Flowing in Group Field Theory Space: a Review

Flowing in Group Field Theory Space: a Review\star\star$\star$This paper is a
contribution to the Special Issue on Tensor Models, Formalism and
Applications. The full collection is available at
http://www.emis.de/journals/SIGMA/Tensor_Models.html

Sylvain CARROZZA

S. Carrozza

Université Bordeaux, LaBRI, UMR 5800, 33400 Talence, France
<EMAIL_ADDRESS>
Received March 08, 2016, in final form July 13, 2016; Published online July
16, 2016

We provide a non-technical overview of recent extensions of renormalization
methods and techniques to Group Field Theories (GFTs), a class of
combinatorially non-local quantum field theories which generalize matrix
models to dimension $d\geq 3$. More precisely, we focus on GFTs with so-called
closure constraint, which are closely related to lattice gauge theories and
quantum gravity spin foam models. With the help of recent tensor model tools,
a rich landscape of renormalizable theories has been unravelled. We review our
current understanding of their renormalization group flows, at both
perturbative and non-perturbative levels.

group field theory; quantum gravity; quantum field theory; renormalization

81T15; 81T16; 83D27; 83C45

##  1 Introduction

From a mathematical perspective, a GFT [9, 43, 57, 70, 71] is a quantum field
theory defined on $d$ copies of a compact Lie group $G$, in which point-like
interactions are replaced by non-trivial combinatorial objects. At the level
of the field theory action, this translates into peculiar non-localities of
the interactions, which are given by pairwise identifications and integrations
of individual group variables of the elementary fields. One could thus say
that a given GFT interaction is local in each pair of copies of the group $G$
thus identified, but non-local from the point of view of the full
configuration space $G^{d}$. This leads to subtleties in the construction and
analysis of such models, which for a long time precluded the definition of
renormalizable theories.

Alternatively, GFTs can be seen as generalized matrix and tensor models, with
group-valued rather than discrete indices. Progress in tensor models (see,
e.g., [25, 26, 53, 55] and references therein) can therefore be directly
imported into the GFT formalism. The continuous structure of the group allows
to considerably enrich this purely combinatorial background and define more
general classes of tensorial theories, which go under the name of Tensorial
Group Field Theories (TGFTs) or simply Tensorial Field Theories (TFTs). The
introduction of derivative couplings, and in particular of non-trivial
propagators, leads to a first class of such models [10, 11, 12, 15, 16, 18,
20, 21, 82, 83]. They are proper field theories whose (perturbative)
definitions already require a non-trivial renormalizability analysis. Since
the group plays a limited role and enjoys no particular physical
interpretation in such models, they are preferably referred to as TFTs.
Second, the group structure can be used to impose particular constraints on
the elementary fields, which endow Feynman amplitudes with the structure of
generalized lattice gauge theories. This latter class of models is usually
referred to as Tensorial Group Field Theories (TGFTs) to emphasize the central
importance of the group111Note however that the distinction between TFTs and
TGFTs is somehow lose, as it relies more on questions of intent and
interpretation than on a mathematically precise definition. These terms are
therefore sometimes used interchangeably in the literature. In the present
review, we will also refer to TFTs as _TGFTs without gaunge invariance_. (both
technically and for the physical interpretation of the amplitudes), or simply
group field theories whenever more general interactions than tensor invariants
are allowed (see Section 2 below).

GFTs were originally introduced in the context of Loop Quantum Gravity (LQG)
[37, 75] for the purpose of resumming Spin Foam amplitudes222Spin foam
amplitudes fall in the class of generalized lattice gauge theories which may
be generated by a GFT. [1, 74] and hence completing the definition of the
dynamics of LQG [80, 85]. While LQG aims at solving the quantum gravity
conundrum through a mere quantization of General Relativity (GR), it naturally
leads to quantum state spaces of geometry in which discrete structures such as
graphs [2] or triangulations [39] are center stage. Such structures percolate
to the dynamical level and lead to interesting quantizations of discretized GR
[8, 40, 41, 45], but the question of whether or not a smooth space-time
structure can be recovered in some limit remains a great challenge in this
approach. In order to address this question, it is in particular crucial to
develop new renormalization tools, which should allow to efficiently explore
the phase space of spin foam models. Two renormalization programmes have
recently emerged to meet this challenge. One is based on an interpretation of
spin foam models as direct space regularizations of quantum gravity [4, 38],
and therefore explores generalizations of lattice renormalization techniques.
The other one interprets spin foam amplitudes as Feynman contributions in the
perturbative expansion of a specific GFT [44, 69, 78], and therefore requires
generalizations of local field theory renormalization techniques to non-local
field theories such as (T)GFTs. We here review this second research programme,
in the particular context of TGFTs with _gauge invariance condition_ (or
equivalently _closure constraint_), which have been extensively studied in the
literature [17, 22, 30, 31, 32, 35, 36, 60, 61, 62, 63, 64, 81, 84]. Though
not as sophisticated as tentative GFT models for $4d$ quantum gravity [7, 8],
such theories generate non-trivial spin foam amplitudes and require key
generalizations of ordinary renormalization methods. They therefore provide a
natural test bed for future and more challenging studies of quantum gravity
models.

Our goal is to present the reader with a bird’s eye view on the recent
literature, and to clearly explain the motivations and the status of the
subject. We will as much as possible refrain from delving into technical
details. Given the rapid development of the subject, we will also not claim to
be exhaustive. The choice of topics to be developed in the main text was at
least to some extent a matter of personal taste.

The plan of this review is as follows. In Section 2 we motivate further the
GFT renormalization programme, as well as the specific class of models which
has been investigated up to now. Perturbative renormalizability is reviewed in
detail in Section 3. A full classification of renormalizable model based on
rigorous power-counting arguments is in particular provided. We then go on to
investigations of the properties of the renormalization group flows of these
models in Section 4. Emphasis is put on functional renormalization methods,
which are of great practical interest even though they have so far only been
applied to GFTs in the crudest truncations.

##  2 From simplicial to tensorial GFTs

###  2.1 Renormalization of GFTs: motivations and basic ingredients

We begin with the introduction of basic GFT structures, which may not be
general enough to encompass all models relevant to full $4d$ quantum gravity,
but will be sufficient in the present context. We define a GFT as a quantum
field theory for a single complex scalar field $\varphi$ leaving on $d$ copies
of a fixed compact Lie group $G$. Unless specified otherwise, we will use a
vector notation for configuration space variables and its Haar measure

$\displaystyle{\bf g}=(g_{1},\ldots,g_{d})\in G^{d},\qquad\mathrm{d}{\bf
g}=\mathrm{d}g_{1}\cdots\mathrm{d}g_{d}.$

We will also use the short-hand notation:

$\displaystyle\overline{\varphi}_{1}\cdot\varphi_{2}=\int\mathrm{d}{\bf
g}\overline{\varphi}_{1}({\bf g})\varphi_{2}({\bf g}),$

for any two square-integrable functions $\varphi_{1}$ and $\varphi_{2}$ on
$G^{d}$. The dynamics of the GFT field is specified by a probability measure

$\displaystyle\mathrm{d}\mu_{C_{\Lambda}}(\varphi,\overline{\varphi})\exp\left(-S_{\Lambda}[\varphi,\overline{\varphi}]\right)$
(2.1)

or equivalently a generating functional

$\displaystyle{\mathcal{Z}}_{\Lambda}[J,\bar{J}]=\int\mathrm{d}\mu_{C_{\Lambda}}(\varphi,\overline{\varphi})\exp\big{(}{-}S_{\Lambda}[\varphi,\overline{\varphi}]+\bar{J}\cdot\varphi+\overline{\varphi}\cdot
J\big{)}.$ (2.2)

The measure $\mathrm{d}\mu_{C_{\Lambda}}$ is the Gaussian measure associated
to the covariance $C_{\Lambda}$, which is a positive operator with kernel:

$\displaystyle C_{\Lambda}({\bf g};{\bf
g}^{\prime})=\int\mathrm{d}\mu_{C_{\Lambda}}(\varphi,\overline{\varphi})\overline{\varphi}({\bf
g})\varphi({\bf g}^{\prime}).$

In other words, $C_{\Lambda}$ is the propagator of the GFT, or equivalently
its free $2$-point function. We explicitly introduced an extra regularization
or scale parameter $\Lambda>0$, as the covariance is in general plagued with
divergences. The role of this parameter is central in our renormalization
programme; we will describe it in greater details once a specific choice of
propagator is made. Perturbations around the Gaussian theory are introduced
through the (interaction part of the) GFT action
$S_{\Lambda}[\varphi,\overline{\varphi}]$, which we will parametrize as

$\displaystyle
S_{\Lambda}[\varphi,\overline{\varphi}]=\sum_{b\in{\mathcal{B}}}t_{b}(\Lambda)I_{b}[\varphi,\overline{\varphi}],$

where ${\mathcal{B}}$ is a set of elementary interactions, $I_{b}$ is the
specific monomial in the fields associated to the interaction $b$, and
$t_{b}(\Lambda)$ is the running coupling constant associated to $b$ at scale
$\Lambda$. One main objective of renormalization is to determine how these
coupling constants should be adjusted to compensate for a change in the cut-
off $\Lambda$. More precisely, as in ordinary field theory, we will require
that the infrared content of the connected $2k$-point functions (or Schwinger
functions)

$\displaystyle\mathsf{S}^{(2k)}_{\Lambda}({\bf g}_{1},\ldots,{\bf g}_{k},{\bf
g}_{1}^{\prime},\ldots,{\bf
g}_{k}^{\prime})=\left.\left(\prod_{i=1}^{k}\frac{\delta}{\delta\bar{J}({\bf
g}_{i})}\frac{\delta}{\delta J({\bf
g}_{i}^{\prime})}\right)\ln{\mathcal{Z}}_{\Lambda}[J,\bar{J}]\right|_{J=\bar{J}=0},$

which also characterize the random measure (2.1), is invariant under a change
of $\Lambda$. Unlike ordinary field theories, we do not have any prior notion
of energy scale which we can rely on to determine what ’infrared’ means in the
GFT context. We will therefore need to adopt a more abstract notion of scale,
a priori unrelated to familiar space-time concepts.

For specific choices of group, propagator and basic interactions, the formal
Feynman expansion of (for instance) the partition function

$\displaystyle{\mathcal{Z}}_{\Lambda}:={\mathcal{Z}}_{\Lambda}[0,0]=\sum_{\mathcal{G}}\prod_{b\in{\mathcal{B}}}\left(-t_{b}(\Lambda)\right)^{n_{b}({\mathcal{G}})}{\mathcal{A}}_{{\mathcal{G}}}(\Lambda),$
(2.3)

generates Feynman diagrams ${\mathcal{G}}$ which are in one-to-one
correspondence with specific (closed) $2$-complexes and are weighted by spin
foam amplitudes ${\mathcal{A}}_{\mathcal{G}}$333Note that symmetry factors may
have to be included in formula (2.3), depending on the detailed definition of
the model and of the Feynman amplitudes. They are not relevant to the present
discussion.. We remind the reader that, more generally, spin foams are
combinatorial objects interpolating between spin-network boundary states,
whose amplitudes are constructed as discrete gravity path-integrals taking the
form of generalized lattice gauge theory amplitudes. They are therefore
interpreted as quantum space-time processes encoding the dynamics of loop
quantum gravity spin-network functionals. In this review, we ignore the extra
combinatorial structure provided by properly closed boundary spin-network
states, since they do not play an essential role in what we want to discuss:
the renormalization of such quantum gravity amplitudes is a particular case of
that of general $n$-point functions (associated to $n$ open boundary spin-
network vertices), we can therefore focus entirely on the latter. In view of
equation (2.3), the GFT formalism provides natural prescriptions for resumming
infinite classes of spin foam amplitudes, with weights parametrized by the GFT
coupling constants. This is the sense in which GFTs allow to complete the
definition of spin foam models, which cannot be claimed to fully specify a
dynamics for quantum gravity unless an extra organization principle for its
amplitudes is clearly spelled out. The summing prescription implemented
through GFT is one such possible organization principle, which effectively
removes a large class of discretization ambiguities entering the definition of
spin foam models (but not all). What remains to be checked is: 1) whether this
formal procedure is mathematically consistent; and 2) whether it is physically
relevant, in the sense that general relativity can be recovered in some limit.
Renormalization will presumably play an important role in order to meet both
of these challenges.

