Tensor models and group field theories: combinatorics, large $N$ and
renormalization

Sylvain<EMAIL_ADDRESS>
aInstitut de Mathématiques de Bourgogne, UMR 5584, CNRS & Université de
Bourgogne, F-21000 Dijon, France.

###### Abstract

We provide a brief overview of tensor models and group field theories,
focusing on their main common features. Both frameworks arose in the context
of quantum gravity research, and can be understood as higher-dimensional
generalizations of matrix models. We describe the common combinatorial
structure underlying such models and review some of the key mathematical
results that have been obtained in this area. Of central importance is the
discovery of a new family of large $N$ expansions for random tensors. It has
driven applications of tensor models to random geometry and non-perturbative
local quantum field theory, and has also enabled the development of rigorous
renormalization methods for group field theories and other non-local quantum
field theories.

##  1 Introduction

With the pioneering work of t’ Hooft [1], it was recognized that a quantum
field theory (QFT) of an $N\times N$ matrix field can be formally expanded
into Feynman amplitudes indexed by _ribbon graphs_ or _combinatorial maps_
[2]. Those two equivalent combinatorial structures formalize the notion of a
graph embedded into a canonically associated surface (considered up to
homeomorphism), and the genus of that surface indexes a _topological large $N$
expansion_ in which $\frac{1}{N^{2}}$ takes the role of a formal small
parameter. In addition to t’ Hooft’s original idea of using this expansion to
probe the non-perturbative sector of QFT, it has found numerous applications
to enumerative combinatorics, and more specifically, to two-dimensional
quantum gravity [3, 4, 5, 6]. Indeed, formal matrix integrals can be used to
efficiently manipulate generating functions of combinatorial maps, and
asymptotic enumeration results for such objects can be inferred by studying
the critical properties of the underlying matrix model [7, 8]. From a physical
point of view, such a critical regime can be interpreted as defining the
continuum limit of a two-dimensional random geometry [9, 10] which is
equivalent to Liouville quantum gravity [11, 12].

Tensor models were first considered in the early nineties [13, 14, 15], in an
attempt to establish an analogous correspondence between random tensors222By
convention, we reserve the word _tensor_ for an object with $D\geq 3$ indices,
to contrast with the case of vectors ($D=1$) and matrices ($D=2$). and
Euclidean quantum gravity in dimension $D\geq 3$. These early works pointed to
a fertile correspondence between the Feynman expansion of such a tensor model
(indexed by higher generalizations of ribbon graphs) and discrete geometries
in dimension $D$. However, no generalization of the large $N$ expansion of
matrix models could be identified at the time. For a while, this hindered
further progress on tensor models and their relationship to random geometry.

In the meantime was introduced the formalism of Group Field Theory (GFT) [16,
17, 18]. A GFT can be understood as an algebraically enriched tensor model
which generates, via its Feynman expansion, discrete spaces weighted by
lattice gauge theory amplitudes (rather than mere combinatorial factors). More
precisely, it is a (perturbative) QFT of a tensor field whose indices take
value into a group $G$, and the action is chosen in such a way as to generate
the amplitudes of a specific lattice gauge theory with structure group $G$.
Particularly studied types of actions are those that generating $BF$ lattice
gauge theory amplitudes [19, 20] and discrete quantum gravity amplitudes in
holonomy-flux variables [21, 22, 23, 24]. The latter can be interpreted as
giving rise to models of random geometry, in which the geometry is function of
both the combinatorial structure of the lattice and its attached group-
theoretic data. From a mathematical point of view, GFTs suffer from
divergences which are of a similar nature as those encountered in local
perturbative QFT, but owing to the non-local character of a GFT action, are
beyond the scope of standard perturbative QFT theorems. Hence, for some time,
it was not clear if and in which sense GFTs could be considered as
mathematically well-defined perturbative QFTs.

In 2009, the formalism of _colored_ tensor models (resp. GFTs) was introduced
by Gurau [25], which eventually led to the discovery of a new large $N$
expansion [26, 27, 28]. The availability of this crucial ingredient, which
generalizes the large $N$ expansion of matrix models but also differs from it
in crucial respects, triggered new developments in both tensor models and
GFTs. On the one hand, the large $N$ expansion has allowed to make precise
statements about the relation of random tensors to random geometry (see Sec.
2.2). On the other hand, the combinatorial tools developed in the context of
tensor models have enabled the proof of a series of renormalization results
adapted to GFT (and other non-local QFTs), which have considerably clarified
their mathematical definition (see Sec. 3.2). Finally, Gurau’s large $N$
expansion and its variants (see Sec. 2.1) were later exploited to explore the
non-perturbative sector of local QFTs with internal symmetry described by
tensor representations (see Sec. 2.3). See [29] for a recent review.

Our review is organized as follows. In Sec. 2, we describe Gurau’s large $N$
expansion of tensor models333Gurau’s first paper on the large $N$ expansion
was aimed at both tensor models and GFTs, but further results in that
direction were for the most part focused on the former.  and some of its
variants, together with their two main applications to date: random geometry
and strongly-coupled QFT. GFTs and closely related families of non-local QFTs
are the focus of Sec. 3. Regarding GFT, we restrict our attention to
renormalization results since they have a strong connection to tensor models.
Other aspects of GFT, which have received continued attention and led to
important developments in recent years are only briefly mentioned in Sec. 3.3.

##  2 Tensor models

###  2.1 Combinatorial structure and melonic large N expansions

In its simplest incarnation, a tensor model takes as input a linear subspace
${\mathcal{T}}$ of multilinear maps acting on some (real, complex or
quaternionic) vector space of dimension $N$, together with the natural group
action of the appropriate unitary group (namely, ${\rm U}(N)$, ${\rm O}(N)$ or
${\rm Sp}(N)$).444We will see shortly that it is advantageous to consider more
constraining symmetries defined by a _product_ of such unitary groups. If
${\mathcal{T}}$ is a space of tensors of order $p$, one chooses an appropriate
basis and simply represents $T\in{\mathcal{T}}$ by a multidimensional array
$T_{a_{1}\cdots a_{p}}$ ($a_{1}\,,\ldots a_{p}\in\\{1,\ldots,N\\}$). One can
then equip ${\mathcal{T}}$ with a formal measure taking the form:

$\mathrm{d}\nu(T)=\mathrm{d}\mu_{\boldsymbol{P}}(T)\,\exp\left(-S_{\rm
int}(T)\right)\,,\qquad S_{\rm int}(T)=\sum_{k}\lambda_{k}I_{k}(T)\,,$ (1)

where: $\boldsymbol{P}$ is a symmetric linear operator on ${\mathcal{T}}$, and
$\mathrm{d}\mu_{\boldsymbol{P}}$ the Gaussian measure of covariance
$\boldsymbol{P}$; the (interaction part of the) action $S_{\rm int}(T)$ is a
polynomial invariant of the tensor $T$, which is typically expanded in terms
of a basis of invariant monomials $\\{I_{k}(T)\\}$; the coupling constants
$\\{\lambda_{k}\\}$ are understood as formal perturbative parameters. In all
the models considered in the present section, $\boldsymbol{P}$ is itself
assumed to be equivariant under the group action on ${\mathcal{T}}$ (see Sec.
3 for models in which this condition is relaxed). One is then interested in
determining the universal properties of such an object in the asymptotic limit
of large dimensionality, namely $N\to+\infty$. With a few notable exceptions
(e.g. [30, 31]), this question has been investigated in the sense of formal
power-series in $\\{\lambda_{k}\\}$. For this reason, we will make no attempt
to provide a probabilistic definition of the formal measure
$\mathrm{d}\nu(T)$, and will allow ourselves to consider any kind of invariant
in the action $S_{\rm int}(T)$ (e.g. real invariants which are unbounded from
below). What makes tensor models interesting from this formal point of view is
that quantities such as the free energy $F\propto\ln\int\mathrm{d}\nu(T)$ can
be interpreted as generating functions of combinatorial objects, namely
combinatorial maps and higher-dimensional generalizations thereof.

