Eur. Phys. J. C (2018) 78: 996
https://doi.org/10.1140/epjc/s10052-018-6483-8

Regular Article - Theoretical Physics

Minimizers of the dynamical Boulatov model

¹ LIPN, UMR CNRS 7030, Institut Galilée, Université Paris 13, Sorbonne Paris Cité, 99, Avenue Jean-Baptiste Clément, 93430, Villetaneuse, France
² International Chair in Mathematical Physics and Applications, ICMPA-UNESCO Chair, 072BP50, Cotonou, Benin
³ Max-Planck-Institute for Gravitational Physics (Albert-Einstein-Institute), Am Mühlenberg 1, 14476, Potsdam-Golm, Germany
⁴ Department of Physics, King’s College London, University of London, Strand, London, WC2R 2LS, UK

^* e-mail: andreas.pithis@kcl.ac.uk

Received: 23 July 2018
Accepted: 28 November 2018
Published online: 7 December 2018

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.