The formal relation between GFTs and spin foam models being an intrinsically
perturbative statement, checking its validity and consistency is essentially
equivalent to proving renormalizability of the GFT. At the very least one
needs to check that – again formally – the set of GFT interactions $I_{b}$ is
stable under a shift of the cut-off $\Lambda$. This translates into a formal
stability of the set of spin foams summed over on the right-hand side of
equation (2.3). But turning this rather vague statement into a sensible
perturbative definition requires that the same stability holds with only
finitely many GFT interactions turned on, and that is equivalent to the
perturbative renormalizability of the GFT. The relevant GFT interactions will
then uniquely determine which (finitely many) elementary spin foam interaction
vertices dominate in this perturbative phase. This is a more concrete and more
rigorous illustration of how the GFT formalism may be powerfully used to
remove spin foam discretization ambiguities and make predictions.

Furthermore, assuming that general relativity may only be recovered in a phase
in which macroscopically large spin-network boundary states acquire large
amplitudes, addressing the second open problem will presumably require to go
beyond perturbation theory. This suggests that the GFT phase space should be
more systematically explored away from its perturbative regime, and the
existence of non-trivial fixed points of the renormalization group
investigated. It is important to realize that such non-trivial fixed points
would correspond to non-perturbative resummations of spin-foam amplitudes and
would therefore be very hard to grasp without recourse to GFT. They will
generate new vacua, supporting new and possibly inequivalent representations
of the GFT, and therefore leading to new GFT phases and phase transitions. The
mechanism of Bose–Einstein condensation has in particular been investigated in
this context, leading to interesting reconstructions of smooth homogeneous and
spherically symmetric space-time geometries from the GFT formalism, with
fascinating applications to cosmology [47, 48, 73] and black holes [72]. If
such a scenario based on a collective reorganization of the spin foam
amplitudes is correct, the field theory language provided by GFT and the
powerful effective methods it entails seems hardly avoidable.

We now briefly outline some basic features of GFT model-building, the
interested reader is referred to reviews on the subject [9, 43, 57, 70, 71]
and references therein for further details. The specific GFTs which generate
quantum gravity spin foam amplitudes (for instance [8, 13, 58]) require $G$ to
be a local symmetry group of space-time (or space), e.g., the Lorentz group
${\rm SO}(1,d-1)$ or its universal covering. In this review, we will only
consider Euclidean groups – in particular ${\rm SU}(2)$ in dimension $d=3$ –
and ignore complications arising from the Lorentzian signature444The
renormalization of GFTs with Lorentzian signature remains largely open and
will likely become an active field of research in the close future. See
however [76] for a first attempt in this direction.. Another important
ingredient is the so-called _gauge invariance condition_ , defined as a global
symmetry of the GFT field under simultaneous translation of its group
variables:

$\displaystyle\forall\,h\in
G,\qquad\varphi(g_{1}h,\ldots,g_{d}h)=\varphi(g_{1},\ldots,g_{d}).$ (2.4)

This condition is common to all known proposals of GFT models for quantum
gravity, and is responsible for the generalized lattice gauge theory form of
the amplitudes appearing on the right-hand side of equation (2.3). Within the
general GFT formalism spelled out at the beginning of this section, we
implement this symmetry by requiring that the covariance $C_{\Lambda}$ is of
the form

$\displaystyle C_{\Lambda}={\mathcal{P}}\tilde{C}_{\Lambda}{\mathcal{P}},$

where $\tilde{C}_{\Lambda}$ is again a positive operator, and ${\mathcal{P}}$
is the projector on translation invariant GFT fields with kernel

$\displaystyle{\mathcal{P}}({\bf g},{\bf
g}^{\prime})=\int_{G}\mathrm{d}h\prod_{\ell=1}^{d}\delta\big{(}g_{\ell}hg_{\ell}^{\prime-1}\big{)}.$

Hence $C_{\Lambda}$ is degenerate and its image lies within the space of
fields verifying (2.4). The gauge invariance condition, also called _closure
constraint_ , is the main dynamical ingredient of GFT models for quantum BF
theory in arbitrary dimension. In dimension $3$, it turns out that ${\rm
SU}(2)$ BF theory can be interpreted as a theory of Euclidean gravity, and
therefore ${\rm SU}(2)$ GFT with closure constraint provides a natural arena
in which to formulate $3d$ Euclidean quantum gravity models. A typical example
is the Boulatov model [29] (which generates Ponzano–Regge spin foam
amplitudes), a more recent version of which will be introduced below. In
higher dimensions, further conditions on the GFT fields – which are known as
_simplicity constraints_ – need to be implemented, possibly leading to further
degeneracies of the covariance. In what follows, we will however ignore such
constraints and focus on examples in which $\tilde{C}_{\Lambda}$ is non-
degenerate. Finally, in most of the quantum-gravity literature, the
$2$-complexes supporting spin foam amplitudes are assumed to be dual to
simplicial decompositions of manifolds. This is a simplification entering the
construction of discrete gravity path-integrals which, though very natural, is
as far as we can tell not very well motivated. At the GFT level, this
corresponds to a choice of action $S_{\Lambda}$ comprising a unique type of
monomials $I_{b}$, which are of order $(d+1)$ and contract the field variables
pairwise following the pattern of a $d$-simplex. Such models have very rigid
combinatorial structure, and therefore their renormalization programme is more
difficult to apprehend. Furthermore, radiative corrections which are not of
the simplicial type are in general generated by the renormalization group
flow, and therefore need to be added to $S_{\Lambda}$ from the outset. The
importance of this simple realization should not be underestimated: in
ordinary quantum field theory, a similar argument implies that one should in
principle allow any number of local interactions in the action; it is then
left to the renormalization group to identify a finite relevant subset of
local operators among all possible interactions. Likewise, a renormalization
programme for GFTs requires the prior identification of an infinite reservoir
of allowed interactions, providing a suitable generalization of ordinary
locality.

The purpose of the next three subsections is to explain how recent
developments in tensor models made such a generalized notion of locality
available and allowed to launch a GFT renormalization programme. We will more
specifically focus on GFTs with gauge invariance condition, and explain
(including a new heuristic argument presented for the first time in this
review) in which sense so-called _colored GFTs_ [49] – which are more recent
and better behaved versions of the simplicial GFTs briefly mentioned before –
can be embedded in a larger and flexible enough class of models. These
theories are known as _tensorial GFTs_ and are the main focus of the rest of
the review.

###  2.2 Simplicial GFT models and tensorial theory space

Colored tensor models [55] and GFTs were introduced in 2009 by Gurau [49] and
have since then overcome two important caveats of older simplicial
constructions [29, 68]: 1) in dimension $d\geq 3$, the combinatorial data of
Feynman diagrams failed to unambiguously encode the combinatorial structure
and topology of the simplicial complexes generated in perturbative expansion;
2) even though power counting results could be derived [14, 44, 66], no
consistent organization of the amplitudes – such as the celebrated $1/N$
expansion of matrix models – could be proposed to make sense of the formal
perturbative expansion. We therefore decided to gloss over the original
simplicial models and work instead with colored structures from the outset.

Within the general GFT set-up we have described, the definition of a colored
GFT model in $d$ dimensions requires the introduction of $d$ auxiliary complex
GFT fields $\\{\varphi_{c}\,|\,c=1,\ldots,d\\}$, with covariance
$\bar{C}_{\Lambda}$ not necessarily identical to $C_{\Lambda}$. The label $c$
is called _color_ , and we conventionally associate the color $0$ to the
original field $\varphi\equiv\varphi_{0}$. The action $S_{\Lambda}$ is then
implicitly defined by

$\displaystyle\exp\big{(}{-}S_{\Lambda}[\varphi,\overline{\varphi}]\big{)}=\int\left[\prod_{c=1}^{d}\mathrm{d}\mu_{\bar{C}_{\Lambda}}(\varphi_{c},\overline{\varphi}_{c})\right]\exp\big{(}{-}S^{\mathrm{col}}_{\Lambda}[\varphi,\overline{\varphi};\varphi_{c},\overline{\varphi}_{c}]\big{)},$
(2.5)

where the colored GFT action is

$\displaystyle
S^{\mathrm{col}}_{\Lambda}[\varphi_{0},\overline{\varphi}_{0};\varphi_{c},\overline{\varphi}_{c}]=\lambda(\Lambda)\int\left[\prod_{0\leq\ell<\ell^{\prime}\leq
d}\mathrm{d}g_{\ell\ell^{\prime}}\right]\prod_{\ell=0}^{d}\varphi_{\ell}({\bf
g}_{\ell})$
$\displaystyle\hphantom{S^{\mathrm{col}}_{\Lambda}[\varphi_{0},\overline{\varphi}_{0};\varphi_{c},\overline{\varphi}_{c}]=}{}+\bar{\lambda}(\Lambda)\int\left[\prod_{0\leq\ell<\ell^{\prime}\leq
d}\mathrm{d}g_{\ell\ell^{\prime}}\right]\prod_{\ell=0}^{d}\overline{\varphi}_{\ell}({\bf
g}_{\ell})$ (2.6)

and we have used the convention that
$g_{\ell\ell^{\prime}}=g_{\ell^{\prime}\ell}$ together with the notation

$\displaystyle{\bf g}_{\ell}=(g_{\ell\ell-1},\ldots,g_{\ell 0},g_{\ell
d},\ldots,g_{\ell\ell+1}).$

The two interactions are interpreted as pairwise gluings of $(d+1)$
$(d-1)$-simplices along $(d-1)$-subsimplices, following to the shape of a
$d$-simplex. This pattern of contractions can be pictorially represented by
white (resp. black) nodes where $(d+1)$ colored half-edges meet as in Fig. 1.
An half-edge of color $\ell$ is associated to a GFT field $\varphi_{\ell}$ or
$\overline{\varphi}_{\ell}$, and is dual to a $(d-1)$-simplex of color $\ell$.
A pair of edges of colors $\ell$ and $\ell^{\prime}$ in turn encodes the
integral over the variable $g_{\ell\ell^{\prime}}$ in formula (2.6); it is
interpreted as a pairwise gluing of two dual $(d-1)$-simplices along a
$(d-2)$-subsimplex. In Fig. 2, we provide an equivalent stranded
representation of the pattern of contractions associated to the $3d$ vertices,
and illustrate how the dual tetrahedra can be reconstructed from the colored
vertices: half-lines are dual to triangles, which are glued pairwise along
their boundary edges.

Figure 1: Colored interaction vertices in dimension $d=3$ (left) and $d=4$
(right). Figure 2: Stranded representation of the interactions in dimension
$d=3$ (left) and dual simplicial picture (right). Colors of boundary triangles
are left implicit, except for green ones. Figure 3: Bubble interactions up to
order $6$ in $d=3$.

Note that half-lines associated to fields $\varphi$ and $\overline{\varphi}$
(with color $0$) are dashed. This is to emphasize that the latter are the true
dynamical variables of the theory; for instance, in equation (2.2) we remark
that only them have been coupled to external sources. The colored fields
$\varphi_{c}$ for $c=1,\ldots,d$ can therefore be (formally) integrated out.
This yields effective interactions $I_{b}$ parametrized by $d$-colored graphs
$b$ involving colors $c=1$ to $d$. These graphs are also called _bubbles_ in
the literature (e.g., in [25, 50]; examples in $d=3$ are provided in Fig. 3.
The coupling constant $t_{b}$ is moreover proportional to555The
proportionality factor depends on the combinatorial structure of the bubbles
alone, not on the other ingredients of the model.
$(\lambda\bar{\lambda})^{N_{b}/2}$, with $N_{b}$ the number of nodes in the
colored graph $b$.