An important observation is that the space of polynomial tensor invariants is
significantly richer for $p\geq 3$ than it is when $p\leq 2$. To see this, it
is convenient to represent tensors and their index-contractions graphically,
as illustrated in Fig. 1: a tensor of order $p$ is represented by a vertex
connected to $p$ half-edges, while the contraction of two indices amounts to
gluing the two corresponding half-edges into an edge. Basic polynomial
invariants can then be represented by closed graphs, which we illustrate in
Fig. 2 for a real vector $\phi$ ($p=1$), an Hermitian matrix $M$ ($p=2$), and
a completely symmetric real tensor of order $T$ ($p=3$). In all three cases, a
generic polynomial invariant can be decomposed as a sum of products of more
basic invariants, which can be represented by connected graphs, and are thus
called _connected invariants_. But the spaces of connected invariants are very
different in each case. The vector $\phi$ admits a single connected invariant:
the inner product $\phi_{a}\phi_{a}$, represented by a single edge.555From now
on, Einstein’s convention of implicit summation over repeated indices is
assumed. By contrast, the matrix $M$ admits an infinite set of connected
invariants, one per order $n\in\mathbb{N}$: the $n^{\rm th}$ power sum of its
eigenvalues, $\mathrm{tr}(M^{n})$, which is represented by a cyclic graph with
$n$ vertices. Furthermore, any two such invariants are independent in the
limit $N\to+\infty$. Finally, the connected invariants of a real symmetric
tensor $T$ of order $3$ are in one-to-one correspondence with $3$-regular
(multi)graphs, whose number grows super-exponentially with the number $n$ of
vertices (or, equivalently, the order of the invariant). Again, any two such
connected invariants become independent when $N$ is sufficiently large. Such a
super-exponential growth of the number of connected invariants is observed for
any $p\geq 3$ and for other types of tensors.666See e.g. [32, 33] for precise
enumeration results. It makes the theory space of tensor models comparatively
more difficult to analyze than those of vector or matrix models. However, as
it turns out, the generating functions produced by specific families of tensor
invariants can be of a simpler nature than those of matrix models, while
remaining richer than those of vector models, which is in part what makes
tensor models physically interesting.

Figure 1: Graphical representation of tensors and their contractions. Figure
2: Connected invariants (organized according to their polynomial order) in the
case of: a real vector ($p=1$); a Hermitian matrix ($p=2$); and a real
symmetric tensor of order $3$ $(p=3)$.

A standard example of matrix model can be obtained by taking ${\mathcal{T}}$
to be the vector space of $N\times N$ Hermitian matrices, while setting
$\boldsymbol{P}=\frac{1}{N}\mbox{$1{\bf l}$}$ (where $1{\bf l}$ is the
identity map on ${\mathcal{T}}$) and
$S(T)=-\frac{\lambda}{4}N\mathrm{tr}(T^{4})$ in equation (1). The primary
objects of interest in such a model are expectation values of invariant
observables relative to the measure $\mathrm{d}\nu$, which can in principle be
reconstructed from correlation functions of the form
$\langle\mathrm{tr}(T^{n_{1}})\cdots\mathrm{tr}(T^{n_{q}})\rangle$. The
perturbative expansion in $\lambda$ of such correlators can in turn be
organized in terms of Feynman graphs, which in this example are $4$-regular
ribbon diagrams: they can be obtained by consistently gluing an arbitrary
number of propagators and vertices with the structure shown in Fig. 3.
Crucially, such ribbon diagrams can be canonically interpreted as
quadrangulations of surfaces with boundaries, and the $N$-dependence of their
Feynman amplitude is entirely determined by the topology of the surface.
Focusing on the free energy, which generates closed and connected
quadrangulations, we obtain the celebrated topological genus expansion:

$\ln\int_{\mathcal{T}}\mathrm{d}\nu(T)=\sum_{g\in\mathbb{N}}N^{2-2g}{\mathcal{F}}_{g}(\lambda)\,,$
(2)

where ${\mathcal{F}}_{g}(\lambda)$ is the generating function of connected
quadrangulations of genus $g$ (and $\lambda$ counts the number of
quadrangles). In the asymptotic limit $N\to\infty$, such a matrix model is
therefore characterized by the planar sector $g=0$, and finite $N$ corrections
can in principle be investigated in a perturbative expansion in $1/N^{2}$.
Interestingly, the structure of such an expansion allows the genus-$g$ free
energy ${\mathcal{F}}_{g}(\lambda)$ to admit better convergence properties
than the full free energy, which enables a non-perturbative treatment of the
coupling constant $\lambda$ at fixed genus (and in particular in the planar
sector). This was the original motivation of ’t Hooft for devising such a
large $N$ expansion in the context of large $N$ gauge theory. Finally, the
topological expansion (2) can be straighforwardly generalized to actions of
the form $S(T)=N\mathrm{tr}(V(T))$, where $V$ is some ($N$-independent)
polynomial. Including a monomial proportional to $T^{n}$ in $V$ amounts to
allowing ribbon vertices of degree $n$, or dually, faces of degree $n$ in the
discrete surfaces generated by the model.

(a) Propagator.

(b) Vertex.

Figure 3: Propagator and interaction vertex of the quartic Hermitian matrix
model: an oriented strand between matrix indices $i$ and $j$ represents a
Kronecker $\delta_{ij}$.

A natural generalization of the previous model to the tensor realm can be
obtain by letting ${\mathcal{T}}$ be the space of real tensors of size $N^{3}$
(equipped with its standard inner product), and $\boldsymbol{P}$ the
orthogonal projector onto its symmetric subspace.777This is equivalent to
taking ${\mathcal{T}}$ to be the space of completely symmetric real tensors
and $\boldsymbol{P}$ the identity, but is somewhat more convenient for our
purpose. A possible generalization of the quartic interaction we considered in
the matrix case is given by the invariant represented in Fig. 4, which is
often times called ’tetrahedral’ due to its combinatorial stucture (we will
come back to this). This leads to the action:

$S_{\rm int}(T)=\frac{-\lambda}{4N^{\alpha}}\,T_{abc}T_{dec}T_{dbf}T_{aef}\,,$
(3)

where $\lambda$ is the perturbative parameter and $\alpha$ is a scaling
parameter that will need to be set to an appropriate value in order to ensure
the existence of a large $N$ expansion.888Note that the tetrahedral invariant
is real and unbounded in both directions, contrary to the positive quartic
interaction we were using in the matrix context. The model we have just
defined was proposed in the very first papers on tensor models [13, 14, 15] as
a way to generate, via the Feynman expansion, three-dimensional ’discrete
spaces’ obtained by gluing an arbitrary numbers of tetrahedra along their
faces, in analogy with matrix models. What is for sure is that the Feynman
graphs of the model can be interpreted as higher generalizations of ribbon
diagrams, in which through each propagator run three strands instead of two;
see Fig. 4. It is however more challenging to provide a precise definition of
what one would mean exactly by ’discrete space’ here. The interplay between
the combinatorial structure of the action and the global structure of the
Feynman diagrams, is indeed much less rigid in higher dimensions than it is in
two dimensions: it is therefore challenging to ensure nice global properties
(such as a manifold structure) from the Feynman rules (which only define local
gluing conditions). Even more problematic, no analogue of the topological
large $N$ expansion of equation (2) could be identified in this model.
Progress on these questions was first achieved in the context of so-called
_colored_ tensor models, in which more constraining symmetries make the
combinatorial structure of the Feynman diagrams more rigid. We now turn to a
description of this important class of models, before briefly returning to
real symmetric tensors at the end of the section.

(a) Propagator ${\boldsymbol{P}}_{a_{1}a_{2}a_{3},b_{1}b_{2}b_{3}}$.

(b) ’Tetrahedral’ invariant and its kernel.

Figure 4: Propagator and interaction vertex of the quartic symmetric tensor
model.

An interesting structural feature of colored tensor models is their relation
to the combinatorial species of _edge-colored graphs_. This relation was first
exploited in the context of models involving a multiplet of complex tensors
[25, 34, 35, 36], whose entries were labeled by an index called _color_. These
were later subsumed by so-called _uncolored_ models [37], which are somewhat
simpler to describe in that they involve a single complex random tensor, and
have as a result imposed themselves as a better starting point to develop a
general theory of random tensors [38]. We will focus on this latter class of
models here, which we will simply refer to as _colored_ tensor models.999Due
to later developments, the historical distinction between _colored_ and
_uncolored_ models can be confusing.

