The Ponzano-Regge model and parametric representation

TL;DR

This paper explores the effects of $$-deformation on the Ponzano-Regge model, introducing a parametric representation and analyzing how correction terms influence the motivic properties of associated graph hypersurfaces.

Contribution

It provides a new parametric representation of the $$-deformed Ponzano-Regge model and investigates the impact of correction terms on the motivic nature of graph hypersurfaces.

Findings

The correction term significantly alters the motivic properties of hypersurfaces.

The hypersurface of the tetrahedron graph is not polynomially countable.

The presence of $$-corrections prevents the hypersurface from fitting integer-coefficient polynomials.

Abstract

We give a parametric representation of the effective noncommutative field theory derived from a $\kappa$-deformation of the Ponzano-Regge model and define a generalized Kirchhoff polynomial with $\kappa$-correction terms, obtained in a $\kappa$-linear approximation. We then consider the corresponding graph hypersurfaces and the question of how the presence of the correction term affects their motivic nature. We look in particular at the tetrahedron graph, which is the basic case of relevance to quantum gravity. With the help of computer calculations, we verify that the number of points over finite fields of the corresponding hypersurface does not fit polynomials with integer coefficients, hence the hypersurface of the tetrahedron is not polynomially countable. This shows that the correction term can change significantly the motivic properties of the hypersurfaces, with respect to the classical case.