The precise form of the effective interaction $I_{b}$ highly depends on the
auxiliary covariance $\bar{C}_{\Lambda}$ and is in general quite involved. Let
us discuss the special and simple situation in which

$\displaystyle\overline{C}_{\Lambda}({\bf g};{\bf
g}^{\prime})=\prod_{\ell=1}^{d}\delta_{\Lambda}\big{(}g_{\ell}g_{\ell}^{\prime{-1}}\big{)},$
(2.7)

where $\delta_{\Lambda}$ is a regularized version of the delta function on
$G$. More precisely, we assume that $\Lambda$ is a sharp cut-off in the
Fourier expansion of $\delta$666For instance, when $G={\rm SU}(2)$, one may
define
$\displaystyle\delta_{\Lambda}(g):=\sum_{j\in\frac{\mathbb{N}}{2}|j(j+1)\leq\Lambda^{2}}(2j+1)\chi_{j}(g),$
where $\chi_{j}$ are the characters of ${\rm SU}(2)$.. Under this condition,
it can be shown that the effective $I_{b}$ are nothing but _tensor invariants_
(up to constant factors and powers of the cut-off $\Lambda$ that we ignore for
the moment). We refer the reader to [50], in which tensor invariants were
first introduced and where their derivation is described in greater details.
Following the literature and by analogy with matrix models, we will use in
this case a trace notation $I_{b}\equiv\operatorname{Tr}_{b}$. A monomial
$\operatorname{Tr}_{b}(\varphi,\overline{\varphi})$ is uniquely determined by
its $d$-colored bubble $b$, under the following rules:

  * •

a white (resp. black) node of $b$ is associated to a field $\varphi$ (resp.
$\overline{\varphi}$);

  * •

an edge of color $\ell$ represents a convolution of two field variables, both
appearing in the $\ell^{\rm{th}}$ copy of the group $G$.

An example is provided in Fig. 4. In this simplified context, tensor invariant
interactions generate an infinite-dimensional GFT theory space in which
colored simplicial models are embedded as a one-parameter family of
models777Colored simplicial models with covariance (2.7) generate coupling
constants $t_{b}$ which are functions of $\lambda\bar{\lambda}$, hence the
one-dimensional character of this subspace of theories.. Bubble interactions
therefore provide a generalized notion of locality of the type we have been
arguing for. Topologically, they represent elementary but non-simplicial cells
with triangulated boundaries. A suggestive $3d$ example is given in Fig.
5888Note that the boundary edges of the double pyramid on the right can be
canonically colored. This illustrates one of the main advantages of the
colored structure: it allows to canonically identify all subsimplices in the
complex, and therefore faithfully encode its topology. See, e.g., [42, 55]..
Note however that bubbles may also be dual to topologically singular999A
topological singularity is defined as a point whose neighbourhood is not
homomorphic to a ball. This notion should not be confused with that of a
metric singularity. elementary cells, such as, e.g., a topological cone over a
non-spherical $(d-1)$-dimensional manifold. This is precisely the case for the
rightmost bubble of Fig. 3, which is dual to a topological cone over the
$2$-torus.

Figure 4: A four-valent $3d$ bubble and its corresponding tensor invariant.
Figure 5: A $3d$ bubble of valency $8$ dual to a double pyramid, which can
equivalently be viewed as a gluing of $8$ colored tetrahedra.

The relevance of the tensorial theory space for GFT renormalization has been
first pointed out in a seminal paper of Ben Geloun and Rivasseau [18], who
proved renormalizability of a tensorial GFT without gauge invariance
condition. This is a context in which the argument we have just presented is
applicable, and the relation between tensor invariant models and colored
simplicial ones is therefore clear. The situation is more ambiguous as soon as
one introduces gauge invariance or other quantum gravity ingredients. In this
case one can make two _a priori_ inequivalent choices.

  1. 1.

The first possibility is to choose the covariances $C_{\Lambda}$ and
$\bar{C}_{\Lambda}$ equal. Both are in particular degenerate, and the
effective interactions $I_{b}$ become quite complicated and hard to manipulate
in concrete calculations. From the point of view of known spin foam models,
which are derived from simplicial discretizations of formal quantum gravity
path-integrals, this is however the most natural construction.

  2. 2.

The second possibility is to assume, as we have done before, that the
auxiliary colored fields $\varphi_{c}$ have trivial covariance
$\bar{C}_{\Lambda}$. One may argue in this case that imposing suitable spin
foam constraints on the remaining dynamical field $\varphi$ will again lead to
legitimate discrete gravity path-integrals, however based on non-simplicial
cellular complexes.

Which of these two alternatives is the most appropriate remains an open
question, and we will not attempt to resolve it in the present article. We
will stick to the second alternative, as GFT renormalization has only been
substantially explored in this framework. But before that, we outline an
additional heuristic calculation which provides a better grasp of the relation
between the two approaches, at least in the context of $3d$ Euclidean quantum
gravity.

###  2.3 Large $\boldsymbol{N}$ expansion and extended tensorial theory space:
heuristic derivation

The colored Boulatov model (studied in, e.g., [5, 6, 33, 49, 51]) is a GFT for
Euclidean quantum gravity in space-time dimension $d=3$. The group $G$ is
therefore taken as the (universal covering) of the local symmetry group of
Euclidean space: $G={\rm SU}(2)$. The propagator with cut-off may be defined
as

$\displaystyle
C_{N}(g_{1},g_{2},g_{3};g_{1}^{\prime},g_{2}^{\prime},g_{3}^{\prime})=\int_{{\rm
SU}(2)}\mathrm{d}h\,\prod_{\ell=1}^{3}K_{1/N^{2}}\big{(}g_{\ell}hg_{\ell}^{\prime-1}\big{)}\underset{N\to+\infty}{\longrightarrow}{\mathcal{P}}(g_{1},g_{2},g_{3};g_{1}^{\prime},g_{2}^{\prime},g_{3}^{\prime}),$

where $K_{\alpha}$ is the heat-kernel on ${\rm SU}(2)$ at time
$\alpha$101010This defines a regularization of the delta function in which
high spin representations are smoothly cut-off:  $\displaystyle
K_{\alpha}(g)=\sum_{j\in\frac{\mathbb{N}}{2}}{\rm e}^{-\alpha
j(j+1)}(2j+1)\chi_{j}(g).$ , and we denote the cut-off by $N$ instead of
$\Lambda$ in reference to the original literature [51, 52, 54]. The auxiliary
covariances appearing in formula (2.5) are furthermore taken equal to the
propagator: $\bar{C}_{N}=C_{N}$. In [51] Gurau showed that, under the
assumption that the coupling constant asymptotically behaves like

$\displaystyle\lambda(N)\underset{N\to+\infty}{\sim}\frac{\lambda_{0}}{N^{3/2}}$

for some fixed $\lambda_{0}$, the colored Boulatov model admits a
$1/N$-expansion. In particular, the partition function can be expanded as

$\displaystyle{\mathcal{Z}}_{N}=N^{6}{\mathcal{Z}}_{0}(\lambda_{0}\bar{\lambda}_{0})+N^{3}{\mathcal{Z}}_{1}(\lambda_{0}\bar{\lambda}_{0})+O(1),$
(2.8)

where ${\mathcal{Z}}_{0}$ and ${\mathcal{Z}}_{1}$ sum over specific infinite
families of Feynman diagrams representing spherical manifolds111111The family
summed over by ${\mathcal{Z}}_{0}$ – the melonic graphs – has been extensively
studied in tensor models (see, e.g., [24, 55, 56] as well as in the present
context [5]. The partition function ${\mathcal{Z}}_{1}$ has as far as we know
not been studied in great details, but it is nonetheless known that it sums
spherical manifolds [52].. What is of crucial interest for us is that singular
topologies, and hence singular effective interactions, are all convergent.
This clear separation between the first leading contributions in $N$ and the
first topologically singular spin foam structures, already established in
[51], was more systematically investigated in [33, 34] by means of different
methods121212In particular, tighter bounds were derived, showing that singular
topologies are suppressed in at least $N^{3(1-S)}$, where $S$ is the number of
singular bubbles. A similar result was shown to hold also in the case of the
$4d$ colored Ooguri model [34].. Singular topologies having no natural space-
time interpretation at this point131313While metric singularities are generic
in general relativity, topological singularities are completely absent of
standard models of classical space-time., this is a welcomed property of the
$1/N$ expansion.

Now, this means that we can truncate the effective action defined in equation
(2.5) to non-singular bubbles without affecting ${\mathcal{Z}}_{0}$ and
${\mathcal{Z}}_{1}$. By definition, such non-singular bubbles have moreover
spherical boundaries, which implies that they lead to bulk amplitudes which
are peaked around trivial holonomies. Let us give an illustration of what this
means by focusing on the simplest possible bubble: the one with $2$ nodes
shown on the leftmost side of Fig. 3. It can be shown that it generates a term
in the action $S_{N}$ of the form

$\displaystyle I_{2}(\varphi,\overline{\varphi})=\int\mathrm{d}{\bf
g}\mathrm{d}{\bf g}^{\prime}\,\overline{\varphi}({\bf g})\varphi({\bf
g}^{\prime})\int[\mathrm{d}h_{i}]^{3}K_{1/N^{2}}\big{(}h_{1}h_{2}^{-1}\big{)}K_{1/N^{2}}\big{(}h_{1}h_{3}^{-1}\big{)}K_{1/N^{2}}\big{(}h_{2}h_{3}^{-1}\big{)}$
$\displaystyle\hphantom{I_{2}(\varphi,\overline{\varphi})=}{}\times\prod_{i=1}^{3}K_{1/N^{2}}\big{(}g_{i}h_{i}g_{i}^{\prime-1}\big{)}.$

Using the gauge invariant condition (2.4), one is free to translate the
$g_{i}$ variables by (say) $h_{3}^{-1}$. This reduces the $h_{i}$ dependence
of the last line to a dependence in $h_{3}^{-1}h_{1}$ and $h_{3}^{-1}h_{2}$.
Performing the change of variables $h_{1}\to h_{3}h_{1}$ and $h_{1}\to
h_{3}h_{2}$, we hence obtain an integral which is completely independent of
$h_{3}$. By normalization of the Haar measure, the new expression of $I_{2}$
is thus

$\displaystyle I_{2}(\varphi,\overline{\varphi})=\int\mathrm{d}{\bf
g}\mathrm{d}{\bf g}^{\prime}\,\overline{\varphi}({\bf g})\varphi({\bf
g}^{\prime})\int\mathrm{d}h_{1}\mathrm{d}h_{2}K_{1/N^{2}}\big{(}h_{1}h_{2}^{-1}\big{)}K_{1/N^{2}}(h_{1})K_{1/N^{2}}(h_{2})$
$\displaystyle\hphantom{I_{2}(\varphi,\overline{\varphi})=}{}\times
K_{1/N^{2}}\big{(}g_{1}h_{1}g_{1}^{\prime-1}\big{)}K_{1/N^{2}}\big{(}g_{2}h_{2}g_{2}^{\prime-1}\big{)}K_{1/N^{2}}\big{(}g_{3}g_{3}^{\prime-1}\big{)}.$
(2.9)

This procedure is nothing else than a gauge fixing and is quite general: there
is a gauge freedom associated to each node in the bubble, which allows to
trivialize the holonomies along a tree of colored edges [46]. Now, in the
large $N$ limit, one realizes that: the heat-kernels appearing in the first
line of equation (2.9) render the integrand sharply peaked around
$h_{1}=h_{2}=\mbox{$1{\bf l}$}$; together with the second line, this implies
that the integrand is also sharply peaked around $g_{i}=g_{i}^{\prime}$. This
simple fact entitles us to expand $\varphi({\bf g}^{\prime})$ in Taylor
expansion around ${\bf g}$:

$\displaystyle\varphi({\bf g}^{\prime})=\varphi({\bf
g})+\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\varphi({\bf
g}(t))+\frac{1}{2}\left.\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\right|_{t=0}\varphi({\bf
g}(t))+\cdots,$

where ${\bf g}^{\prime}(t)$ is an affinely parametrized geodesic from ${\bf
g}$ to ${\bf g}^{\prime}$ in ${\rm SU}(2)^{3}$. This reduces $I_{2}$ to an
infinite sum over tensor invariant contractions of the fields _and their
derivatives_. More precisely, this procedure can only generate ${\rm
SU}(2)$-invariant differential operators acting on each copy of ${\rm SU}(2)$
and one therefore obtains:

$\displaystyle I_{2}(\varphi,\overline{\varphi})=a(\Lambda)\int\mathrm{d}{\bf
g}\,\overline{\varphi}({\bf g})\varphi({\bf g})+b(\Lambda)\int\mathrm{d}{\bf
g}\,\overline{\varphi}({\bf
g})\left(-\sum_{\ell=1}^{3}\Delta_{\ell}\right)\varphi({\bf g})+\cdots,$
(2.10)

where $\Delta_{\ell}$ is the Laplace operator acting on the
$\ell^{\mathrm{th}}$ copy of ${\rm SU}(2)$, and $a(\Lambda)$, $b(\Lambda)$ are
computable functions. Higher order terms will involve invariant differential
operators of arbitrary order.