A _complex colored tensor model_ [38] is a theory of a complex random tensor
$T_{a_{1}\cdots a_{D}}$ of size $N^{D}$, living in the $D$-fundamental
representation of $\mathrm{U}(N)^{D}$. This means there is one independent
unitary symmetry for each index of $T$: for any
$U^{(1)},\ldots,U^{(D)}\in\mathrm{U}(N)$, we have the transformation law

$T_{a_{1}\cdots a_{D}}\to U^{(1)}_{a_{1}b_{1}}\cdots
U^{(D)}_{a_{D}b_{D}}T_{b_{1}\cdots b_{D}}\,.$ (4)

The formal measure is defined as in equation (1) with the covariance taken to
be the identity, so that

$\int\mathrm{d}\mu_{\boldsymbol{P}}(\bar{T},T)\bar{T}_{a_{1}\cdots
a_{D}}T_{a_{1}\cdots b_{D}}=\prod_{k=1}^{D}\delta_{a_{k}b_{k}}\,,$ (5)

while the action is assumed to be an invariant of the form:

$S_{\rm
int}(\bar{T},T)=-\sum_{\mathcal{B}}\lambda_{\mathcal{B}}N^{-\alpha({\mathcal{B}})}\mathrm{Tr}_{\mathcal{B}}(\bar{T},T)\,.$
(6)

In the last formula, the sum is taken over (a finite set of) _connected and
bipartite $D$-colored graphs_ $\\{{\mathcal{B}}\\}$, which index a generating
set of invariants $\\{\mathrm{Tr}_{\mathcal{B}}(\bar{T},T)\\}$ under the group
action (4); see Fig. 5 for some examples. Connected $D$-colored graphs, also
known as _bubbles_ in this context, have two types of vertices of coordination
$D$ (black and white), representing a tensor $T$ or its conjugate $\bar{T}$,
and $D$ types of edges labeled by an index in $\\{1,\ldots,D\\}$, called
_color_. An edge labeled by $k\in\\{1,\ldots,D\\}$ always connects a black to
a white vertex, and represents the invariant contraction of two indices in
position $k$. By a construction originally due to Pezzana which matured into a
combinatorial approach to PL manifolds known as _crystallization theory_ [39]
(see also [36, 40]), any bubble diagram admits a dual interpretation as a
colored triangulation of an orientable _pseudomanifold_ 101010This is a
generalization of the notion of topological manifold that allows mild types of
topological singularities. of dimension $D-1$.111111This topological
interpretation is the reason why the number of indices of a colored random
tensor is denoted by the letter $D$. For instance, the first four bubbles
shown in Fig. 5 represent triangulations of the $2$-sphere, while the fifth is
dual to a triangulation of the $2$-torus. As soon as $D\geq 3$, the colored
structure of the bubbles is essential to their dual topological interpretation
[34]: the color labels allow to unambiguously specify how subsimplices of
arbitrary co-dimension should be identified, which is illustrated in the right
panel of Fig. 6. Taking the topological cone over each such
$(D-1)$-dimensional pseudomanifold yields an elementary building block of
$D$-dimensional space (a $3$-ball in the first four examples of Fig. 5, a
pseudomanifold with a pointlike singularity in the fifth example). This
construction provides a precise higher-dimensional generalization of the
correspondence between matrix invariants and $n$-gons (which is sharper, and
therefore more satisfactory than what we had in the symmetric tensor model
with action (3)). In the Feynman expansion, colored invariants are glued
together by propagator edges, which we can assign a new color label ($0$ by
convention). With this convention in place, Feynman diagrams have the
structure of $(D+1)$-colored graphs. They are therefore dual to
pseudomanifolds of dimension $D$, which are precisely obtained by gluing
together the previously mentioned topological cones along their boundaries.
Having the same type of combinatorial structure encoding both tensor
invariants and Feynman graphs is a remarkable property of colored tensor
models. See Fig. 7 for an example of Feynman graph, and for its
reinterpretation as a (colored) stranded diagram. Finally, coming back to
equation (6), the parameters $\\{\alpha({\mathcal{B}})\\}$ need to be set to
appropriate values in order to ensure the existence of a large $N$ limit. This
choice is not unique, but we will focus here on the so-called _melonic_ large
$N$ scaling [37], which is governed by a combinatorial quantity known as the
_Gurau degree_ $\omega$ [26, 27, 28, 36].

Given a connected $(D+1)$-colored graph ${\mathcal{G}}$ (e.g. a Feynman
diagram), let $p({\mathcal{G}})\in\mathbb{N}$ be its number of white (resp.
black) vertices and $F({\mathcal{G}})$ its number of _faces_ , defined as the
number of bi-colored cycles in ${\mathcal{G}}$. The Gurau degree of
${\mathcal{G}}$ is a _positive integer_ , defined by [26, 27]

$\omega({\mathcal{G}})=D-F({\mathcal{G}})+\frac{D(D-1)}{2}p({\mathcal{G}})\,.$
(7)

The positivity of $\omega$ is not explicit in this formula, but can be
recognized by re-expressing (7) as [28]

$\omega({\mathcal{G}})=\frac{1}{(D-1)!}\sum_{\sigma}g({\mathcal{J}}({\mathcal{G}},\sigma))\,,$
(8)

where $\sigma$ runs over cyclic permutations of $\\{0,\ldots,D\\}$,
${\mathcal{J}}({\mathcal{G}},\sigma)$ is a combinatorial map canonically
associated to the pair $({\mathcal{G}},\sigma)$ – dubbed _jacket_ of
${\mathcal{G}}$ in [41] –, and
$g({\mathcal{J}}({\mathcal{G}},\sigma))\in\mathbb{N}$ is its topological
genus. The positivity of $\omega$ makes it an interesting higher-dimensional
analogue of the standard genus of an orientable surface. This is reinforced by
the realization that $\omega$ is nothing but the orientable genus when $D=2$
(which can be seen from both (7) and (8)). However, an important difference
between $D=2$ and $D\geq 3$ is that $\omega$ is _not a topological invariant_
when $D\geq 3$. See [36, 38, 40] for further detail.

Figure 5: Representation of colored tensor invariants as $D$-colored graphs
$(D=3)$.

(a) $D=3$.

(b) $D=4$.

Figure 6: Interpretation of $D$-colored graphs as dual
$(D-1)$-pseudomanifolds. Figure 7: A Feynman diagram represented as a
$(D+1)$-colored graph ($D=3$), where color-$0$ edges are dashed to emphasize
they represent propagators (by contrast, other colored edges are internal to
interaction vertices). It can be reinterpreted as a stranded graph upon
substituting color-$0$ edges by stranded propagators, as shown in the right
panel, and its amplitude is determined by interpreting each strand as a
Kronecker delta.

Now, if one sets121212A bubble ${\mathcal{B}}$ being a connected $D$-colored
graph, its degree is defined by (7) after substituting $D$ for $D-1$.

$\alpha({\mathcal{B}}):=\frac{2}{(D-2)!}\omega({\mathcal{B}})$ (9)

for any bubble ${\mathcal{B}}$ appearing in equation (6), one can prove the
existence of a sensible large $N$ expansion generalizing the one of matrix
models [37]. In particular, one finds the following expression for the
partition function

$\ln\int\mathrm{d}\nu(\bar{T},T)=\sum_{\omega\in\mathbb{N}}N^{D-\frac{2}{(D-1)!}\omega}{\mathcal{F}}_{\omega}(\\{\lambda_{\mathcal{B}}\\})\,,$
(10)

where ${\mathcal{F}}_{\omega}(\\{\lambda_{\mathcal{B}}\\})$ is the generating
function of connected $(D+1)$-colored graph, enumerated by number of bubbles
of each type (the variable $\lambda_{\mathcal{B}}$ counting the number of
bubbles of type ${\mathcal{B}}$). The leading order sector consists in
connected $(D+1)$-colored graphs of vanishing degree, which are known as
_melonic graphs_ [35]. This is the analogue of the planar sector of matrix
models, and any melonic graph is indeed mapped via Pezzana’s theorem to a
triangulation of the $D$-sphere. However, since $\omega$ is not topological in
$D\geq 3$, not every colored triangulation of the $D$-sphere is melonic. In
fact, the set of melonic graphs is quite restricted: it can be generated
recursively, from the unique colored graph on two vertices, by a finite set of
elementary graph insertions along edges.131313The ”melonic” qualifier is
motivated by the shape of such an elementary insertion, which is reminiscent
of a cataloupe melon; see Fig. 8. Melonic graphs are therefore in bijection
with combinatorial trees (more specifically $(D+1)$-ary trees). This will
become important when we discuss applications of melonic large $N$ expansions.

(a)

(b)

Figure 8: Two families of melonic diagrams: (a) ’local’ melons from complex
colored models; (b) ’bilocal’ melons from models with ’tetrahedral’
interaction, as e.g. in the real $\mathrm{O}(N)^{3}$ model (each vertex in the
picture represents a ’tetrahedral’ interaction, whose internal structure has
been omitted).

