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

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. I. THE CASE FOR UNIVERSAL, BACKGROUND-INDEPENDENT QUANTUM GRAVITYTo describe physically relevant spacetimes, such as black holes or Friedmann-Lemaitre-Robertson-Walker spacetimes, it is necessary to go beyond General Relativity, where these spacetimes feature singularities. Such singularities are expected to be resolved once quantum fluctuations of spacetime are properly accounted for. Yet, this is a particularly challenging task if it is to be compatible with the background-independence at the heart of our modern understanding of gravity. Backgroundindependence implies that no configuration of spacetime should be singled out a priori from all configurations that enter the path integral. This is incompatible with perturbative techniques around a fixed spacetime, which single out a special background to perturb around, and which do not provide a predictive quantum field theory of gravity. Instead, one is led to introduce an infinite number of independent local counterterms to cancel divergences at each loop order [1][2][3]. This motivates that background independence should be taken seriously in the search for an ultraviolet complete definition of the gravitational path integral. Thus, we aim at an implementation of the path integral without auxiliary geometric background structures 1 . A promising route to construct a background independent path integral consists of making a transition to discrete building blocks, as in dynamical triangulations [5], Regge calculus [6], matrix/tensor models [7-9], spin foams [10] and causal sets [11]. This allows to construct a discrete approximation of all random geometries (and potentially additional configurations with no interpreta- * eichhorn@cp3.sdu.dk † j.lumma@thphys.uni-heidelberg.de ‡ adpjunior@id.uff.br § a.sikandar@thphys.uni-heidelberg.de 1 An alternative route makes use of an auxiliary background structure at the technical level while ensuring the independence of physical results from this background structure, see, e.g., [4].tion as a spacetime geometry) that enter the path integral for quantum gravity. These discrete building blocks are typically not viewed as physical, "fundamental" building blocks of spacetime. Rather, they are auxiliary, unphysical entities, allowing to define a regularized path integral in analogy to lattice gauge theories for non-gravitational quantum field theories. One might object that quantum spacetime might be fundamentally discrete, wh...

The phase diagram of the multi-matrix model with ABAB interaction from functional renormalization

“…Secondly, its generalization to higher dimensions is closely connected to a model that is being explored in the context of the AdS/CFT conjecture [6,7]. Thirdly, in matrix models a tentative connection to an asymptotically safe fixed point in the vicinity of two dimensions has been found [8] and conjectured in higher dimensions [9]. Fourth, this class of models provides a combinatorial approach to dynamical triangulations, complementing computer simulations of the latter [10][11][12].…”

“…However, apart from the case in d = 2 [1], their continuum limit so far only leads to geometrically degenerate configurations [2,3], the same way as in EDT, or planar ones [101]. Recent results indicate the possibility for non-trivial universality classes [9,15]; however, the geometric properties of the corresponding phases have not been investigated yet. If a universal continuum limit within a phase with desirable geometric properties can be taken, asymptotic safety can be confirmed in a background-independent fashion and with straightforward access to the scaling exponents.…”

“…The application of the FRG in the discrete quantum gravity context facilitates a backgroundindependent form of coarse graining where the number of degrees of freedom serves as a scale for a Renormalization Group flow. This program was started by analyzing matrix models for 2d Euclidean quantum gravity in [8,13] and has by now been extended to tensor models for Euclidean quantum gravity in 3 and 4 dimensions [9,14,15,109], see [16] for a review. Related developments for non-commutative geometry [110,111] and tensorial group field theories [18][19][20][21][22][23][24][25][26][27][28][29][30] exist.…”

“…This rich set of connections motivates a joint study of matrix and tensor models and in particular the search for a universal continuum limit in these. Functional Renormalization Group techniques have been developed for these models [8,13,14] and are being applied in [9,[15][16][17] and related tensorial group field theories [18][19][20][21][22][23][24][25][26][27][28][29][30], with the potential to discover a universal continuum limit beyond perturbation theory, see also [31,32] for related developments with the Polchinski equation.…”

Critical Reflections on Asymptotically Safe Gravity

“…FRG tools which interpret the tensor size N as an appropriate notion of "pre-geometric" (i.e., background-independent) coarse-graining scale [257], allow to recover the well-known continuum limit in two-dimensional quantum gravity within systematic uncertainties related to truncations [258]. First tentative hints for universal critical behavior in models with 3-and 4-dimensional building blocks have been found [259,260]. The importance of symmetry-identities has been emphasized in [261].…”