The previous argument can be applied in full generality. Any effective vertex
$I_{b}$ associated to a non-singular bubble $b$ can be expanded into tensor
invariant contractions of the fields and their derivatives. Moreover, only
invariant differential operators are allowed in this expansion. By picking up
a basis of such operators, one can thus define a generalized space of bubbles
$\overline{{\mathcal{B}}}\supset{\mathcal{B}}$ labelling generalized trace
invariants $\operatorname{Tr}_{\overline{b}}(\varphi,\overline{\varphi})$. The
original bubble interactions
$\operatorname{Tr}_{{b}}(\varphi,\overline{\varphi})$ (with
$b\in{\mathcal{B}}$) therefore generate a small subset of generalized tensor
invariants, those which do not involve any non-trivial differential operator.
What our analysis proves is that, up to convergent and topologically singular
contributions in the $1/N$ expansion (2.8), the colored Boulatov model
generates a one-parameter family of effective actions in the space of
generalized tensor invariants. They therefore provide a suitable GFT theory
space for the implementation of the first strategy proposed at the end of the
previous subsection, and also shows that the second approach is actually a
truncation of the first.

Finally, we point out that the same argument can be implemented for the
colored Ooguri model, and leads to the definition of generalized tensor
invariant interactions for ${\rm Spin}(4)$ in dimension $4$. It remains
however to understand how this heuristic calculation may be generalized to
$4d$ models with simplicity constraints, which have not yet been shown to
admit $1/N$ expansions.

##  3 Perturbatively renormalizable TGFTs with closure constraint

###  3.1 A general class of models: local ’potential’ approximation

We are now ready to introduce the class of TGFTs with gauge invariant
condition, as defined in the literature. We are still in the general set-up
introduced at the beginning of the preceding section. Namely, we consider a
complex GFT field $\varphi$ over $d\geq 3$ copies of a compact Lie group $G$.
Its free $2$-point function is assumed to be of the form

$\displaystyle C_{\Lambda}({\bf g};{\bf
g}^{\prime})=\int_{1/\Lambda^{2}}^{+\infty}\mathrm{d}\alpha\int_{G}\mathrm{d}h\,\prod_{\ell=1}^{d}K_{\alpha}\big{(}g_{\ell}hg_{\ell}^{\prime{-1}}\big{)},$
(3.1)

which is a regulated version of the formal operator
$C={\mathcal{P}}\Big{(}{-}\sum\limits_{\ell=1}^{d}\Delta_{\ell}\Big{)}^{-1}{\mathcal{P}}$,
$K_{\alpha}$ and $\Delta_{\ell}$ being respectively heat-kernels and Laplace
operators on $G$. Equation (3.1) is known as the Schwinger representation of
the propagator, and $\alpha$ is accordingly called a Schwinger parameter. As
for the interaction action $S_{\Lambda}$, we assume that it is generated by
tensor invariants141414Note that in this review we will always include mass
terms in the action rather than the covariance, but the opposite convention is
sometimes found in the literature.:

$\displaystyle
S_{\Lambda}[\varphi,\overline{\varphi}]=\sum_{b\in{\mathcal{B}}}t_{b}(\Lambda)\,\operatorname{Tr}_{b}[\varphi,\overline{\varphi}].$
(3.2)

Furthermore, since color labels have been introduced as purely auxiliary
objects, we will also assume that the action is invariant under permutations
of the colors. This implies specific dependences between some of the coupling
constants $t_{b}$ appearing in equation (3.2).

From the point of view of the extended GFT space described at the end of the
last section, this is the exact analogue of a local potential approximation in
ordinary quantum field theory. Indeed, locality is here embodied by tensor
invariance, which entitles us to call the action $S_{\Lambda}$ a local
potential151515Even though we cannot define a potential function, due to the
combinatorially non-trivial nature of tensorial locality.. The only non-local
terms are restricted to the first non-trivial differential operators appearing
in the Taylor expansion of the most general propagator (see equation (2.10)):
the Laplace operators $\Delta_{\ell}$.

Note that in this review we focus more precisely on _connected_ tensor
invariants. Non-connected bubble contributions may also be included in the
action (3.2), and sometimes cannot be dispensed with. For instance, the
diagram shown in Fig. 6 is associated to the non-connected tensor invariant:

$\displaystyle\left(\int\mathrm{d}{\bf g}\,\overline{\varphi}({\bf
g})\varphi({\bf g})\right)\cdot\left(\int\mathrm{d}{\bf
g}\,\overline{\varphi}({\bf g})\varphi({\bf g})\right),$

that is to the product of the tensor invariants encoded by its connected
components. Such interactions have for instance appeared in [84], and have
been more systematically studied in [60]. In the models we will more
particularly discuss below, non-connected interactions are irrelevant (as
implicitly shown in, e.g., [35], but more systematically derived in [60]), and
for simplicity we have decided to ignore them altogether.

Figure 6: A bubble with two connected components.

In order to legitimize the class of TGFTs thus defined as a perfectly honest
arena for renormalization, we need to comment a bit more on the notion of
scale in this context. For definiteness, let us specialize to $G={\rm SU}(2)$,
which is one of the most relevant example as far as quantum gravity is
concerned. The heat-kernel regularization we have introduced amounts to a
smooth regularization of the quantity

$\displaystyle p^{2}=\sum_{\ell=1}^{d}j_{\ell}(j_{\ell}+1),$

where $\\{j_{\ell}\\}$ are the spin labels associated to the harmonic
expansion of the fields. Modes associated to $p\geq\Lambda$ are exponentially
suppressed, while the theory is essentially untouched at small $p$. By analogy
with ordinary field theories, we may call $p$ momentum. The fact that we need
to regularize large momenta is dictated by the theory itself: this region of
the GFT state space is where most of the degrees of freedom lie, and where
they produce divergences. Therefore the renormalization group may only flow
from large to small cut-off161616We remind the reader that the renormalization
group is actually not a group: it has a fundamentally directed character since
its whole purpose is to erase (irrelevant) physical information.. Because of
that, and by analogy with high energy particle physics, it is conventional in
the TGFT literature to dub large (resp. small) momenta ’ultraviolet’ (resp.
’infrared’); we will stick to this nomenclature.

The purpose of our renormalization programme may now be explicitly stated: the
goal is to develop the necessary tools for determining the functional
dependence of the coupling constants $t_{b}(\Lambda)$, under the condition
that the infrared sector of the GFT is kept fixed. We will in particular aim
at a complete classification of perturbatively renormalizable models. This
programme is interesting per se, in the sense that it proposes to extend the
scope of renormalization theory to quantum field theories with exotic notions
of locality. From a quantum gravity perspective its relevance is on the other
hand conditioned by a key conjecture: that it is possible to assume that there
is a large separation of scales between the cut-off and the support of
interesting $3d$ Euclidean quantum gravity states. It is reassuring to see
that this hypothesis is at least superficially consistent. Given that spins
label the eigenvalues of the LQG length operator in $3d$ and that small spins
are associated to small lengths, we may for example expect that smooth quantum
gravity states can be approximated by (large superpositions of) spin-networks
comprising a large number of nodes and edges (or, equivalently, dual triangles
and dual edges), but only bounded spins. This also suggests that the study of
these smooth quantum gravity states will necessitate a non-perturbative
treatment of the GFT renormalization group171717Another related observation is
that quantum gravity may require the inclusion of differential operators of
arbitrary orders, as found in the generalized tensor invariant space evoked
previously.. The present section is devoted entirely to perturbative
questions, while some non-perturbative aspects will be discussed in the
following one.

###  3.2 Power-counting theorem and classification of models

We now outline the power-counting arguments leading to the full classification
of renormalizable TGFTs with closure constraint. A very nice feature of TGFTs
is that they are amenable to general multiscale methods developed in the
context of constructive field theory [77], allowing rigorous proofs of
renormalizability at all orders.

The Feynman amplitudes of these models are labelled by $(d+1)$-colored graphs
in which only dashed (or, equivalently, color-$0$) lines may be open. The
$d$-colored connected components without dashed lines, in other words the
bubbles, are the interaction vertices, while dashed lines are propagators.
Given a Feyman graph ${\mathcal{G}}$, we will denote by $L({\mathcal{G}})$,
$V({\mathcal{G}})$ and $N({\mathcal{G}})$ its set of (internal) dashed lines,
bubble vertices and external legs181818As is customary in the literature, we
will allow ourselves to use the same notation for the cardinals of these
sets.. Accordingly, the amplitude associated to a graph ${\mathcal{G}}$ is
determined by the following Feynman rules: each node is associated to an
integration over $G^{d}$; each colored line internal to a bubble represents a
delta function on $G$; and finally, dashed lines must be replaced by kernels
of $C_{\Lambda}$. We provide a $3d$ example in Fig. 7.

Figure 7: Example of TGFT Feynman graph with $N=4$, $V=4$ and $L=9$ in $3d$
(left). The amplitude is reconstructed from the drawing by means of the
Feynman rules given on the right-hand side.

The multiscale analysis191919In the present context, see for instance [30, 35,
36, 84]. For a more general and in-depth discussion in local field theory,
[77] is recommended. relies on a discrete slicing of the propagator:

$\displaystyle
C_{\Lambda}=C_{M^{\rho}}=\sum_{i\in\mathbb{N}|i\leq\rho}{\mathsf{C}}_{i},$
(3.3)

where

$\displaystyle{\mathsf{C}}_{0}:=\int_{1}^{+\infty}\mathrm{d}\alpha\int_{G}\mathrm{d}h\prod_{\ell=1}^{d}K_{\alpha}\big{(}g_{\ell}hg_{\ell}^{\prime{-1}}\big{)},$
$\displaystyle\forall\,i\geq
1,\quad{\mathsf{C}}_{i}:=\int_{M^{-2i}}^{M^{2(i-1)}}\mathrm{d}\alpha\int_{G}\mathrm{d}h\prod_{\ell=1}^{d}K_{\alpha}\big{(}g_{\ell}hg_{\ell}^{\prime{-1}}\big{)}.$

$M>1$ is a fixed but arbitrary slicing parameter and we have assumed that the
UV cut-off is of the form $\Lambda=M^{\rho}$ with $\rho\in\mathbb{N}$. Each
covariance ${\mathsf{C}}_{i}$ is then essentially responsible for the
propagation of the modes with $M^{i-1}\lesssim p\lesssim M^{i}$. This
procedure induces a decomposition of Feynman amplitudes

$\displaystyle{\mathcal{A}}_{\mathcal{G}}=\sum_{\mu}{\mathcal{A}}_{{\mathcal{G}},\mu},$
(3.4)

according to scale attributions $\mu:=\\{i_{l}\in\mathbb{N},\,l\in
L({\mathcal{G}})\\}$. In this formula, ${\mathcal{A}}_{{\mathcal{G}},\mu}$ is
an amplitude constructed from the sliced propagators $\\{{\mathsf{C}}_{i}\\}$
rather than the full propagator $C_{\Lambda}$: that is, to each line $l$, we
now associate the propagator ${\mathsf{C}}_{i_{l}}$. The multiscale strategy
then consists in looking for estimates of each of the amplitudes
${\mathcal{A}}_{{\mathcal{G}},\mu}$ separately, rather than of the full
amplitude ${\mathcal{A}}_{\mathcal{G}}$. The scale attributions $\mu$ have the
considerable advantage that they allow to optimize the naive bounds one would
derive for ${\mathcal{A}}_{\mathcal{G}}$, and as a consequence to more
precisely understand the origin of the divergences. For instance, given a
graph ${\mathcal{G}}$ with scale attribution $\mu$, one can realize that
divergences may only be generated by _high subgraphs_ : these are defined as
subgraphs ${\mathcal{H}}\subset{\mathcal{G}}$ which have internal scales
higher than the scales of their external legs. In order to implement a
renormalization procedure, one first and foremost needs to understand the
structure of divergent high subgraphs, and study their behaviour in the limit
in which the separation of scales between internal lines and external legs is
large.