Other large $N$ limits dominated by families of melonic diagrams (as we will
see, with slightly different combinatorial properties) have been obtained in
closely related tensor models. These include the original colored tensor
model, built from a $(D+1)$-tuple of complex tensors [26, 27, 28]. In the
context of models involving a single tensor, there is some freedom in the
choice of symmetry group, while preserving the colored structure of the
Feynman graphs. Early on, the so-called _multi-orientable_ tensor model [42,
43, 44, 45, 46], involving a complex $3$-tensor, was shown to admit a melonic
limit with an action proportional to a ’tetrahedral’ interaction of the form
$\bar{T}_{abc}T_{dec}\bar{T}_{dbf}T_{aef}$ (note the similarity with (3)). The
$\mathrm{U}(N)\times\mathrm{O}(N)\times\mathrm{U}(N)$ invariance of this model
paved the way to a more systematic exploration of _real colored tensor models_
obeying $\mathrm{O}(N)^{D}$ invariance. The $D=3$ case was first considered in
[47] (with the first two interactions shown in Fig. 10), and later generalized
to higher $D$ in [48]. At the combinatorial level, the main difference between
real and complex colored models is that real invariants (resp. real Feynman
diagrams) are labeled by connected $D$-colored graphs (resp. $(D+1)$-colored
graphs) with _no bipartite restriction_ imposed; see Fig. 10 for some
examples. In particular, for $D=3$, the action (3) obeys the desired
$\mathrm{O}(N)^{3}$ symmetry and can rightly be represented as a non-bipartite
$3$-colored graph (${\mathcal{B}}_{T}$ in Fig. 10). The latter can be
interpretated under Pezzana’s duality as a triangulation of a non-orientable
pseudomanifold.141414More precisely, it has the topology of the real
projective plane. From this point of view, it is therefore somewhat misleading
to call it ’tetrahedral’, which explains our use of quotation marks. Focusing
on the $\mathrm{O}(N)^{3}$ model with interaction (3) for simplicity, one can
prove that, with the large $N$ scaling defined by setting
$\alpha=\frac{3}{2}$, a large $N$ expansion exists. In particular, the
partition function expands as [47, 49]

$\ln\int\mathrm{d}\nu(T)=\sum_{\omega\in\frac{\mathbb{N}}{2}}N^{3-\omega}{\mathcal{F}}_{\omega}(\lambda)\,,$
(11)

where $\omega$ is a combinatorial quantity associated to Feynman graphs –
again called _degree_ –, and ${\mathcal{F}}_{\omega}(\lambda)$ is the
generating function of connected $4$-colored graphs with degree $\omega$. The
degree of a graph ${\mathcal{G}}$ is defined as
$\omega({\mathcal{G}}):=3+3p({\mathcal{G}})-F({\mathcal{G}})$ (which agrees
with (7)), and can again be re-expressed as a sum of genera of ribbon graphs
canonically associated to ${\mathcal{G}}$. However, owing to the absence of
bipartite structure in the Feynman graphs, those ribbon graphs do not need to
be orientable, which explains why $\omega$ is in general a positive _half_
-integer. As in complex models, leading order Feynman graphs ($\omega=0$) have
a melonic structure, albeit of a slightly different type: from the point of
view of the Feynman expansion, a melonic insertion has a _bilocal_ structure,
while melonic decorations where always _local_ (that is, tadpole-like) in
complex models. This difference is illustrated in Fig. 8. Generalizing this
construction to arbitrary $D>3$ is not that straightforward, but interactions
supporting a ’bilocal’ melonic large $N$ expansion have been identified for
any _prime_ $D\geq 3$ in [48, 50]: they are associated to certain _complete_
colored graphs (the ’tetrahedral’ interaction being itself represented by a
complete graph on four vertices), and the somewhat surprising prime condition
on $D$ is the result of combinatorial constraints that need to be imposed on
the colorings of such complete graphs.

Finally, colored tensor models with more than one asymptotic parameter have
been discussed in the literature. For instance, reference [51] introduced a
$3$-index random tensor obeying
$\mathrm{U}(N)\times\mathrm{O}(D)\times\mathrm{U}(N)$,151515$D$ here has
nothing to do with the number of indices of the tensor. which can be analyzed
in a double asymptotic expansion in the (formally) small parameters $1/N^{2}$
and $1/\sqrt{D}$. This model can equivalently be understood as a colored
multi-matrix model involving a large number ($D\to+\infty$) of matrices. The
so-called _large $D$ expansion_ of this model [51, 52] was extended to higher
order tensors in [48], and a double-scaling limit in which $\frac{N^{2}}{D}$
is kept fixed was studied in [53, 54, 55]. An interesting aspect of these
models is that they allow to embed a tensor-like melonic expansion of the form
(11) into the more standard genus expansion of a matrix model.

Finally, we briefly come back to the model described around equation (3),
whose symmetry was described by a single copy of $\mathrm{O}(N)$ acting on a
completely symmetric real tensor. This early proposal received renewed
attention after it was convincingly argued in [56] that it should support a
(bilocal) melonic large $N$ expansion if the random tensor is not only taken
to be symmetric, but also _traceless_. This crucial observation, supported by
explicit empirical checks, was proven rigorously in [57] (see also [58]), then
generalized to the other two _irreducible_ $3$-tensor representations of
$\mathrm{O}(N)$ [57, 59]. These results were then extended to $5$-index
irreducible $\mathrm{O}(N)$ tensors ($p=5$) [60], which suggests that bilocal
melonic limits might exist for arbitrary $p\geq 3$ (or, at the very least,
prime $p$). Similar techniques have allowed to partially extend the results of
[51] about the large $D$ expansion of colored matrix models to Hermitian
matrices [61]. In all those works (and particularly so in [57] and [60]), the
absence of colored structure makes the combinatorial proofs traditionally
employed in the analysis of tensor models rather involved. This calls for the
development of alternative methods, which might be better suited to the
analysis of irreducible random tensors. Finally, tensor models with symplectic
symmetry have been considered in [62] ($\mathrm{Sp}(N)$) and [63, 64]
($\mathrm{Sp}(2N,\mathbb{R})$).

###  2.2 Random geometry

It was realized in the 70s and 80s that matrix models can be used to model
continuum random surfaces, or 2d Euclidean quantum gravity [3, 4, 5, 6]. In a
first step, the topological large $N$ expansion is used to generate an
ensemble of random discretizations of two-dimensional orientable surfaces as
in (2). Taking the strict $N\to+\infty$ limit of this partition function
(appropriately rescaled by $1/N^{2}$) produces a generating function
${\mathcal{F}}_{0}(\lambda)$ of discretizations of the $2$-sphere. Each such
discretization can be endowed with a natural metric,161616For instance, one
can attribute the same flat reference metric to all building blocks, and in
this way endow the surface with a piecewise flat metric. In the dual ribbon
graph representation, one may instead rely on the graph distance. so one can
think of the matrix model as producing a random discrete metric space. A
(somewhat heuristic) continuum limit can then be obtained by tuning $\lambda$
to the (real, positive) critical value $\lambda_{c}$ of
${\mathcal{F}}_{0}$.171717This makes sense because ${\mathcal{F}}_{0}$ indeed
has a finite radius of convergence, even though the full partition function
one started from had vanishing radius of convergence. The basic idea is that,
as $\lambda\to\lambda_{c}^{-}$ the sum over surfaces becomes dominated by
discrete geometries involving an arbitrarily large number of elementary
building blocks. The upshot is that the _critical regime_ of matrix integrals
in the $N\to+\infty$ limit may be thought of as encoding continuum random
surfaces of genus $0$.181818As a side note, a non-trivial sum over topology
can be recovered by performing a so-called _double-scaling limit_ , in which
the $N\to+\infty$ and $\lambda\to\lambda_{c}$ limits are performed jointly
while keeping the quantity $N(\lambda-\lambda_{c})^{\alpha}$ fixed, for some
appropriate $\alpha>0$. The probabilistic object behind this limit, known as
the _Brownian map_ [9, 10], has been constructed rigorously as the
$n\to+\infty$ limit of various families of uniform random planar maps with
fixed number $n$ of vertices.191919Each map is equipped with the graph
distance, suitably rescaled by $n^{-1/4}$ in the $n\to+\infty$ limit. The
result is a random metric space which is (almost surely) homeomorphic to a
$2$-sphere, has spectral dimension $2$ and Hausdorff dimension
$4$.202020Convergence is defined in the sense of Gromov-Hausdorff in the space
of all compact metric spaces. Furthermore, the limit is universal in the sense
that it is largely independent of the details of the combinatorial maps being
considered (which in the matrix model picture, is related to the choice of
potential), and it is furthermore equivalent to Liouville quantum gravity [11,
12].

