Minimizers of the dynamical Boulatov model

TL;DR

This paper analyzes the solutions of the Euler-Lagrange equation in the dynamical Boulatov model, revealing a surprisingly rich solution space with distinct classes of solutions and conditions for global minima.

Contribution

It provides a complete characterization of solutions on invariant functions and uncovers the structure of minima in the model's action.

Findings

Solution space forms a vector space despite non-linearity

Identifies three classes of solutions: saddle points, minima, and maxima

Discovers parameter regions with degenerate global minima

Abstract

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.