Without going too much into details, a general Abelian power-counting theorem
[14, 35, 84] can be derived which, when $G$ is commutative, provides a
combinatorial characterization of the divergent subgraphs. When $G$ is non-
Abelian, further subtleties enter the picture because, although the Abelian
power-counting still holds as a bound [27, 28], it is not necessarily optimal
in this case. However, since it turns out _a posteriori_ that all the
divergent graphs encountered in TGFTs do saturate the Abelian bounds202020This
is due to the fact that the associated $2$-complexes are simply connected, see
again [27, 28]., we will simply ignore this subtlety, and the interested
reader is referred to [30, 35] for more details.

Before introducing the notion of degree of divergence, we need to define the
very central notion of _face_. In the present context, a face $f$ of a graph
${\mathcal{G}}$ is defined as a maximal bicolored path in ${\mathcal{G}}$,
with the restriction that one of the two colors must be $0$. For convenience,
we will simply attribute the color $\ell$ to a face consisting of a path with
colors $0$ and $\ell$. We will say that a line $l\in L({\mathcal{G}})$
pertains to $f$ ($l\in f$) if it coincides with one of the dashed lines of the
path. Faces can furthermore be _open_ (or, equivalently, _external_) or
_closed_ (or, equivalently, _internal_), depending on whether they are
connected to external legs of ${\mathcal{G}}$ or not. We will denote by
$F({\mathcal{G}})$ (resp. $F_{\rm ext}({\mathcal{G}})$) the set of closed
(resp. open) faces of ${\mathcal{G}}$. Conventionally, we may also orient
dashed lines positively from white to black nodes, and orient the faces
accordingly. This allows to introduce an adjacency matrix $\varepsilon_{lf}$,
of size $L\times F$ and with only $0$ or $1$ entries: $\varepsilon_{lf}=1$ if
$l\in f$, and $\varepsilon_{lf}=0$ otherwise. Faces are particularly important
because, given the form of the propagator and the Feynman rules, the integrand
of an amplitude factorizes over its faces. More precisely, each closed face
$f$ yields a factor

$\displaystyle K_{\underset{l\in f}{\sum}\alpha_{l}}\left(\underset{l\in
f}{\overrightarrow{\prod}}h_{l}\right),$

where $\alpha_{l}$ and $h_{l}$ are respectively the Schwinger parameter and
holonomy associated to the propagator line $l$, and the product over
holonomies is taken accordingly to the orientation of $f$. See Fig. 8 for an
example. An open face $f$ yields on the other hand a factor

$\displaystyle K_{\underset{l\in
f}{\sum}\alpha_{l}}\left(g_{s(f)}\left[\underset{l\in
f}{\overrightarrow{\prod}}h_{l}\right]g_{t(f)}^{-1}\right),$

where $g_{s(f)}$ and $g_{t(f)}$ are boundary variables associated to the
fields sitting at the source ($s(f)$) and target ($t(f)$) ends of $f$.

Figure 8: An internal face of color $1$ and length $3$, and its associated
amplitude integrand.

######  Proposition 3.1.

The superficial degree of divergence of a $($non-vacuum$)$ graph
${\mathcal{G}}$ is

$\displaystyle\omega({\mathcal{G}}):=-2L({\mathcal{G}})+D\left(F({\mathcal{G}})-R({\mathcal{G}})\right)\geq
0,$

where $R({\mathcal{G}})$ is the rank of the adjacency matrix
$\varepsilon_{lf}$ of ${\mathcal{G}}$, and $D$ is the dimension of $G$.

The superficial degree of divergence (which we abbreviate to “degree of
divergence” or simply “degree” in the sequel) captures the UV asymptotic
behaviour of the amplitudes. For a single-slice amplitude at scale $i$ (i.e.,
$\mu=\\{i_{\ell}=i,\,l\in L({\mathcal{G}})\\}$), one can show that
${\mathcal{A}}_{{\mathcal{G}},\mu}$ has an exponential scaling of the form
$M^{\omega({\mathcal{G}})i}$ when $i\to+\infty$212121When
$\omega({\mathcal{G}})=0$ the divergences are logarithmic.. This corresponds
to the situation in which ${\mathcal{G}}$ contains a single high subgraph –
itself. The fact that the divergences are in this case essentially controlled
by the combinatorial quantity $(F-R)$ was already proven in [14], though in a
slightly different context. The analysis of more general scale attributions is
based on a step-by-step estimation of the contributions of high subgraphs,
from higher to smaller scales, and was first detailed in [35, 84]. We will not
need to go into such details here, which are only relevant for the full
rigorous proof of renormalizability. We only point out that the concept of
high subgraphs allows a very natural treatment of overlapping divergences,
which otherwise lead to somewhat challenging recursive constructions (see [77]
and references therein).

Once one understands how Feynman amplitudes diverge in the UV, one may try to
devise simple criteria of renormalizability, for instance in terms of the
dimensions $d$ (space-time) and $D$ (group). To this effect, it is important
to find a more practical expression of the divergence degree, which puts
combinatorial quantities such as the number of external legs $N$ to the
forefront. The presence of the rank $R$, which is a direct consequence of the
gauge invariance condition we imposed on the fields, makes it more involved
than in TGFTs without this ingredient. Invoking elementary combinatorial
relations, one easily proves that222222The variable $\rho$ of this equation is
a combinatorial quantity associated to a graph, and has nothing to do with the
cut-off appearing in equation (3.3).

$\displaystyle\omega=D(d-2)-\frac{D(d-2)-2}{2}N+\sum_{k\in\mathbb{N}^{*}}[(D(d-2)-2)k-D(d-2)]n_{2k}+D\rho,$
(3.5)

where $N$ is the number of external legs and $n_{2k}$ the number of bubbles of
valency $k$. The whole non-trivial dependence in the rank has been included in
the combinatorial quantity

$\displaystyle\rho:=F-R-(d-2)(L-V+1).$

The key missing ingredient leading to a general classification of models is a
bound on $\rho$. The following proposition, which was first derived in [35],
serves this purpose.

######  Proposition 3.2.

Let ${\mathcal{G}}$ be a non-vacuum graph. Then

$\displaystyle\rho({\mathcal{G}})\leq 0,$

and $\rho({\mathcal{G}})=0$ if and only if ${\mathcal{G}}$ is a melonic graph.

We refer the reader to, e.g., [19, 35] for a more precise combinatorial
characterization of melonic graphs in this context. As illustrated in Fig. 9,
melonic graphs tend to maximize the number of internal faces. Proposition 3.2
can alternatively be taken as a definition of melonic graphs. We only mention
two important properties. First, melonic graphs are associated to and
generated by a specific subset of bubbles, which are accordingly called
_melonic bubbles_. In Fig. 3, all bubbles are melonic except for the rightmost
one, showing that in dimension $d=3$ the first non-trivial interactions are
necessarily melonic. Second, melonic graphs and melonic bubbles have both
trivial topology: in dimension $d$ they represent $d$-balls, and are therefore
topologically suitable building blocks of $(d+1)$-dimensional space-time.

Figure 9: Melonic (right) and non-melonic (left) $4$-point graphs in $d=3$.
$F=2$ for the former and only $1$ for the latter.

Proposition 3.2 shows that melonic bubbles and melonic graphs lead to the most
severe divergences. Following the literature, we now proceed with a
classification of what we may call _melonic models_. We define them as TGFTs
which: 1) include melonic interactions; and 2) are perturbatively consistent
under renormalization. We emphasize that the first hypothesis is non-trivial,
and is somewhat implicitly assumed in the literature. We will come back to
this interesting aspect below.

Power-counting renormalizability requires the degree of divergence to be
bounded from above. Moreover, in the presence of divergences, $\omega$ should
decrease with the number of external legs, in such a way that only finitely
many $n$-point functions need to be renormalized. From these conditions alone,
one can derive a full classification of melonic models allowed by the power-
counting analysis, in terms of the dimensions $d$ and $D$, and the maximal
valency $v_{\max}$ of the renormalizable bubbles. The complete list,
established in [35]232323The classification of [35] includes only just-
renormalizable models, we have added super-renormalizable and finite models
for completeness., is reported in Table 1. Models of type A to E are candidate
just-renormalizable GFTs, and are in principle the most interesting ones: they
have infinitely many divergent Feynman amplitudes, which leads to universal
properties of the flows. Models of type F and G are on the other hand super-
renormalizable, which means that their divergences are generated by a finite
family of single-vertex graphs (also known as tadpoles). Finally, models of
type H are finite and are therefore not very interesting from the point of
view of renormalization: the renormalization group is in this case unable to
provide a physical hierarchy for the amplitudes and interactions, which may
well all contribute with roughly the same intensity.

Table 1: List of power-counting renormalizable melonic models. Type | $d$ | $D$ | $v_{{\max}}$ | $\omega$ | Explicit examples
---|---|---|---|---|---
A | 3 | 3 | 6 | $3-N/2-2n_{2}-n_{4}+3\rho$ |  $G={\rm SU}(2)$ [35, 31]
B | 3 | 4 | 4 | $4-N-2n_{2}+4\rho$ |  $G={\rm SU}(2)\times{\rm U}(1)$ [32]
C | 4 | 2 | 4 | $4-N-2n_{2}+2\rho$ | –
D | 5 | 1 | 6 | $3-N/2-2n_{2}-n_{4}+\rho$ |  $G={\rm U}(1)$ [84, 81]
E | 6 | 1 | 4 | $4-N-2n_{2}+\rho$ |  $G={\rm U}(1)$ [84, 81, 22]
F | 3 | 2 | arbitrary | $2-2V$ | –
G | 4 | 1 | arbitrary | $2-2V$ |  $G={\rm U}(1)$ [36, 63]
H | 3 | 1 | arbitrary | $1-L-V<0$ |  $G={\rm U}(1)$ [62]

A striking feature of this classification is that the only combination of $d$
and $D$ which is compatible with a quantum space-time interpretation of the
amplitudes is $d=D=3$. Indeed, only in this case is the would-be space-time
dimension $d$ consistent with the dimension $D$ of the local symmetry group
$G$. This is quite remarkable: we have first motivated the general class of
TGFTs with gauge invariance condition from Euclidean quantum gravity in three
dimensions, and reciprocally, pure quantum field theory arguments allow us to
in a sense _derive_ dimension $3$ as the only consistent one.

This prompted an in-depth study of the $d=3$ model with $G={\rm SU}(2)$, which
was proven renormalizable at all orders in [35]. Its flow equations were then
studied in greater details in [31]. Note that, because of the subtleties
associated to non-Abelian amplitudes, this particular example required extra
care, which we are glossing over in this review. Still in $d=3$, a
renormalizable model of type B based on the group ${\rm SU}(2)\times{\rm
U}(1)$ has been considered in [32]. Examples of Abelian ${\rm U}(1)$ models of
type D and E were proposed in [84] and also proven renormalizable at all
orders. Interestingly, and as is clearly allowed by the power-counting
arguments we have reviewed, the $\varphi^{6}$ model of type D requires the
inclusion of a non-melonic and non-connected interaction of the form
$(\overline{\varphi}\cdot\varphi)^{2}$, sometimes called “anomalous”. The
model of type E, which according to our power-counting arguments might also
have necessitated the inclusion of non-melonic bubbles242424Graphs with
$\rho=-1$ or $-2$ might in principle still lead to divergences., remains
consistent with only melonic bubbles included252525Note that we are not
claiming that non-melonic bubbles cannot be consistently included, only that
the model is consistent without them. The construction of non-melonic phases
remains a largely unexplored and interesting research direction.. The beta
functions of these two models were then studied in [81], and the functional
renormalization group of the model of type D was investigated in [22].

