Functional Renormalisation Group Approach for Tensorial Group Field Theory: a Rank-3 Model

TL;DR

This paper develops a Functional Renormalisation Group framework for tensorial group field theories, analyzing a rank-3 model over U(1)^3, identifying fixed points, and exploring phase transitions related to the size of the group manifold.

Contribution

It introduces a general FRG formalism for tensorial group field theories and applies it to a specific rank-3 model, revealing fixed points and phase transition insights.

Findings

Existence of Gaussian and non-Gaussian fixed points in UV and IR regimes.

Identification of critical exponents and flow diagrams.

Evidence of phase transition only at large group manifold radius.

Abstract

We set up the Functional Renormalisation Group formalism for Tensorial Group Field Theory in full generality. We then apply it to a rank-3 model over U(1) x U(1) x U(1), endowed with a linear kinetic term and nonlocal interactions. The system of FRG equations turns out to be non-autonomous in the RG flow parameter. This feature is explained by the existence of a hidden scale, the radius of the group manifold. We investigate in detail the opposite regimes of large cut-off (UV) and small cut-off (IR) of the FRG equations, where the system becomes autonomous, and we find, in both case, Gaussian and non-Gaussian fixed points. We derive and interpret the critical exponents and flow diagrams associated with these fixed points, and discuss how the UV and IR regimes are matched at finite N. Finally, we discuss the evidence for a phase transition from a symmetric phase to a broken or condensed phase, from an RG perspective, finding that this seems to exist only in the approximate regime of very large radius of the group manifold, as to be expected for systems on compact manifolds.