*Editorial* **Editorial for the Special Issue "Progress in Group Field Theory and Related Quantum Gravity Formalisms"**

### **Sylvain Carrozza 1, Steffen Gielen 2,\* and Daniele Oriti 3**


Received: 16 January 2020; Accepted: 17 January 2020; Published: 20 January 2020

**Abstract:** This editorial introduces the Special Issue "Progress in Group Field Theory and Related Quantum Gravity Formalisms" which includes a number of research and review articles covering results in the group field theory (GFT) formalism for quantum gravity and in various neighbouring areas of quantum gravity research. We give a brief overview of the basic ideas of the GFT formalism, list some of its connections to other fields, and then summarise all contributions to the Special Issue.

**Keywords:** quantum gravity; group field theory

### **1. The Group Field Theory Formalism for Quantum Gravity**

Group field theory (GFT) sits at the intersection of various formalisms within the wider field of quantum gravity [1–3]. The basic idea behind GFT is to extend the framework of random matrix and tensor models, where a sum over triangulations is generated as the perturbative expansion of a theory of matrices or tensors, by including additional group-theoretic data to be interpreted as the discrete parallel transports of a connection formulation for gravity. These are the same variables that are fundamental to the definition of loop quantum gravity and spin foam models. GFT are thus a proposal for formulating the dynamics of quantum states built out of the kinematical data of loop quantum gravity, and thus for completing and extending the loop quantisation programme.

A straightforward example of a GFT that illustrates these aspects is the Boulatov model [4] for three-dimensional Riemannian quantum gravity. This model is defined by the action

$$\begin{split} S\_{\text{Bool}}[\varphi] &= \; \frac{1}{2} \int \mathrm{d}^3 \mathcal{g} \, \varphi^2(\mathcal{g}\_1, \mathcal{g}\_2, \mathcal{g}\_3) \\ &- \frac{\lambda}{4!} \int \mathrm{d}^6 \mathcal{g} \, \mathcal{q}(\mathcal{g}\_1, \mathcal{g}\_2, \mathcal{g}\_3) \, \mathcal{q}(\mathcal{g}\_1, \mathcal{g}\_4, \mathcal{g}\_5) \, \mathcal{q}(\mathcal{g}\_2, \mathcal{g}\_5, \mathcal{g}\_6) \, \mathcal{q}(\mathcal{g}\_3, \mathcal{g}\_6, \mathcal{g}\_4), \end{split} \tag{1}$$

where the GFT field *ϕ* is a real-valued function on three copies of SU(2) with an additional permutation symmetry,

$$
\varphi: \mathrm{SU}(2)^3 \to \mathbb{R}, \quad \varphi(\mathfrak{g}\_1, \mathfrak{g}\_2, \mathfrak{g}\_3) = \varphi(\mathfrak{g}\_2, \mathfrak{g}\_3, \mathfrak{g}\_1) = \varphi(\mathfrak{g}\_3, \mathfrak{g}\_1, \mathfrak{g}\_2), \tag{2}
$$

and "gauge invariance" under the diagonal left action of the group on all the arguments of the field,

$$
\varphi(\emptyset, \emptyset, \emptyset\_2, \emptyset\_3) = \varphi(h\emptyset\_1, h\emptyset\_2, h\emptyset\_3) \quad \forall h \in \text{SL}(2) \,. \tag{3}
$$

The action consists of a quadratic "kinetic" term, with trivial propagator, and an interaction term with a somewhat unusual ("nonlocal") pairing of arguments. In fact, the group nature of the domain of the dynamical fields and such non-local pairing of arguments in the interactions (shared with matrix and tensor models) can be understood as defining properties of the formalism, the other ingredients (e.g., symmetries, choice of group, kinetic and interaction terms) being a specification of models within the general framework. If one now considers the perturbative expansion of the partition function

$$Z\_{\text{Bool}} = \int \mathcal{D}\boldsymbol{\varphi} \,\, e^{-S\_{\text{Bool}}[\boldsymbol{\varphi}]},\tag{4}$$

in powers of the coupling *λ*, due to this peculiar structure of the interaction term, the Feynman graphs arising in such an expansion are dual to three-dimensional simplicial complexes, i.e., discrete combinatorial spacetimes. Concretely, one finds