Abelian super-renormalizable models of type G actually provided the first
examples of renormalizable TGFTs with gauge invariance [36]. Since only
finitely many divergent graphs are generated in this case, renormalization
could be implemented by means of a generalization of the standard Wick
ordering prescription. Constructive aspects of a $\varphi^{4}$ model of this
type, as well as of a finite model of type H, were more recently studied in
[62, 63]. This led in both cases to a Borel resummation of the perturbative
expansion, thus proving its analytical existence. It is a very interesting
step towards a full non-perturbative definition of just-renormalizable TGFTs
with closure constraint, including the more physically relevant $d=3$ and
$G={\rm SU}(2)$ situation.

Examples of renormalizable models of type C and F have not been explicitly
exhibited in the literature. There is however no doubt that such example
exists, for instance with the group $G={\rm U}(1)^{2}$. Indeed, the arguments
and tools from [36] and [84] are directly applicable to this Abelian group. In
particular, the analysis of non-melonic graphs proposed in [84] allows to
demonstrate that melonic bubbles alone lead to a consistent model in situation
C. We further conjecture that it is also possible to consistently include
$\varphi^{4}$ necklace bubbles [23] in this context, which will generate
divergent but non-melonic $2$-point graphs (with $\rho=-1$). See Fig. 10.

Figure 10: Non-melonic $2$-point graph with $\rho=-1$ in $d=4$. Its single
vertex is a necklace bubble, which suggests that such an interaction may
consistently be included in a TGFT model of type C. The graph has a degree
$\omega=0$ and is therefore logarithmically divergent in this case.

###  3.3 Renormalization, subtraction schemes and contractible graphs

Once power-counting renormalizability has been checked, several standard
quantum field theory techniques may be applied to prove full-fledged
renormalizability. The main physical idea is to re-express the Feynman
expansion in terms of new physically meaningful perturbative parameters. The
bare coupling constants are indeed associated to processes occurring at
arbitrarily large energies, and have therefore no empirical content. Instead,
one should parametrize the theory with the values of $n$-point functions at an
arbitrarily chosen but physically accessible low energy scale. In this new
expansion, Feynman amplitudes converge, and the divergences of the bare
amplitudes are interpreted as spurious effects resulting from a misplaced
parametrization of the field theory.

Different renormalization prescriptions may be used, leading to slightly
different (but equivalent) definitions of renormalized quantities. One can for
example rely on the celebrated Bogoliubov–Parasiuk–Hepp–Zimmermann (BPHZ)
scheme, which amounts to using $p^{2}=0$ as a reference energy scale. This
standard textbook procedure is rather simple at first loop orders, but may
appear somewhat mysterious when it comes to overlapping divergences262626We
remind the reader that, given a graph ${\mathcal{G}}$, two divergent subgraphs
${\mathcal{H}}_{1},{\mathcal{H}}_{2}\in{\mathcal{G}}$ are said to overlap if
neither of the three relations is verified:
${\mathcal{H}}_{1}\cap{\mathcal{H}}_{2}=\varnothing$,
${\mathcal{H}}_{1}\subset{\mathcal{H}}_{2}$,
${\mathcal{H}}_{2}\subset{\mathcal{H}}_{1}$.. The effective expansion, based
on multiscale methods, is more in the spirit of Wilson’s brilliant
reformulation of renormalization. It consists in a step-by-step recursive
definition of effective coupling constants $t_{b,i}$, which measure the
amplitudes of physical processes associated to the index scale $i$. The
contributions of divergent graphs at a given scale are reabsorbed into the
coupling constants at lower scales, resulting in a discrete renormalization
group flow from higher to lower scales:

$\displaystyle t_{b,i-1}-t_{b,i}=\beta_{i}(\\{t_{b^{\prime},j}\,|\,i\leq
j\leq\rho\\}).$

The $n$-point functions may then be expressed as formal multi-series in the
$t_{b,i}$’s, with convergent coefficients in the limit $\rho\to+\infty$.
Moreover, the effective expansion provides a new perspective on the more
standard renormalized BPHZ expansion: at fixed multiscale parameter $\mu$
(3.4), divergent graphs can simply not overlap, and this simple realization
greatly clarifies the reason why the BPHZ procedure converges at all orders in
perturbation theory. We refer the reader to [77] for a detailed discussion of
both the BPHZ expansion and the effective expansion in the context of local
scalar field theories.

Both effective and renormalized expansions have been successfully generalized
and applied to TGFTs with gauge invariance condition [35, 36, 84]. Reviewing
these constructions in detail would take us too far into technicalities, we
therefore only expose the core argument explaining why these techniques can be
applied at all. In ordinary quantum field theory, renormalization relies on
the key realization that high energy processes look essentially local (as seen
by an observer operating at much lower energy scales). This is true
irrespectively of how complicated these high energy processes are, and is the
main reason why the contributions of high divergent subgraphs at a given scale
can always be absorbed into redefinitions of the coupling constants at lower
momenta. Possibly severe complications arise in our TGFT context: first, the
non-standard notion of locality encapsulated in tensor invariant interactions
renders the analysis obviously more intricate; second, and more importantly,
the main combinatorial building blocks of the amplitudes generated by TGFTs
with closure constraint are the faces, which are intrinsically non-local
objects.

For concreteness, let us consider the example of Fig. 11. On the left one
finds a typical effective contribution to the $4$-point function generated by
a melonic graph in $3d$. In terms of the Schwinger parameters $\alpha_{1}$ and
$\alpha_{2}$, the integrand of its amplitude is:

$\displaystyle\int\mathrm{d}h_{1}\mathrm{d}h_{2}\left[K_{\alpha_{1}+\alpha_{2}}(h_{1}h_{2})\right]^{2}\int\left[\prod_{i<j}\mathrm{d}g_{ij}\right]K_{\alpha_{1}}\big{(}g_{11}h_{1}g_{31}^{-1}\big{)}K_{\alpha_{2}}\big{(}g_{21}^{-1}h_{2}g_{41}\big{)}$
$\displaystyle\qquad{}\times\delta\big{(}g_{12}g_{22}^{-1}\big{)}\delta\big{(}g_{13}g_{22}^{-1}\big{)}\delta\big{(}g_{42}g_{32}^{-1}\big{)}\delta\big{(}g_{43}g_{33}^{-1}\big{)}\varphi({\bf
g}_{1})\overline{\varphi}({\bf g}_{2})\overline{\varphi}({\bf
g}_{3})\varphi({\bf g}_{4}).$

The question is whether this expression can be approximated by an elementary
tensor invariant in the sector $\alpha_{1},\alpha_{2}\to 0$272727We remind the
reader that the Schwinger parameters should be thought of as inverse squared
momenta $p^{-2}$.. Though it is not that obvious at first sight, the answer is
yes. We can resort to a similar line of arguments as the one which led us to
the expansion (2.10). The gauge symmetry associated to the amplitudes allows
to gauge-fix one of the two holonomies and reduce this expression to:

$\displaystyle\int\mathrm{d}h\left[K_{\alpha_{1}+\alpha_{2}}(h)\right]^{2}\int\left[\prod_{i<j}\mathrm{d}g_{ij}\right]K_{\alpha_{1}}\big{(}g_{11}hg_{31}^{-1}\big{)}K_{\alpha_{2}}\big{(}g_{21}^{-1}g_{41}\big{)}$
$\displaystyle\qquad{}\times\delta\big{(}g_{12}g_{22}^{-1}\big{)}\delta\big{(}g_{13}g_{22}^{-1}\big{)}\delta\big{(}g_{42}g_{32}^{-1}\big{)}\delta\big{(}g_{43}g_{33}^{-1}\big{)}\varphi({\bf
g}_{1})\overline{\varphi}({\bf g}_{2})\overline{\varphi}({\bf
g}_{3})\varphi({\bf g}_{4}).$

It is then obvious that, in the large scale limit, $h$ is peaked around the
identity and therefore $g_{11}$ (resp. $g_{21}$) is identified to $g_{31}$
(resp. $g_{41}$). A Taylor expansion of the external variables of color $1$
along their external faces then allows to approximate the amplitude by the
tensor invariant (and melonic) interaction shown on the right-hand side of
Fig. 11, up to a scale-dependent constant $\nu(\alpha_{1},\alpha_{2})$.

Figure 11: Approximation of a tracial graph as an effective trace invariant
contribution plus corrections.

We say that a graph is _contractible_ if its bulk holonomies can be
trivialized in the UV region, as illustrated in the example before282828This
is equivalent to simple-connectedness of the $2$-complex, i.e., flat
connections are trivial up to gauge.. Only if all divergent graphs are
contractible can we reabsorb UV divergences into tensor invariant effective
interactions. It may however happen that disconnected effective bubbles are
generated in this way. A contractible graph generating a connected tensor
invariant interaction is furthermore called _tracial_. We now understand that
the reasons why renormalization theory meaningfully applies to all the
candidate models we have introduced in the previous subsection are that: 1)
melonic graphs are tracial; and 2) all non-melonic divergent corrections are
contractible. It is important to understand that these are highly non-trivial
facts, which rely on intimate relations between the topology of colored graphs
and the scaling of their amplitudes. On the one hand, trivial topology in the
bulk of divergent graphs ensures that our tensor invariant truncation is
stable under renormalization. But there is more as, in return, this consistent
renormalization scheme guarantees that no topological singularities can be
generated by radiative corrections. For instance, the rightmost interaction of
Fig. 3 can be consistently set to $0$ in the $d=3$ model on ${\rm SU}(2)$,
even though it is allowed by our power-counting arguments [35]. Interestingly,
divergent non-melonic graphs with non-melonic bubbles such as the one shown in
Fig. 10 are also tracial, which suggests again that necklace terms may
sometimes be consistently included into the picture.

##  4 Renormalization group and non-trivial fixed points

The purpose of this section is to illustrate some of the interesting
properties of the renormalization group flows of TGFTs with gauge invariance
condition. The goal of renormalization group investigations is two-fold: 1)
understand the fate of the renormalized coupling constants in the deep UV, and
hence determine whether the theory is consistent to arbitrary high scales or
not; 2) systematically explore the theory space away from the perturbative
fixed point, and investigate in particular the existence of non-trivial fixed
points. The first objective can be first addressed within a perturbative
scheme, and is embodied in this context by the question of _asymptotic
freedom_. In the unfavourable case in which asymptotic freedom does not hold
(which means that the renormalized coupling constants do not converge to $0$
in the UV), there is still the possibility that the theory may be UV completed
by means of a non-perturbative UV fixed point. The second objective is central
to the whole GFT approach to quantum gravity and requires a non-perturbative
treatment anyway. We therefore decide to focus on the Functional
Renormalization Group (FRG), and more precisely on the Wetterich equation,
which can conveniently be used to address both types of questions and will
presumably play an important role in the future. The related Polchinski
equation was on the other hand investigated in [59, 60, 61].

###  4.1 Effective average action and Wetterich equation

Irrespectively of one’s preferred formulation of the renormalization group,
consistent flow equations may only be formulated for _dimensionless_ coupling
constants $u_{b}(\Lambda)$, which are appropriate rescalings of
$t_{b}(\Lambda)$ by powers of $\Lambda$. This important aspect may be
formalized by the notion of _canonical dimension_ $d_{b}$ of tensor invariant
bubbles, which has been discussed at length in [22, 31, 60]. For melonic
models (with closure constraint), it is defined as

$\displaystyle d_{b}:=D(d-2)-(D(d-2)-2)\frac{N_{b}}{2},$

where $N_{b}$ is the valency of the bubble $b$ (i.e., its number of nodes).
Hence the dimensionless coupling constants are defined as

$\displaystyle u_{b}(\Lambda):=\frac{t_{b}(\Lambda)}{\Lambda^{d_{b}}},$

and $b$ is renormalizable (or, equivalently, perturbatively relevant) if and
only if $d_{b}\geq 0$. In view of the expression (3.5) for the divergence
degree $\omega$, it appears that the most divergent graphs contributing to the
running of $t_{b}$ diverge like $\Lambda^{d_{b}}$, and yield corrections to
the dimensionless couplings $u_{b}$ of order $1$ (as they should).