Random tensor models were initially motivated by the challenge of generalizing
the success story of matrix models to Euclidean quantum gravity in $D\geq 3$
dimension. Of all attempts made until now, the framework of complex colored
tensor models seems to be the soundest from a topological point of view,
thanks to its connection to colored diagrams, which can be rigorously
interpreted as higher-dimensional discretized (pseudo)manifolds.212121In
$D=2$, it was proven that uniform colored triangulations of the $2$-sphere do
converge to the Brownian map [65], so we have no a priori reason to reject
colored triangulations as an insufficiently general combinatorial structure to
explore random geometry in $D\geq 3$. Moreover, the large $N$ limit of (10)
generates an ensemble of colored triangulations of the $D$-sphere, as was the
case in $D=2$. The main difference is that these are limited to melonic
triangulations, which constitute a quite restricted set of triangulated
spheres. The fact that they are in bijection with trees actually leads to a
rather simple behaviour as compared to what is found in matrix models. For
instance, the generating function $G(\lambda)$ of rooted melonic
$(D+1)$-colored triangulations, which governs the genus-$0$ sector of the
original colored tensor model (as well as an ’uncolored’ version (6) in which
an infinite set of bubble interactions have been switched on [37]), obeys the
simple polynomial equation:

$G(\lambda)=1+\lambda G(\lambda)^{D+1}\,,$ (12)

which one may recognize as the equation for the generating function of Fuss-
Catalan numbers. The critical properties of $G$, as well as the asymptotic
enumeration of (rooted) melonic $(D+1)$-colored triangulations, can readily be
extracted from this equation [35], leading to

$G(\lambda)\underset{\lambda\to\lambda_{c}^{-}}{\sim}K\sqrt{\lambda_{c}-\lambda}\,,\qquad\\#\\{\mathrm{rooted}\;\mathrm{colored}\;\mathrm{triangulations}\;\mathrm{of}\;\mathrm{size}\;n\\}\underset{n\to+\infty}{\sim}K\lambda_{c}^{n}n^{-3/2}\,.$
(13)

The square-root singularity of $G(\lambda)$ and the exponent $-3/2$ in the
second relation are directly related [66],222222By contrast, the number of
rooted planar maps with $n$ vertices scales asymptotically with an exponent
$-5/2$, which is typical of planar combinatorial objects. and indicative of a
continuum limit governed by Aldous’ _continuous random tree_ [67, 68] (also
known as the _branched polymer phase_ in the physics literature [69, 70]).
This is a random metric space of non-integer spectral dimension $4/3$ (and
Hausdorff dimension $2$), which can be obtained as a scaling limit of uniform
random trees of size $n$ (equipped with the graph distance multiplied by a
corrective factor $n^{-1/2}$). It was confirmed in [71] that uniform melonic
triangulations of size $n$ similarly converge to Aldous’ continuum random
tree, _for any topological dimension_ $D\geq 3$. This is a somewhat
disappointing but rather remarkable universality result: while tensor models
in the melonic scaling (9) offer an interesting generalization of the
topological large $N$ expansion of matrix models, unlike the latter, the
random geometries they generate tend to degenerate into tree-like metric
spaces in the continuum limit. More generally, it is reasonable to expect that
the same result should hold for any other model exhibiting a large $N$ limit
dominated by tree-like combinatorial species. Henceforth, a number of attempts
have been made to evade such tree-like combinatorial behaviour, which we now
briefly describe.

First, double-scaling limits in which the large $N$ and critical limits are
taken simultaneously have been constructed [72, 73, 74], with the idea that
incorporating subleading contributions of tensor models into the continuum
limit might be sufficient to change its universality class. However, the
structure of fixed-degree Feynman graph, as unraveled in the complex colored
setting in [75], seems to invalidate this strategy for the standard melonic
scaling of (9): for any $\omega\in\mathbb{N}$, there is only a finite number
of connected Feynman graphs of degree $\omega$ modulo insertions of melonic
two-point subgraphs and ladder four-point subgraphs (such equivalent classes
of graphs are known as _schemes_). Given that the generating function of
ladder diagrams can be resummed explicitly as a geometric series (see Fig. 9),
there seems to be no room to evade the universality class of branched
polymers. Furthermore, similar results have been obtained within alternative
melonic large $N$ expansions, such as e.g. the $\mathrm{O}(N)^{3}$ model [45,
76, 77, 78, 79].

However, as argued in Sec. 2.1, the theory space of tensor models is much
vaster than that of matrix models. In particular, other scalings than the
melonic one can be considered to obtain alternative but consistent large $N$
limits.232323In fact, due to the proliferation of invariants with arbtrarily
complex combinatorial structures, determining the optimal scaling of a bubble
${\mathcal{B}}$ (i.e. the smallest value of $\alpha({\mathcal{B}})$ in (10)
compatible with the existence of a large $N$ expansion) is a challenging
question in general. See [40] and references therein for further detail. For
instance, it is straightforward to emulate matrix-like large $N$ limits in the
context of a tensor model with $D$ even [80], which demonstrates that the
Brownian map universality class can be obtained within tensor models. More
complicated examples of non-melonic large $N$ limits have been constructed in
[80, 81, 82, 40], but all of them lead to a critical regime which is either
tree-like or planar-like, or that mixes tree-like and planar-like structures.
In $D=3$, a theorem due to Bonzom [83] indicates that any reasonable bubble
interaction (that is, a $3$-colored graph of genus $g=0$, representing the
triangulated boundary of a $3$-ball) will lead to the universality class of
branched polymers.

Further strategies might be considered. For instance, if one attempts to
generate random geometries from other tensor model observables than the
generating function of connected Feynman graphs, the universality class of the
continuum limit might change. Indeed, it was observed in the colored multi-
matrix model of [53] (considered in some triple-scaling limit) that
restricting the sum over graphs to $2$-particle irreducible (or, equivalently,
$3$-edge connected) contributions was sufficient to turn a melonic critical
regime into a model of random maps coupled to the Ising model. Whether such a
combinatorial mechanism might prove useful to generate a higher-dimensional
random geometry from random tensors remains to be explored.

For now, getting out of the universality classes of continuous random trees
and random maps remains a challenge for tensor models. More broadly, obtaining
a non-trivial random metric space with integer spectral dimension $D\geq 3$ as
a scaling limit of discrete geometries is an outstanding open question for any
approach; see e.g. the extensive literature on (causal) dynamical
triangulations ([84] and references therein), or [85, 86] for more specific
proposals aiming at overcoming this roadblock.

###  2.3 Large N (local) quantum field theory

What limits the scope of melonic large $N$ limits in the purely combinatorial
context of random geometry (a universal tree-like behaviour at criticality)
makes them quite useful in the context of large $N$ quantum mechanics and QFT.
More precisely, the _bilocal_ melonic limits described in Sec. 2.1 have been
taken advantage of to explore the strong coupling regime of Q(F)T by means of
analytical methods [87, 88, 89, 90], a feat which is typically out of reach
even at large $N$ (at least in the absence of additional structures, such as
integrable ones).

It was first realized by Witten [91], then by Klebanov and Tarnopolsky [92],
that tensor models are able to emulate the physics of the Sachdev-Ye-Kitaev
model [93, 94, 95]. The latter describes a _disordered_ quantum-mechanical
system of $N$ Majorana fermions $\psi_{a}$ ($a\in\\{1,\ldots,N\\}$), with all-
to-all interaction Hamiltonian $H\sim
J_{abcd}\psi_{a}\psi_{b}\psi_{c}\psi_{d}$. Taking the random couplings
$J_{abcd}$ to be independent centered Gaussian variables with $N^{3}\langle
J_{abcd}^{2}\rangle=\lambda^{2}$ leads to a consistent large $N$ limit (for
the quenched theory). If one takes in addition the strong coupling limit
$\lambda\gg 1$, an emergent time-reparametrization symmetry makes the model
explicitly solvable. This symmetry and the way in which it is (spontaneously
and explicitly) broken, leading to an effective Schwarzian dynamics [95, 96],
is structurally identical to what is found in a two-dimensional model of
gravity known as Jackiw-Teitelboim gravity [97]. As first described by Kitaev
[94], this observation leads to a rather explicit holographic duality between
SYK-like quantum mechanics and $\mathrm{AdS}_{2}$ quantum gravity, which has
received a lot of attention in recent years, notably for its applications to
the theory of quantum chaos and quantum black holes (see e.g. the review
[98]). Interestingly for us here, a key property of the SYK model is that its
large $N$ limit is dominated by _melonic_ Feynman diagrams (see e.g. [99]),
which can be explicitly resummed in the strong coupling regime. To be more
precise, this melonic limit is of a _bilocal_ type, which turns out to be
essential to generate the interesting physics we just hinted at. The
pioneering works of Witten and Klebanov-Tarnopolsky recognized that the
bilocal melonic limits of tensor models can also support SYK-like physics,
with the advantage that _no disorder_ needs to be introduced. Witten’s model
[91] was formulated in the original colored framework proposed by Gurau [26],
while the Klebanov-Tarnopolsky model [92] relied on the ’uncolored’
combinatorial structure of the $\mathrm{O}(N)^{3}$ model [47]. Since the
latter has imposed itself as the large $N$ structure of choice in later works,
we focus on the Klebanov-Tarnopolsky model here. It describes $N^{3}$ Majorana
fermions $\psi_{abc}$ ($a,b,c\in\\{1,\ldots,N\\}$), with action