$$Z\_{\text{Bool}} = \sum\_{\Gamma} \lambda^{V(\Gamma)} \sum\_{\{j\_f\} \in \text{Irrep } f \in \Gamma} \prod\_{v \in \Gamma} (2j\_f + 1) \prod\_{v \in \Gamma} \begin{Bmatrix} j\_{\overline{v}\_1} & j\_{\overline{v}\_2} & j\_{\overline{v}\_3} \\ j\_{\overline{v}\_4} & j\_{\overline{v}\_5} & j\_{\overline{v}\_6} \end{Bmatrix},\tag{5}$$

which is a sum over graphs Γ and, for each Γ, over assignments of irreducible representations *jf* of SU(2) to each face of Γ. For each such assignment of *jf* one finds an amplitude which is a product over 'face amplitudes' (2*jf* + 1) and 'vertex amplitudes' given by a Wigner 6*j*-symbol for the six faces that meet at a vertex (the number six arising from the six group elements integrated over in the interaction).

Each graph Γ is dual to an oriented 3d simplicial complexes *C*, where each vertex *v* ∈ Γ is dual to a tetrahedron *T* ∈ *C*, and each face *f* ∈ Γ dual to a link *l* ∈ *C*. The interesting observation is now that the amplitude appearing in the expansion (5) is nothing but the Ponzano–Regge state sum [5] of the triangulated manifold *C*, multiplied with an overall weight *λ<sup>V</sup>*(Γ) ≡ *λNT*(*C*) depending on the number *NT*(*C*) of tetrahedra in *C*. The Ponzano–Regge state sum defines a discrete path integral for three-dimensional quantum gravity on a given triangulation *C*, written on a basis which is the analog of spherical harmonics; see e.g., [6] for details and a discussion of how to rigorously define such a state sum. In these variables, one obtains what is known as a spin foam model in the loop quantum gravity literature, i.e., a covariant definition of the quantum dynamics of spin networks. The same Feynman amplitudes can also be expressed directly in group variables, where they take the form of a lattice gauge theory for 3*d* BF theory (equivalent to pure 3*d* gravity with no cosmological constant). An expression in which the same amplitudes coincide with the discrete path integral for 3*d* quantum gravity in triad and connection variables can also be given.

In summary, the perturbative expansion of the Boulatov model generates a sum over discrete (simplicial) spacetimes with a discrete quantum gravity path integral assigned to each spacetime, augmented by a sum over discrete topologies:

$$Z\_{\rm Bou} = \sum\_{\mathbb{C}} \lambda^{N\_{\Gamma}(\mathbb{C})} Z\_{\rm PR}(\mathbb{C}) \,. \tag{6}$$

The Ponzano–Regge state sum defines a topological field theory which is triangulation independent for fixed topology, so that the sum over triangulations of the same 3*d* manifold merely leads to repeated factors of the same state sum appearing in this expansion. However, in models that are not topological, summing over all simplicial complexes would restore discretisation independence.

This correspondence between perturbative expansions of appropriate GFT models and discrete path integrals for quantum gravity, as well as the connection to spin foam models, extend to other cases, in particular candidate models for quantum gravity in four spacetime dimensions. Spin foam models of immediate interest for loop quantum gravity [7], for which a more detailed understanding in terms of simplicial geometry is available, can be obtained from the expansion of a GFT for a field with four arguments valued in the Lorentz group or SU(2), with additional geometricity conditions imposed on the kinetic or interaction kernels and combinatorially nonlocal interactions of *ϕ*5 type [8].

In what we have presented so far, the GFT approach appears to give simply a reformulation of known expressions for spin foam models and other discrete quantum gravity path integrals, which can be obtained by other means. However, being able to define them in terms of a quantum field theory—albeit an unusual one in that the field *ϕ* does not live on spacetime but on an abstract group manifold—provides further avenues to explore spin foam and other models, beyond studying the GFT perturbative expansion.