We insist on the fact that the notion of canonical dimension just defined is
_well-suited to melonic models only_ , as we have implicitly assumed that the
most divergent contributions come from melonic graphs. A model which would for
instance contain only necklace interactions would bring us out of the melonic
world described in this review, and henceforth yield a different notion of
canonical dimension292929This possibility is actively explored in the context
of $4d$ models with Barrett–Crane simplicity constraints [Carrozza S., Lahoche
V., Oriti D., work in progress]..

In the Wetterich–Morris [67, 86] approach to the functional renormalization
group, a one-parameter family of deformed generating functionals is introduced

$\displaystyle\mathcal{Z}_{\Lambda,k}[J,\bar{J}]:=\int\mathrm{d}\mu_{C_{\Lambda}}(\varphi,\overline{\varphi})\exp\big{(}{-}S_{\Lambda}[\varphi,\overline{\varphi}]-\overline{\varphi}\cdot
R_{k}\cdot\varphi+\bar{J}\cdot\varphi+\overline{\varphi}\cdot J\big{)}.$

The new operator $R_{k}$ has the function of regularizing the field modes
below an infrared cut-off $k$ while leaving the high energy sector unaffected.
This allows the introduction of a new generating functional, the _effective
average action_ $\Gamma_{k}$, which is the Legendre transform of
$W_{k}[J,\bar{J}]:=\ln(\mathcal{Z}_{k,\Lambda}[J,\bar{J}])$, appropriately
shifted by the $2$-point counter-term $\overline{\varphi}\cdot
R_{k}\cdot\varphi$ (see [22] for a detailed discussion). Interestingly, for
suitable choices of $R_{k}$, $\Gamma_{k}$ can be shown to interpolate between
the bare action $S_{\Lambda}=\Gamma_{\Lambda}$ in the UV, and the full
effective action $\Gamma_{0}$ (or in other words the generating functional of
one-particle irreducible graphs) in the infrared, which we illustrate in Fig.
12.

Figure 12: The effective average action interpolates between the bare action
in the ultraviolet and the full effective action in the infrared.

Finally, as derived in detail in [22] for TGFTs with gauge invariant
condition, the effective average action verifies a Wetterich equation

$\displaystyle
k\partial_{k}\Gamma_{k}[\varphi,\overline{\varphi}]=\int\mathrm{d}\textbf{g}_{1}\mathrm{d}\textbf{g}_{2}\mathrm{d}\textbf{g}_{3}k\partial_{k}R_{k}(\textbf{g}_{1};\textbf{g}_{2})\big{(}\Gamma_{k}^{(2)}+R_{k}\big{)}^{-1}(\textbf{g}_{2},\textbf{g}_{3}){\mathcal{P}}(\textbf{g}_{3};\textbf{g}_{1}),$
(4.1)

where

$\displaystyle\Gamma^{(2)}_{k}[\varphi,\overline{\varphi}]({\bf g};{\bf
g}^{\prime}):=\frac{\delta^{2}\Gamma_{k}}{\delta\varphi({\bf
g})\delta\overline{\varphi}({\bf g}^{\prime})}[\varphi,\overline{\varphi}]$

is the full interacting propagator at scale $k$ and ${\mathcal{P}}$ is as
before the projector onto gauge invariant fields. Equation (4.1) defines a
formal flow in an infinite-dimensional space of theories, a rigorous
mathematical definition of which is for the time being out of reach. The
standard procedure used in the literature to make sense of the Wetterich
equation consists in choosing a finite-dimensional ansatz for the effective
average action, and then systematically projecting the formal flow equation
down to this finite-dimensional subspace of theories. The local potential
approximation introduced in the previous section is one such possible ansatz,
and yields a system of one-loop flow equations. Standard one-loop perturbative
equations may be recovered with a truncation which only includes
renormalizable interactions, while the computation of higher order loops
requires the inclusion of non-renormalizable corrections to the potential. The
same method can be used in the non-perturbative regime, with the important
caveat that truncations are much harder to justify in this case. The
functional renormalization group equation provides little analytical control
over error terms, and one must therefore resort to more empirical
justifications, based mainly on numerical tests of convergence of the
truncation procedure. It turns out at the end of the day that the FRG provides
reliable and effective methods for discovering and computing the properties of
non-trivial fixed points in ordinary statistical field theories (see, e.g.,
[3]). This is what makes them particularly precious in GFTs, as they have the
potential to unravel new and more physical phases. Applications of the FRG to
TGFTs being rather recent, only the simplest truncations have been considered
so far, and the question of their reliability remains to be further explored.

###  4.2 Example: perturbative treatment of rank-3 models

Let us start with a perturbative application of the formalism just introduced,
which already suggests a variety of different properties for the
renormalization group flows of TGFTs. We restrict our attention to $d=3$
renormalizable models of the type A ($D=3$) and B ($D=4$). Interestingly, the
situation is reminiscent of that of an ordinary local scalar field, which is
renormalizable up to quartic interactions in space-time dimension $4$, and up
to order $6$ interactions in dimension $3$. The same statement holds for $d=3$
melonic models once the space-time dimension is traded for the group dimension
$D$. This remarkable fact allows a simple but informative comparison of the
qualitative features of the flow equations of TGFTs against those of ordinary
local field theories.

Let us start with a model of type $B$ and choose for instance the group ${\rm
SU}(2)\times{\rm U}(1)$ as in [32]. This theory is renormalizable up to order
$4$, therefore a natural ansatz for the effective average action is303030We
use the short-hand notation $\Delta:=\sum\limits_{\ell=1}^{3}\Delta_{\ell}$.

$\displaystyle\Gamma_{k}(\varphi,\overline{\varphi})=-Z(k)\,\overline{\varphi}\cdot\Delta\varphi+Z(k)u_{2}(k)k^{2}\;\vbox{\hbox{\includegraphics[scale={0.6}]{int2}}}+Z(k)^{2}u_{4}(k)\;\vbox{\hbox{\includegraphics[scale={0.6}]{int4}}}\;,$

where for convenience we directly represented tensor interactions by their
associated colored graphs313131Note that each colored graph in this equation
is to be thought of as representing the equivalent class of bubbles with the
same combinatorial structure up to permutation of the color labels. There is
for instance a sum of three distinct melonic interactions at order $4$,
corresponding to three inequivalent permutations of the color labels., and we
have introduced a wave-function parameter $Z(k)$. In this group dimension, the
mass term has canonical dimension $2$ and the marginal $\varphi^{4}$
interactions have as they should dimension $0$. It was shown in [32] that the
perturbative renormalization group flow reduces in this truncation to

$\displaystyle k\frac{\partial u_{2}(k)}{\partial k}=-2u_{2}(k)-3\pi
u_{4}(k),$ $\displaystyle k\frac{\partial u_{4}(k)}{\partial
k}=-2\pi{u_{4}(k)}^{2}.$ (4.2)

One notices a major difference with ordinary scalar field theories: the
derivative of the $\varphi^{4}$ coupling is negative, which means that it
decreases towards $0$ in the ultraviolet. We therefore obtain an
asymptotically free and UV complete perturbative definition of the theory!
This is due to a quite general and remarkable property of TGFTs at large. As
was first remarked in [10] in the context of TGFTs without gauge invariance
condition, and later on generalized to models with closure constraint in [81],
wave-function counter-terms generally dominate over vertex renormalization
ones, and are ultimately responsible for changes in the signs of some of the
coefficients of the flow equations. A beautiful explanation based on a
symmetry argument has been furthermore proposed in [79], thereby proving that
asymptotic freedom is a completely general feature of quartic renormalizable
TGFTs. In particular, Abelian models of type E are asymptotically free, as
argued for in [81].

Let us now move on to the $d=3$ model with $G={\rm SU}(2)$, which is
renormalizable up to order $6$ interactions. Accordingly, one can choose the
following ansatz for the effective average action:

$\displaystyle\Gamma_{k}(\varphi,\overline{\varphi})=-Z(k)\overline{\varphi}\cdot\Delta\varphi+Z(k)u_{2}(k)k^{2}\;\vbox{\hbox{\includegraphics[scale={0.6}]{int2}}}+Z(k)^{2}u_{4}(k)k\;\vbox{\hbox{\includegraphics[scale={0.6}]{int4}}}$
$\displaystyle\hphantom{\Gamma_{k}(\varphi,\overline{\varphi})=}{}+Z(k)^{3}u_{6,1}(k)\;\vbox{\hbox{\includegraphics[scale={0.6}]{int61}}}+Z(k)^{3}u_{6,2}(k)\;\vbox{\hbox{\includegraphics[scale={0.6}]{int62}}}\;,$

and we note the change of dimensionality of the $\varphi^{4}$ interactions
with respect to the previous situation. The one-loop perturbative flow can in
this case be approximated by323232These flow equations were not explicitly
evaluated in [32], but they immediately follow from the computations reported
there.

$\displaystyle k\frac{\partial u_{2}(k)}{\partial
k}\approx-2u_{2}(k)-au_{4}(k),$ $\displaystyle k\frac{\partial
u_{4}(k)}{\partial k}\approx-u_{4}(k)-b(u_{6,1}(k)+2u_{6,2}(k)),$
$\displaystyle k\frac{\partial u_{6,1}(k)}{\partial k}\approx-
cu_{4}(k)u_{6,1}(k),$ $\displaystyle k\frac{\partial u_{6,2}(k)}{\partial
k}\approx-du_{4}(k)u_{6,2}(k),$ (4.3)

where $a$, $b$, $c$ and $d$ are strictly positive constants. This is a rather
complicated system of equations, but we notice again the negative signs in the
last two equations. It is therefore tempting to conjecture that, if one
assumes that all the coupling constants are positive, then the marginal
constants $u_{6,1}$ and $u_{6,2}$ both converge to $0$ in the ultraviolet. As
shown in [31]333333This paper relied on different methods, in the language of
the multiscale expansion. It also went further in that $2$-loop contributions
were included to account for quadratic terms (in $u_{6,1}^{2}$, $u_{6,1}^{2}$
and $u_{6,1}u_{6,2}$) which have been neglected in (4.3). The main conclusions
are however the same., this is actually misleading because, given the form of
the flow equations, it can be proven that $u_{4}(k)$ necessarily reaches
negative values for large $k$, which has the effect of making the
$\varphi^{6}$ interactions grow again (see [31]). Hence, one is forced to
conclude that this model _cannot_ be asymptotically free if we assume that
both positive marginal coupling constants are positive343434The situation is a
more subtle when $u_{6,1}u_{6,2}<0$, we do not exclude the possibility that
one could define an asymptotically free theory with positive action in this
particular case.. This example shows that the question of asymptotic freedom
can be tricky in $\varphi^{6}$ renormalizable models, because the
$\varphi^{4}$ super-renormalizable coupling constants have a non-trivial
influence on the marginal $\varphi^{6}$ interactions. In particular, the
combinatorial TGFT of [10] and the type $D$ model of [81], which have both
been argued to be asymptotically free on the basis of an analysis which
neglected the flow of super-renormalizable constants, may possibly suffer from
a similar back-reaction effect.