$S=\int
dt\left(\frac{\mathrm{i}}{2}\psi_{abc}(t)\partial_{t}\psi_{abc}(t)+\frac{\lambda}{4N^{3/2}}\psi_{abc}(t)\psi_{dec}(t)\psi_{dbf}(t)\psi_{aef}(t)\right)\,.$
(14)

Note that the interaction part of the action is given by a ’tetrahedral’
invariant as in (3) (corresponding to the colored bubble ${\mathcal{B}}_{T}$
of Fig. 10), the main difference being the non-trivial time-dependence. We
have in particular a non-trivial propagator, and a deterministic interaction
which is local in time. The structure of the large $N$ limit is identical to
the one described in (11), except that the Feynman amplitudes and fixed-degree
partition functions are more complicated. In particular, in the $N\to+\infty$
we have a closed equation for the full two-point function [92, 94, 95]

$G(t_{1},t_{2})=G_{\mathrm{free}}(t_{1},t_{2})+\lambda^{2}\,\int\mathrm{d}t\mathrm{d}t^{\prime}\,G_{\mathrm{free}}(t_{1},t)\,[G(t,t^{\prime})]^{3}\,G(t^{\prime},t_{2})\,,$
(15)

where $G(t_{1},t_{2}):=N^{-3}\langle
T(\psi_{a_{1}a_{2}a_{3}}(t_{1})\psi_{a_{1}a_{2}a_{3}}(t_{2}))\rangle$ and
$G_{\rm free}(t_{1},t_{2})$ denotes the same quantity at $\lambda=0$. This
equation is structurally similar to (12), and actually results from the same
melonic Schwinger-Dyson equation, as represented in Fig. 9. In the strong-
coupling regime $\lambda\gg 1$, the second term on the right-hand side can be
shown to dominate over the first one [94, 95], which leads to the simplified
equation

$\lambda^{2}\,\int\mathrm{d}t\,G(t_{1},t)\,[G(t,t_{2})]^{3}=-\delta(t_{1}-t_{2})\,.$
(16)

Remarkably, this equation is invariant under reparametrizations of time

$t\to f(t)\,,\qquad
G(t_{1},t_{2})\to|f^{\prime}(t_{1})f^{\prime}(t_{2})|^{1/4}G(f(t_{1}),f(t_{2}))\,,$
(17)

and admits a conformal solution
$G(t_{1},t_{2})\propto\mathrm{sgn}(t_{1}-t_{2})|t_{1}-t_{2}|^{-2\Delta}$ of
conformal dimension $\Delta=1/4$. This solution spontaneously breaks the
reparametrization symmetry down to $\mathrm{SL}(2,\mathbb{R})$, leading to
Goldstone modes $f\in\mathrm{Diff}(S^{1})/\mathrm{SL}(2,\mathbb{R})$.242424We
are assuming an $S^{1}$ topology for the (Euclidean) time direction, which is
appropriate for a thermal path-integral. The effective dynamics of $f$ can in
turn be determined by analysing the explicit breaking of the reparametrization
symmetry introduced by $G_{\rm free}(t_{1},t_{2})$ in (17). This can be done
analytically thanks to another structurally important feature of bilocal
melonic large $N$ expansions: their leading order four-point functions can be
resummed explicitly as geometric series of ladder diagrams. This allows to
compute the leading inverse coupling correction to the conformal dynamics,
which leads to an effective dynamics for the Goldstone modes governed by the
so-called _Schwarzian action_

$S_{\rm
eff}(f)\sim\frac{1}{\lambda}\int\mathrm{d}t\,\\{f,t\\}\,,\qquad\\{f,t\\}=\frac{f^{\prime\prime\prime}(t)}{f^{\prime}(t)}-\frac{3}{2}\left(\frac{f^{\prime\prime}(t)}{f^{\prime}(t)}\right)^{2}\,.$
(18)

The observation that the same Schwarzian dynamics governs boundary modes of
Jackiw-Teitelboim gravity on the hyperbolic disk [97], has led to the
development of the much studied holographic correspondence between SYK-like
many-body models and quantum gravity on $\mathrm{AdS}_{2}$ [98].

Figure 9: In the $N\to+\infty$ limit, the full two-point function of a melonic
QFT obeys a closed Schwinger-Dyson equation (top), while the full four-point
function is a sum of ladder diagrams (bottom).

An interesting feature of the Klebanov-Tarnopolsky model of (14) over the
standard SYK model is that it naturally fits in the framework of local QFT,
and can in particular be generalized to higher spacetime dimensions. This has
been taken advantage of in recent years to construct a new family of large $N$
QFTs [88, 89, 90] which enjoy similarly nice analytical properties as the SYK
model: namely, the full two-point function obeys a closed Schwinger-Dyson
equation at leading order, while information about four-point functions (as
well as higher-point correlators) can be recovered from a resummation of
certain classes of ladder diagrams; see Fig. 9. In particular, one can
consider a real bosonic tensor field $\phi_{abc}(x)$ in $d$ spacetime
dimensions, with $\mathrm{O}(N)^{3}$-invariant (and color symmetric) action

$\displaystyle S(\phi)$
$\displaystyle=\frac{1}{2}\int\mathrm{d}^{d}x\phi_{abc}(x)\left((-\Delta)^{\zeta}+m^{2\zeta}\right)\phi_{abc}(x)$
(19)
$\displaystyle+\frac{1}{4}\int\mathrm{d}^{d}x\left(\frac{\lambda}{N^{3/2}}\mathrm{Tr}_{{\mathcal{B}}_{T}}(\phi(x))+\frac{\lambda_{P}}{N^{2}}\mathrm{Tr}_{{\mathcal{B}}_{P}}(\phi(x))+\frac{\lambda_{D}}{N^{3}}\mathrm{Tr}_{{\mathcal{B}}_{D}}(\phi(x))\right)\,,{}$

where ${\mathcal{B}}_{P}$, ${\mathcal{B}}_{D}$ and ${\mathcal{B}}_{T}$ are the
quartic invariants represented in Fig. 10, and $\zeta\in]0,1]$ is some fixed
parameter. $B_{T}$ is nothing but the ’tetrahedral’ interaction, while
${\mathcal{B}}_{P}$ (’pillow’) and ${\mathcal{B}}_{D}$ (’double-trace’) need
to be included for consistency of the renormalization group flow. This class
of models was first considered in [100] with the standard
$\left(-\Delta+m^{2}\right)^{-1}$ propagator (that is, setting $\zeta=1$ in
(19)) and in $d=4-\epsilon$. It was shown that, at leading order in $1/N$ and
$\epsilon$, the $\beta$-functions of the model admit a non-trivial fixed point
with $\lambda$, $\lambda_{T}$ and $\lambda_{P}$ all real of order
$\sqrt{\epsilon}$. This defines an infrared conformal field theory, which is
however unstable due to the presence of an operator of complex dimension [100,
101]. This instability motivated a modification of the model [102] in which:
1) the propagator is taken to be _long-range_ , meaning that $\zeta<1$ in
(19); 2) the ’tetrahedral’ coupling $\lambda$ is assumed to be _purely
imaginary_. More precisely, setting $\zeta=d/4$ makes the quartic interaction
marginal, and as a consequence of the long-range propagator, there is no wave-
function renormalization. This simplification, together with the melonic
structure, allows to compute flow equations non-perturbatively in the coupling
constants, and to construct explicit renormalization group trajectories that
flow to a unitary CFT in the IR [103, 104, 105]. This result provides a well-
controlled mathematical framework in which to test key paradigms of QFT: for
instance, it was proven that the long-range model in $d=3$ obeys the
$F$-theorem [106]. Finally, the model (19) with purely imaginary ’tetrahedral’
coupling constant was also considered for $\zeta=1$ (short-range) and $d=4$,
where it was observed that the $\beta$-functions are consistent with a well-
behaved and _asymptotically free_ theory in the ultraviolet [107]. Further
results on tensor quantum mechanics and tensor QFT can be found in e.g. [108,
109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123,
124, 125].

