Phase transitions in TGFT: functional renormalization group in the cyclic-melonic potential approximation and equivalence to \(\mathrm{O}(N)\) models. (English) Zbl 1509.83014
Summary: In the group field theory approach to quantum gravity, continuous spacetime geometry is expected to emerge via phase transition. However, understanding the phase diagram and finding fixed points under the renormalization group flow remains a major challenge. In this work we tackle the issue for a tensorial group field theory using the functional renormalization group method. We derive the flow equation for the effective potential at any order restricting to a subclass of tensorial interactions called cyclic melonic and projecting to a constant field in group space. For a tensor field of rank \(r\) on U(1) we explicitly calculate beta functions and find equivalence with those of \(\mathrm{O}(N)\) models but with an effective dimension flowing from \( r-1\) to zero. In the \( r-1\) dimensional regime, the equivalence to \(\mathrm{O}(N)\) models is modified by a tensor specific flow of the anomalous dimension with the consequence that the Wilson-Fisher type fixed point solution has two branches. However, due to the flow to dimension zero, fixed points describing a transition between a broken and unbroken phase do not persist and we find universal symmetry restoration. To overcome this limitation, it is necessary to go beyond compact configuration space.
MSC:
| 83C45 | Quantization of the gravitational field |
| 81T32 | Matrix models and tensor models for quantum field theory |
| 81T17 | Renormalization group methods applied to problems in quantum field theory |
| 81T16 | Nonperturbative methods of renormalization applied to problems in quantum field theory |
| 83C20 | Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory |
| 83-10 | Mathematical modeling or simulation for problems pertaining to relativity and gravitational theory |
| 81-10 | Mathematical modeling or simulation for problems pertaining to quantum theory |
Keywords:
tensorial group field theory (TGFT); renormalization group; global symmetries; nonperturbative effects; models of quantum gravityReferences:
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