In particular, one can study perturbative and non-perturbative renormalization of GFTs, and look for theories that can be defined consistently at all scales, and hence become candidates for a fundamental theory. Relying on results and methods developed in the context of tensor models [9] which share the same basic combinatorial structure as GFTs, a tentative but mathematically precise GFT renormalization framework has been developed [10,11]. It has allowed us to demonstrate the perturbative renormalizability—that is, the consistency and predictivity—of simple but non-trivial 'tensorial' GFT actions, and has led to further investigations of the phase structure of GFTs at the non-perturbative level. While the precise physical interpretation of such abstract and background-independent fixed points remains to be elucidated, it is hoped that these technological advances will find suitable extensions to realistic four-dimensional GFT models of quantum gravity.

Regarding the perturbative expansion of GFT itself, tensorial GFT actions are of even broader interest because they admit a 1/*N* expansion. As in the widely studied case of matrix models for two-dimensional quantum gravity, the 1/*N* expansion allows to partially re-sum the perturbative expansion, and thereby provides crucial control over the critical regime of the theory. As a result, the study of tensorial GFT models has seen a number of interesting developments in recent years [12].

The renormalization analysis is also related to the search for a continuum limit in GFTs which can be pursued with quantum field theory methods, addressing the key open question of a continuum limit (or sum over discretizations) in spin foam models and loop quantum gravity. This continuum limit may be given by a non-perturbative phase (often suggested to be of condensate type) in which the GFT field acquires a nonvanishing expectation value, which would be where relevant continuum physics is found (in particular, the number of building blocks diverges). The possible condensate phase of GFTs has been studied with methods coming from condensed matter theory, and has been applied to the description of cosmology and black holes within GFT [13,14]. For example, the emergen<sup>t</sup> cosmological dynamics of the universe, whose microscopic description is given by a GFT model, is extracted from the condensate hydrodynamics rephrased in terms of suitable geometric observables. These effective cosmological dynamics show both the correct classical limit at large volumes and a rather generic bouncing dynamics in place of the classical big bang singularity. Moreover, under further assumptions, they match the effective dynamics found in loop quantum cosmology.

Motivated by these applications to cosmology, formal investigations of the algebraic structure of GFTs have been initiated, aiming at a more refined account of GFT condensate states, and of the condensation mechanism itself. Even more ambitiously, this research direction lays the groundwork for a reformulation and extension of thermal physics to background-independent quantum gravity [15,16].

This Special Issue consists of contributions related to the different avenues of research within the GFT program and to neighboring areas of interest. As we have made clear, loop quantum gravity, spin foam models, and more generally discrete and combinatorial approaches to quantum gravity are closely related to GFT and thus work in these fields has direct implications for GFT. Vice versa, results in the GFT formalism could be of both inspiration and direct application in other quantum gravity formalisms. Looking further afield, submissions from research fields with relevance to more specific aspects of GFT research were also encouraged; these included, for instance, fundamental cosmology, quantum information or condensed matter theory, but also mathematical and formal aspects.

### **2. Contributions to the Special Issue**

The Special Issue consists of 14 published manuscripts; ten research articles and four review articles. The research articles (listed in chronological order of publication) cover the following topics:


application of GFT to cosmology. Such bouncing cosmologies have also been seen in models of (limiting curvature) mimetic gravity, in which one modifies gravity by including a scalar field; therefore, the precise relation of mimetic gravity and the cosmological sector of quantum gravity has recently attracted interest. This paper presents a reconstruction procedure by which, starting from a given cosmological effective dynamics from quantum gravity, one can obtain a classical mimetic gravity action (given in terms of a particular function *f*(*φ*)) that reproduces this cosmological solution, in the isotropic and homogeneous sector. This might then be seen as a candidate for an effective field theory for quantum gravity approaches such as GFT. The effective field theory is then used to study anisotropies and inhomogeneities.


The Special Issue also includes four review articles, namely


**Funding:** The work of S.G. is supported by the Royal Society under a Royal Society University Research Fellowship (UF160622) and a Research Grant for Research Fellows (RGF\R1\180030). The work of D.O. is supported by the Deutsche Forschung Gemeinschaft (DFG).

**Acknowledgments:** The gues<sup>t</sup> editors would like to thank all the authors for their contributions and the reviewers for the constructive reports. Their work helped the editors to collect this Special Issue. S.C. acknowledges support from Perimeter Institute. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Economic Development, Job Creation and Trade.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the writing of the manuscript, or in the decision to publish the results.