Figure 10: Quartic invariants of $\mathrm{O}(N)^{3}$ melonic QFTs.

##  3 Group field theories

###  3.1 Nonlocal field theories of a tensorial type

The framework of GFT [16, 17, 18] can be understood as a further
generalization of tensor models, in which tensor ’indices’ are taken to live
in a Lie group $G$, which we will assume to be compact. This extra group
structure can be exploited to equip the stranded graphs of tensor models with
lattice gauge theory data, and weigh their amplitudes accordingly. In a
similar way as the Feynman expansion of a tensor model can be interpreted as
generating a random (discrete) geometry, the Feynman expansion of a GFT can be
understood as defining a lattice gauge theory on a random lattice.
Prototypical examples are provided by the Boulatov and Ooguri models [19, 20].
They are theories of a single field ${\varphi}\in L^{2}(G^{D})$ obeying the
so-called _gauge invariance condition_ (or _closure constraint_):

$\forall h\in
G\,,\qquad{\varphi}(g_{1}h,g_{2}h,\ldots,g_{D}h)={\varphi}(g_{1},g_{2},\ldots,g_{D})\,.$
(20)

In the template of equation (1), this condition can be imposed by choosing
$\boldsymbol{P}$ to be the orthogonal projector on field configurations
obeying the closure constraint, whose integral kernel is:

$\boldsymbol{P}(g_{1},g_{2},\ldots,g_{D};g_{1}^{\prime},g_{2}^{\prime},\ldots,g_{D}^{\prime})=\int_{G}\mathrm{d}h\prod_{k=1}^{D}\delta\left(g_{k}hg_{k}^{\prime{\mbox{\tiny-1}}}\right)\,,$
(21)

and $h$ in this formula can ultimately be interpreted, on a given Feynman
graph, as parallel transport between two interaction vertices connected by a
propagator line. The interaction originally considered by Boulatov [19], with
$D=3$ and $G=\mathrm{SU}(2)$, was a ’tetrahedral’ interaction analogous to
(3),252525The Ooguri model is similarly defined in $D=4$, with an interaction
following the combinatorial pattern of a $4$-simplex. or in its colored
version

$S_{\rm
int}({\varphi})=\frac{-\lambda}{4}\int_{G^{6}}{\varphi}(g_{1},g_{2},g_{3}){\varphi}(g_{4},g_{5},g_{3}){\varphi}(g_{4},g_{2},g_{6}){\varphi}(g_{1},g_{5},g_{6})\,.$
(22)

Complex variants of this model, based on bipartite colored bubble
interactions, started to be considered in the original papers on melonic large
$N$ expansions [25, 26], and made the structure of the random lattice better
defined from a topological standpoint [34, 36]. In this framework, the
Boulatov model generates $BF$ lattice gauge theory amplitudes which are
topological invariants (modulo divergences that need to be regularized in an
appropriate way). In the special case $G=\mathrm{SU}(2)$, those can be
interpreted as three-dimensional Euclidean quantum gravity path-integrals
expressed in holonomy-flux variables, and are mapped upon Fourier transform to
the Ponzano-Regge state-sum model [126]. This correspondence between the
Boulatov and Ponzano-Regge models has motivated numerous further works
exploring the relation between GFT and quantum gravity, and more specifically
so in the context of loop quantum gravity and spin foam models [21, 22, 127,
128, 129, 23, 24, 130, 131].262626This is where the term _group field theory_
actually originates from. From this perspective, GFT provides a tentative
albeit natural completion of spin foam models,272727See [132] for an
alternative approach. one that enables the use of standard QFT concepts and
methods (e.g. renormalization) to explore their phase structure. The most
studied spin foam models of four-dimensional quantum gravity (see e.g. [133,
134] for reviews), which share certain features of the simpler models we will
restrict our attention to, can all be embedded into the GFT framework.

The GFT formalism aims at defining QFTs _of_ spacetime, rather than _on_
spacetime. It is therefore not constrained by the same notion of (spacetime)
locality that is so fundamental in standard QFT. Indeed, from a mathematical
standpoint, a GFT can be understood as a _non-local_ QFT on the manifold
$G^{D}$, in which interactions obey a particular kind of non-locality
(sometimes referred to as _combinatorial non-locality_): for instance, the
interaction (22) involves an integration on $G^{6}$, which makes it non-local,
but it has a rather specific structure in which individual group elements in
$G$ interact locally by pairs. This pairwise locality is the essence of GFT;
it constrains the kind of interactions one can legitimately consider in a
similar way as the ordinary notion of spacetime locality does in standard QFT.

Like any other QFT, a GFT is typically plagued with divergences which arise
from the short scale structure of the configuration space $G^{D}$. These must
be regularized and renormalized away in some consistent manner. Two approaches
have been considered to achieve this goal. In the first, one looks for an
asymptotic large $N$ expansion analogous to the ones described in Sec. 2 for
purely combinatorial models, where $N$ is identified to a cut-off within a
suitable regularization scheme. For instance, one may consider a regularized
version of the propagator (21)

$\boldsymbol{P}_{1/N^{2}}(g_{1},g_{2},\ldots,g_{D};g_{1}^{\prime},g_{2}^{\prime},\ldots,g_{D}^{\prime})=\int_{G}\mathrm{d}h\prod_{k=1}^{D}K_{1/N^{2}}\left(g_{k}hg_{k}^{\prime{\mbox{\tiny-1}}}\right)\,,$
(23)

where $K_{1/N^{2}}$ is the heat-kernel at time $1/N^{2}$ on $G$, which imposes
a smooth cut-off of order $N$ on dimensions of irreducible representations
labelling harmonic modes.282828E.g. when $G=\mathrm{SU}(2)$, we have
$K_{\alpha}(g)=\sum_{j\in\frac{\mathbb{N}}{2}}\exp\left(-\alpha
j(j+1)\right)\chi_{j}(g)\,,$ (24) where $\chi_{j}$ is the character of the
spin-$j$ irreducible representation of $\mathrm{SU}(2)$. This has the effect
of regularizing any possible divergence of models of the Boulatov-Ooguri type,
and one may then investigate which asymptotic scalings of their bubble
interactions lead to consistent large $N$ limits. After pioneering steps were
made in this direction [135, 136, 41, 127, 137, 138, 139], the first large $N$
results of Gurau opened the way to a thorough investigation of complex and
colored versions of the Boulatov-Ooguri models, which were found to obey a
consistent large $N$ expansion dominated by melonic diagrams [26, 27, 28, 140,
141, 142]. The critical properties of the leading-order sector have also been
analyzed in some respects [143], but remain significantly less understood than
for colored tensor models. In particular, the natural metric interpretation of
GFT Feynman graphs is function of their combinatorial structure, but also of
the group data attached to their edges. This makes a precise analysis of GFT
models of random geometry difficult. To our knowledge, no definitive results
have been obtain regarding the metric properties of their continuum limit.

The second approach to a rigorous definition of GFTs embraces their
interpretation as QFTs, and looks for an appropriate generalization of the
full machinery of (perturbative and non-perturbative) renormalization. An
important challenge is that no a priori notion of spacetime scale is available
in GFT, and neither is the usual notion of locality, so it is not immediately
clear what one would even mean by a flow of local interactions parametrized by
a scale. This problem was first addressed and resolved in the context of GFTs
in which no gauge invariance condition has been imposed, which are known in
the literature as (_tensor_ or) _tensorial field theories_ [144, 145, 146]
(not to be confused with the local field theories of Sec. 2.3!). Such models
do not exploit the group structure to produce a sum over lattice gauge theory
or quantum gravity amplitudes, and have in particular no direct connection to
spin foams, but they remain mathematically interesting in their own right.

###  3.2 Renormalization

In standard QFT, the renormalization group method relies on an harmonious
interplay between spacetime scales and locality, in the sense that high energy
radiative corrections to local interactions remain local as seen from low
energy. In tensorial field theories, it was proposed that colored tensorial
invariance should play the role of locality, meaning that, in complete analogy
with tensor models, basic interactions should be labeled by connected colored
graphs encoding pairwise convolutions of the group variables of the field. In
an ordinary QFT, the energy scale ladder can be put in correspondence with the
non-trivial spectrum of the propagator. In the GFT context, this motivates the
introduction of a more abstract notion of scale, defined as an element of the
spectrum of the propagator. Obviously, such a definition is relevant only in
situations when the propagator has a rich spectrum, which is not the case of
e.g. the Boulatov-Ooguri propagator (21) (since it is a projector). This
observation, together with other arguments [147], motivated the introduction
of tensorial field theories and GFTs with non-trivial propagators (such as
e.g. $(-\Delta+m^{2})^{-1}$ where $\Delta$ is the Laplace-Beltrami operator on
$G^{D}$).

