrams adapted to iterations of background field effective action computations. As a simple combinatorial illustration, we show how these techniques can be used to recover the universality of the Tutte polynomial and its relation to the $q$-state Potts model. As a more sophisticated example, we use ordered diagrams with decorations and external structures to solve the Polchinski's exact renormalization group equation. Finally, we work out an analogous construction for the Schwinger-Dyson equations, which yields a bijection between planar $ϕ^{3}$ diagrams and a certain class of decorated rooted trees.

hep-th↗
arXiv2008

Seesaw and noncommutative geometry

The 1-loop corrections to the seesaw mechanism in the noncommutative standard model are computed. Other consequences of the Lorentzian signature in the inner space are summarised.

hep-th↗
arXiv2003

Non-linear sigma-models in noncommutative geometry: fields with values in finite spaces

We study sigma-models on noncommutative spaces, notably on noncommutative tori. We construct instanton solutions carrying a nontrivial topological charge q and satisfying a Belavin-Polyakov bound. The moduli space of these instantons is conjectured to consists of an ordinary torus endowed with a complex structure times a projective space $CP^{q-1}$.

math.QA↗
arXiv2000

Some Properties of Non-linear $σ$-Models in Noncommutative Geometry

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

hep-th↗
arXiv2000

Renormalization of noncommutative Yang-Mills theories: A simple example

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

hep-th↗
arXiv1998

On Kreimer's Hopf algebra structure of Feynman graphs

We reinvestigate Kreimer's Hopf algebra structure of perturbative quantum field theories with a special emphasis on overlapping divergences. Kreimer first disentangles overlapping divergences into a linear combination of disjoint and nested ones and then tackles that linear combination by the Hopf algebra operations. We present a formulation where the Hopf algebra operations are directly defined on any type of divergence. We explain the precise relation to Kreimer's Hopf algebra and obtain thereby a characterization of their primitive elements.

hep-th↗
arXiv1997

Classification of Finite Spectral Triples

It is known that the spin structure on a Riemannian manifold can be extended to noncommutative geometry using the notion of a spectral triple. For finite geometries, the corresponding finite spectral triples are completely described in terms of matrices and classified using diagrams. When tensorized with the ordinary space-time geometry, finite spectral triples give rise to Yang-Mills theories with spontaneous symmetry breaking, whose characteristic features are given within the diagrammatic approach: vertices of the diagram correspond to gauge multiplets of chiral fermions and links to Yukawa couplings.

hep-th↗
arXiv1996

Towards a $Z'$ Gauge Boson in Noncommutative Geometry

We study all possible $U(1)$-extensions of the standard model (SM) in the framework of noncommutative geometry (NCG) with the algebra $\hhh\op\cc\op\cc\op M_3(\cc)$. Comparison to experimental data about the mass of a hypothetical $Z'$ gauge boson leads to the necessity of introducing at least one new family of heavy fermions.

hep-th↗