As mentioned already, if a model is not asymptotically free, one may still
contemplate the idea of finding a non-perturbation UV completion of it. This
is an interesting but notoriously hard question to investigate, since this
requires to establish the existence of a non-perturbative fixed point of the
renormalization group. An elementary standard method often invoked in
statistical physics to test this assumption is the $\varepsilon$-expansion
[87], which has the advantage of being essentially perturbative. In scalar
field theories for instance, one can formally define statistical models in
dimension $4-\varepsilon$, which smoothly interpolate between dimension $4$
and dimension $6$. A TGFT generalization of this construction was proposed in
[32]. The procedure consists in defining a $d=3$ TGFT on the group ${\rm
SU}(2)\times{\rm U}(1)^{D-3}$ for arbitrary $D\geq 3$, and then analytically
continue the parameter $D:=4-\varepsilon$ to the interval $3\leq D\leq 4$.
When $\varepsilon$ is small enough, one may assume that the $\varphi^{4}$
truncation remains pertinent:

$\displaystyle\Gamma_{k}(\varphi,\overline{\varphi})=-Z(k)\overline{\varphi}\cdot\Delta\varphi+Z(k)u_{2}(k)k^{2}\;\vbox{\hbox{\includegraphics[scale={0.6}]{int2}}}+Z(k)^{2}u_{4}(k)k^{\varepsilon}\;\vbox{\hbox{\includegraphics[scale={0.6}]{int4}}}\;.$

Note that $u_{4}$ acquired a small canonical dimension $\varepsilon$. This has
the effect of slightly modifying the flow equations (4.2) to

$\displaystyle k\frac{\partial u_{2}(k)}{\partial k}\approx-2u_{2}(k)-3\pi
u_{4}(k),\qquad k\frac{\partial u_{4}(k)}{\partial k}\approx-\varepsilon
u_{4}(k)-2\pi{u_{4}(k)}^{2}.$ (4.4)

Accordingly, one formally finds a new solution to the fixed point equation:

$\displaystyle
u_{2}^{*}\approx\frac{3}{4}\varepsilon+{\mathcal{O}}\big{(}\varepsilon^{2}\big{)},\qquad
u_{4}^{*}\approx-\frac{1}{2\pi}\varepsilon+{\mathcal{O}}\big{(}\varepsilon^{2}\big{)}.$

Qualitatively, the renormalization group flow (4.4) is therefore as
represented in Fig. 13. Extrapolating to $\varepsilon=1$, this suggests the
existence of a TGFT analogue of the Wilson–Fisher fixed point of $3d$ local
scalar field theory in the ${\rm SU}(2)$ model of [35]. This hypothesis should
of course be taken with a grain of salt, since the first terms in the
$\varepsilon$-expansion may only give a crude idea of what is really going on
at $\varepsilon=1$. Note also that $u_{4}^{*}$ has the ‘wrong’ sign (again
because wave-function counter-terms dominate in TGFT), which might be taken as
a sign that the formal fixed point we found is only a spurious effect. This
anyway provides solid motivations for performing a non-perturbative study of
the Wetterich equation directly in the case $\varepsilon=1$ [Carrozza S.,
Lahoche V., work in progress].

Figure 13: Qualitative phase portrait of the renormalization group flow
equations (4.4), with arrows oriented from high to small scales.

###  4.3 Non-perturbative aspects and truncations

As far as TGFTs with gauge invariance are concerned, the non-perturbative
aspects of the Wetterich equation have been studied in two complementary
papers [17, 22].

In [22], the role of gauge invariance was carefully analyzed and the Wetterich
equation (4.1) was formally derived. The $d=6$ melonic model on ${\rm U}(1)$
(type E) was studied in the $\varphi^{4}$ melonic truncation

$\displaystyle\Gamma_{k}(\varphi,\overline{\varphi})=-Z(k)\overline{\varphi}\cdot\Delta\varphi+Z(k)u_{2}(k)k^{2}\;\vbox{\hbox{\includegraphics[scale={0.7}]{int2_6d}}}+Z(k)^{2}u_{4}(k)\;\vbox{\hbox{\includegraphics[scale={0.7}]{melon_bubble_6d}}}\;.$

For such Abelian models, it is convenient to use a Litim cut-off operator
$R_{k}$ [65] which, in momentum space, is defined by the kernel

$\displaystyle
R_{k}({\mathbf{p}};{\mathbf{p}})=Z(k)\big{(}k^{2}-{\mathbf{p}}^{2}\big{)}\Theta(k^{2}-{\mathbf{p}}^{2})\prod_{\ell=1}^{6}\delta_{p_{\ell}p_{\ell}^{\prime}},$

where $p_{\ell}\in\mathbb{Z}$ label ${\rm U}(1)$ representations. The merit of
this type of cut-off is that it greatly simplifies the structure of the
truncated flow equations, leading to beta functions which are algebraic
fractions in the coupling constants. In the perturbative regime one can check
again than the model is asymptotically free, as first proven in [81] with
different methods. An important aspect of TGFTs on compact Lie groups is that,
because of finite size effects, the renormalization group flow is not
autonomous. The beta functions of the dimensionless coupling constants
explicitly depend on the infrared cut-off $k$, which complicates the search
for non-trivial fixed points. The UV ($k\gg 1$) and IR ($k\to 0$) asymptotic
regimes were analyzed separately in [22]. One finds in both a non-perturbative
fixed point analogous to the Wilson–Fisher fixed point of statistical field
theory. The quantitative values of the coupling constants are slightly
different in the two regimes, but the qualitative structure of the phase
portrait is the same. In particular, and unlike the formal fixed point found
in the $\varepsilon$-expansion of $3d$ models, the coupling constant $u_{4}$
at the fixed point is positive (and $u_{2}$ is negative).

A non-compact version of this model, based on the Abelian group $\mathbb{R}$,
was investigated in [17]. The renormalization group analysis requires in this
case a further regularization of infinite volume divergences. This was
implemented by a compactification of $\mathbb{R}$ into ${\rm U}(1)$, thus
resulting in a regularized theory identical to [22]. The authors could then
define an appropriate thermodynamic limit, capturing the properties of the
model in the limit in which the volume of ${\rm U}(1)$ is scaled to infinity.
Once more, a fixed point of the Wilson–Fisher type is found in this
truncation, consistently with the results of [22]. It remains to be seen
whether this qualitative behaviour will survive closer scrutiny. One will in
particular need to check its regularization independence and its stability
under extensions of the truncation. This preliminary result may nonetheless be
interpreted as a hint of a phase transition between a symmetric and a broken
phase of a condensate type, in which the field $\varphi$ acquires a vacuum
expectation value. In view of recent applications of Bose–Einstein
condensation, which is a phase transition with an order parameter of the same
type, these are particularly interesting results. They may help justifying
scenarii which have been recently proposed in the GFT literature, with
important applications in cosmology [47, 48, 73] and black hole physics [72].

Non-Abelian models may also be explored in this formalism, and a particularly
interesting one is again the $d=3$ theory on ${\rm SU}(2)$ [35]. Checking its
perturbative behaviour requires to push the truncation to order $6$ at least,
and even to higher orders if one wants to also account for perturbative
$2$-loop contributions. Moreover, the Litim cut-off is not very convenient in
this setting, because of the gauge invariant condition. There was no
difficulty in the Abelian model mentioned before because the closure
constraint translates into a simple momentum constraint $p_{1}+\dots+p_{6}=0$,
which can be explicitly dealt with. With ${\rm SU}(2)$ the situation is not so
simple: the gauge invariance condition encodes complicated recoupling
relations among the ${\rm SU}(2)$ harmonic modes, and results in quite
challenging expressions. It is therefore better to work in direct space and to
rely on a heat-kernel regularization, as was already done in the perturbative
Section 3. Such a construction is under way [Carrozza S., Lahoche V., work in
progress]: even if the flow equations obtained within this renormalization
scheme are not algebraic fractions, they are nonetheless computable and can be
integrated out numerically. We should therefore soon be able to directly
compute the properties of the flow of model [35] in a $\varphi^{6}$
truncation, and compare them with the features we extrapolated from the formal
$\varepsilon$-expansion.

##  5 Summary and outlook

We hope we have managed to convince the reader that GFT renormalization is an
active field of research which has already born interesting fruits. For one
thing, the fact that such non-local field theories can be defined and analyzed
by means of renormalization methods may at first sound like a contradiction in
the terms. Locality is indeed a primary concept in relativistic quantum field
theories, and is absolutely key to the formulation of renormalization theory.
While at the fundamental level, GFT in a sense goes away with space-time
altogether, it is remarkable that tensor invariance may successfully be used
as a substitute for locality. On top of providing a (for a long time missing)
structure encoding the topology of GFT interactions and Feynman diagrams, it
introduces just enough of flexibility to allow for a GFT theory space which is
stable under renormalization.

The inclusion of the GFT gauge invariance condition (2.4) into renormalizable
tensor field theories was a necessary and by no means obvious step in the
direction of quantum gravity applications. We have explained in some detail
the remarkable interplay between topology and renormalization which makes it
possible (Section 3.3): in perturbative expansion, the most divergent spin
foam amplitudes turn out to be supported on simply connected $2$-complexes,
which allows to trivialize the bulk holonomies associated to ultraviolet
scales, and therefore reabsorb the associated divergences into effective
tensor invariant coupling constants.

This produced a rather large class of renormalizable TGFTs with closure
constraint (Table 1), and consequently a natural test bed for GFT
renormalization. Renormalization group studies have in particular shown that
asymptotic freedom is a generic feature of $\varphi^{4}$ perturbative models,
which may therefore be interpreted as ultraviolet complete theories. Moreover,
the non-perturbative truncations investigated so far suggest that non-trivial
fixed points with properties analogous to that of the Wilson–Fisher fixed
point of local scalar field theories are generic. This opens the way to the
study of GFT phases and phase transitions.

We conclude with a non-exhaustive list of open problems which we consider
particularly interesting.

Inclusion of non-melonic interactions. The renormalizable models presented in
this review are governed by melonic interactions and melonic radiative
corrections. We have however pointed out on several occasions that non-melonic
bubbles (such as the necklace bubble of Fig. 10) may also be included and
potentially lead to the definition of yet other perturbative phases. This
question deserves to be explored more systematically, as it should in
particular help us understand to which extent the polymer phases typically
generated by melonic families of graphs – which have a tree-like structure –
may be escaped [56].

Local potential approximation and beyond. A heuristic derivation of an
extended tensor theory space, which includes interactions with arbitrary
$G$-invariant differential operators, has been proposed in Section 2.3. This
suggests to interpret all the renormalizable models studied so far as local
potential approximations within this extended tensor theory space, and
therefore to explore the properties of more general truncations. Our heuristic
argument also shows that in $3d$ and with $G={\rm SU}(2)$, the generalized
tensor theory space contains in principle the colored Boulatov model. This
might open the way to a proper quantum gravity interpretation of this three-
dimensional theory.

Lorentzian signature. The whole literature on GFT and GFT renormalization is
focused on models with compact Lie groups, which may at best result in
consistent formulations of Euclidean quantum gravity. The GFT formulation of
quantum gravity models with Lorentz signature necessitates to go beyond this
framework. While a renormalization scheme taking the additional infrared
divergences associated to non-compact linear groups is already available [16,
17], the physically relevant ${\rm SL}(2,\mathbb{R})$ and ${\rm
SL}(2,\mathbb{C})$ are much more challenging. Indeed, the invariant Laplace
operators on such groups are not positive, leading to complications associated
to the definition of natural propagators. To a large extent, these theories
resemble quantum fields on Minkowski space-time, therefore Euclidean
multiscale methods are not easily applicable. Alternative techniques, based
for instance on Epstein–Glaser renormalization (as it is applicable in any
signature) or on a Wick rotation, will be necessary to explore this question
in greater detail.

$4d$ quantum gravity constraints. Eventually one will need to check whether
the now well-understood closure constraint can be consistently complemented
with spin foam simplicity constraints. If a renormalizable $4d$ model can be
defined in this context, we will obtain a consistent perturbative sum over
spin foam transition amplitudes, and therefore a tentative definition of the
dynamics of LQG. All the renormalization group methods that have been
developed for and tested on our simpler toy-models will then be of great
practical use, in particular to determine whether a sector of the quantum
dynamics reproduces general relativity in a suitably defined classical limit.

### Acknowledgements

The author acknowledges support from the ANR JCJC CombPhysMat2Tens grant.

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