This strategy was first proposed in the pioneering work [144], where it was
proven that the abstract notions of locality and scale thus defined can
support a consistent perturbative renormalization scheme for tensorial field
theories. Moreover, a first all-order perturbative renormalizability theorem
was obtained by means of rigorous multiscale renormalization methods [148].
Further results of this kind were obtained in subsequent works, such as [146,
149, 150, 151, 152, 153]. Interestingly, there is a close analogy between the
perturbative renormalization analysis of tensorial field theories and the
large $N$ analysis of tensor models. The _superficial degree of divergence_ of
a Feynman graph ${\mathcal{G}}$ can be expressed in terms of Gurau’s degree
(7), while scaling dimensions of perturbative coupling constants are analogous
to the scaling parameters $\alpha({\mathcal{B}})$ from (6). It is also crucial
in the renormalization analysis that divergent graphs tend to have a simple
combinatorial structure (such as a melonic one). Hence, from a qualitative
point of view, it is possible to interpret the renormalization group flow of a
tensorial field theory as dynamically generating a consistent asymptotic large
$N$ scaling, similar to those that are being imposed by hand in the tensor
model setting.

The perturbative renormalization analysis of tensorial field theories was
generalized in [154, 155, 156, 157] to GFT proper, that is to say to tensor
field theories with gauge invariance condition (20). To be more precise, the
class of GFT models considered in those works are complex models with
regularized propagator292929This is a regularized version of
$(-\Delta+m^{2})^{-1}$, in which the eigenvalues of $-\Delta+m^{2}$ of order
$\Lambda^{2}$ or larger have been smoothly cut-off.

$\boldsymbol{P}_{\Lambda}(g_{1},g_{2},\ldots,g_{D};g_{1}^{\prime},g_{2}^{\prime},\ldots,g_{D}^{\prime})=\int_{1/\Lambda^{2}}\mathrm{d}\alpha\,e^{-\alpha
m^{2}}\int_{G}\mathrm{d}h\prod_{k=1}^{D}K_{\alpha}\left(g_{k}hg_{k}^{\prime{\mbox{\tiny-1}}}\right)\,,$
(25)

and interactions labelled by connected $D$-colored diagrams. The _superficial
degree of divergence_ of a Feynman graph ${\mathcal{G}}$ has been determined
to be of the form [156]

$\omega({\mathcal{G}})=-2L({\mathcal{G}})+d_{G}\left(F({\mathcal{G}})-R({\mathcal{G}})\right)\,,$
(26)

where $L({\mathcal{G}})$ is the number of propagator lines of ${\mathcal{G}}$,
$F({\mathcal{G}})$ its number of bi-colored cycles involving the color $0$ (or
_faces_), $R({\mathcal{G}})$ is the rank of the adjacency matrix between its
lines and faces, and $d_{G}$ the dimension of $G$. Divergences arise from
subgraphs with $\omega\geq 0$, and formula (26) can be exploited to classify
the set of potentially renormalizable models in terms of the free parameters
$D$, $d_{G}$, as well as the list of bubble interactions included in the
action [156, 158]. And just like for models with no gauge invariance, the
multiscale renormalization methods of [148] turn out to be well-suited to the
proof of all-order perturbative renormalizability theorems, because they allow
to efficiently deal with the renormalization of overlapping divergent
subgraphs. In particular, it was shown that with $D=3$ and $G=\mathrm{SU}(2)$,
the sextic model involving the first four interactions shown in Fig. 5 is
perturbatively renormalizable at all orders [156], and its discrete multiscale
flow equations were analyzed in detail in [159]. The fact that ultraviolet
divergences can be reabsorbed into effective interactions which still obey the
criterion of tensorial locality is non-trivial in this model; it relies on a
nice interplay between the topology of the Feynman graphs (as defined via
Pezzana’s duality) and the lattice gauge theory structure of their amplitudes.
We refer to the review [158] and references therein for more details on this
class of models.

Other interesting results have been obtained about the renormalization of
tensorial field theories and GFTs. To begin with, it was established in [149]
that tensor field theories can generate renormalization group flows which are
_asymptotically free_ in the utraviolet. This may appear surprising given
their superficial similarity to standard scalar field theories (which are not
compatible with asymptotic freedom), but it turns out to be a consequence of
the non-trivial combinatorial structure of tensorial interactions. In [160],
it was shown that wave-function renormalization tends to dominate over vertex
renormalization, and that, as a result, asymptotic freedom is a robust feature
of quartic models. The situation is more complicated in models involving
marginal interactions of higher order, as exemplified by the sextic model of
[156], which fails to be asymptotically free for somewhat subtle reasons
[160]. Nonetheless, it became clear from these results that tensor field
theory provides non-trivial examples of QFTs which are mathematically
interesting in their own right, independently from their eventual relation to
GFT or quantum gravity. Since then, tensor field theory has been used as a
test bed for constructive renormalization programs: for instance, the
multiscale renormalization methods of [148, 161, 162] have been employed to
construct renormalizable tensorial field theories in [163, 164, 165], and more
recently, such models were revisited through the lenses of stochastic analysis
[166]. In another direction, the perturbative renormalization struture
underlying tensor field theories was shown to fit in the general Hopf-
algebraic framework due to Connes and Kreimer [167, 168, 169, 170]. Finally, a
number of efforts have been made to map out the non-perturbative fixed points
of tensor field theories and GFTs by means of Functional Renormalization Group
(FRG) methods. This program started out with the analysis of [171, 172], which
focused on models without gauge invariance condition, and was then applied to
proper GFTs in [173, 174, 175, 176, 177, 178]. As far as quantum gravity is
concerned, it is hoped that non-perturbative FRG methods in appropriate
truncations will help map out the phase diagrams of GFTs in the infrared
regime, and eventually allow to locate non-perturbative fixed points which
have the potential to lead to interesting continuum limits. We refer to the
recent literature [179, 180, 181, 182] for a detailed account and status of
this search (see also [183, 184] for applications of functional
renormalization to the search of consistent large $N$ limits).

We conclude this section with two open questions regarding the renormalization
of GFTs. First, while rigorous perturbative renormalization theorems are
already available for simple classes of GFTs (tensor field theories, and GFTs
resembling the Boulatov-Ooguri models), can those results and the methods they
rest on be extended to four-dimensional quantum gravity proposals (e.g. of the
type [23, 24, 130, 185, 131])? Second, how could one go about proving the
existence of the non-perturbative fixed points which are empirically observed
in truncated FRGs?

###  3.3 Further developments

We conclude by listing some important research directions on GFT which we
could not give justice to in this short review.

In recent years, there has been a strong push towards applications of the GFT
formalism to cosmology, and other symmetry-reduced regimes of general
relativity. A mechanism of Bose-Einstein condensation adapted to GFT was
investigated and argued to predict effective Friedmann equations from full-
fledged quantum gravity [186, 187, 188]. A similar construction was also
applied to black holes in [189, 190]. This approach is grounded in the Fock
space formalism of [130]; it allows to extract a low energy effective dynamics
from GFT by means of well-established methods originating from standard
quantum many-body physics. We refer to the reviews [191, 192, 193] for more
complete accounts of GFT cosmology.

The same many-body physics interpretation has also allowed to analyze GFT
wave-functions through the lenses of entanglement theory. In particular,
holographic entanglement area laws [194, 195, 196] have been derived in GFT,
thus establishing interesting connections with holographic tensor networks
[197, 198].

While the GFTs we focused on in the previous two subsections assumed a
_compact_ Lie group $G$, GFT models of _Lorentzian_ quantum gravity typically
require $G$ to be _non-compact_ (see e.g. [199, 131]). From the point of view
of renormalization, this leads to an additional infrared problem, which was
approached by means of FRG methods in [172, 175]. Interestingly, in recent
works [200, 201], the non-compact nature of $G$ was taken advantage of to
develop new mean-field approximations, which provide useful information about
those models while a complete renormalization group treatment awaits.

Finally, we note that a certain type of effective melonic quantum dynamics was
recently suggested to arise from a generalized Boulatov model [202]. This may
lead to interesting new connections between GFT and the large $N$ QFT
framework described in Sec. 2.3, which are a priori quite different (both in
intent and content).

## Acknowledgements

I would like to thank Razvan Gurau, Daniele Oriti and Vincent Rivasseau for
their pioneering works, their contagious enthusiasm for this subject, and for
teaching me much about it in the first place.

The IMB receives support from the EIPHI Graduate School (contract
ANR-17-EURE-0002).

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