Universal critical behavior in tensor models for four-dimensional quantum gravity

Astrid Eichhorn,Arslan Sikandar,Johannes Lumma,Antonio D Pereira

doi:10.1007/jhep02(2020)110

Abstract

Four-dimensional random geometries can be generated by statistical models with rank-4 tensors as random variables. These are dual to discrete building blocks of random geometries. We discover a potential candidate for a continuum limit in such a model by employing background-independent coarse-graining techniques where the tensor size serves as a pre-geometric notion of scale. A fixed point candidate which features two relevant directions is found. The possible relevance of this result in view of universal results for quantum gravity and a potential connection to the asymptotic-safety program is discussed.

Highlights

We argue that robust physical predictions should be extracted from the gravitational path integral in a similar manner
The interactions of the rank-4-tensor model are dual to building blocks of four-dimensional space, whereas the tensors themselves are dual to building blocks of three-dimensional space
We provide first hints for such continuum limits for real rank-4 models by discovering a potential universality class that is new from a tensormodel point of view but appears to be not incompatible — within the respective systematic uncertainties — with results for the Reuter universality class in quantum gravity

Summary

Tensor models as a framework for background-independent quantum gravity

One potential route to evaluate the effect of quantum gravitational fluctuations in a background-independent setting is provided by the tensor-model approach [15,16,17,18,19,20,21,22,23]. In the Feynman diagram expansion of the tensor model, these interactions are glued together along propagators, forming triangulations of four-dimensional space In this way, tensor models encode the discretized configurations of spacetime that enter the path-integral in a combinatorial way. In quantum-gravity models based on regularizations in terms of discrete building blocks, this implies the independence of the continuum limit from the choice of building blocks (at least within certain classes, defined, e.g., by the emergent symmetries). As in [37], in [56], a certain type of large N critical behavior of rank-3-models has been found to agree (within the estimated systematic errors) with that of matrix models for two-dimensional quantum gravity through a form of dimensional reduction in which the universal continuum limit becomes independent of the microscopic dimensionality of the building blocks. As the key result of our paper we will find hints for universal critical behavior in tensor models that is not incompatible with the Reuter universality class (given the systematic uncertainties on both sides)

Functional renormalization group techniques for tensor models

The model

Interacting fixed point