(No) phase transition in tensorial group field theory

Andreas G.A Pithis,Johannes Thürigen

doi:10.1016/j.physletb.2021.136215

Abstract

Continuum spacetime is expected to emerge via phase transition in discrete approaches to quantum gravity. A promising example is tensorial group field theory but its phase diagram remains an open issue. The results of recent attempts in terms of the functional renormalization group method remain inconclusive since they are restricted to truncations of low order. We overcome this barrier with a local-potential approximation for U(1) tensor fields at arbitrary rank r focusing on a specific class of so-called cyclic-melonic interactions. Projecting onto constant field configurations, we obtain the full set of renormalization-group flow equations. At large cut-offs we find equivalence with r−1 dimensional O(N) scalar field theory in the large-N limit, modified by a tensor-specific, relatively large anomalous dimension. However, on small length scales there is equivalence with the corresponding scalar field theory with vanishing dimension and, thus, no phase transition. This is confirmed by numerical analysis of the full non-autonomous equations where we always find symmetry restoration. The essential reason for this effect is isolated zero modes. This result should therefore be true for tensor field theories on any compact domain and including any tensor-invariant interactions. Thus, group field theories with non-compact degrees of freedom will be necessary to describe a phase transition to continuum spacetime.

Highlights

In various approaches to a quantum theory of gravity, continuum spacetime is expected to emerge from a microscopic theory of discrete geometries in terms of a phase transition [1–4]
Like matrix models relate to two-dimensional gravity [7], such a theory with tensor fields of rank r generates discrete r-dimensional geometries which converge to continuum geometries at criticality
Important first insights have been gained for tensorial group field theory (TGFT) using the method of the functional renormalization group (FRG) [8–11], first applied to discrete geometry systems in Ref. [12]

Summary

INTRODUCTION

In various approaches to a quantum theory of gravity, continuum spacetime is expected to emerge from a microscopic theory of discrete geometries in terms of a phase transition [1–4]. The renormalization group is a useful tool to understand how physical theories evolve along scales and allows to chart their phase diagrams In this way, important first insights have been gained for TGFT using the method of the functional renormalization group (FRG) [8–11], first applied to discrete geometry systems in Ref. Various insights in the phase structure of TGFT have been gained which are based on truncations of the theory space to low order [13–20] or apply to the autonomous UV limit [21]. In these works one typically finds nonGaussian fixed points but due to the named limitations these results need further verification. We argue that the isolated zero mode due to the compactness of the field domain is the essential reason for symmetry restoration and conjecture that this phenomenon applies to any compact domain and the full theory space including arbitrary tensor-invariant interactions

FUNCTIONAL RENORMALIZATION GROUP IN THE MELONIC POTENTIAL APPROXIMATION

VANISHING DIMENSION AT SMALL SCALES AND SYMMETRY RESTORATION

CONCLUSIONS