Phase transitions with spontaneous symmetry breaking are expected for group field theories as a basic feature of the geometogenesis scenario. The following paper aims to investigate the equilibrium phase for group field theory by using the ergodic hypothesis on which the Gibbs-Boltzmann distributions must break down. The breaking of the ergodicity can be considered dynamically by introducing a fictitious “time” inducing a stochastic process described through a Langevin equation, from which the randomness of the tensor field will be a consequence. This type of equation is considered particularly for complex just-renormalizable Abelian model of rank $d=5$, and we study some of their properties by using a renormalization group considering a “coarse graining” both in time and space.

• Received 3 October 2022
• Accepted 24 March 2023

DOI:https://doi.org/10.1103/PhysRevD.107.086009