Combinatorial decompositions for deformed or decorated classes of maps

Abstract

The perturbative expansion of tensorial field theories in Feynman graphs can be interpreted as weighted generating series of some piecewise linear varieties. This simple fact establishes a link between two a priori distinct fields: the combinatorics of discrete manifolds on one hand and tensorial field theories on the other hand. In this thesis, we study different aspects revolving around this connection between combinatorics and field theory. First, we consider constellations model, which generalize maps and their algebraic properties. This makes them suited to probe the b-deformation, a deformation of the algebra of symmetric functions. We will study the constraints satisfied by the generating series of cubical b-deformed constellations. Second, we analyze the double scaling limit of particular tensor models of order 3. For tensor of order greater than two, the nature of the 1/N-expansion is qualitatively different from the matrix case of order 2. In particular, only the leading order graphs are fully characterized. Despite this fact, it is possible to identify graphs of subleading orders contributing to the double scaling limit by implementing the scheme decomposition for Feynman graphs of these theories. An analysis of the singularity of the schemes then allows us to give a complete characterization of the graphs contributing to the double scaling limit. Finally, we investigate a particular link between a tensor and a vector field theory which both admit a melonic limit. Namely, we will show that we can obtain the vectorial Amit-Roginski model by considering perturbations around a classical solution of the Boulatov model, a tensorial theory. We give sufficient conditions on the classical solution so that the effective action for the perturbation around this solution takes the form of the Amit-Roginski action.