Flowing in Group Field Theory Space: a Review
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≥3. More precisely, we focus on GFTs with so-called closure cons...
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irk-123456789-1477672019-02-16T01:26:11Z Flowing in Group Field Theory Space: a Review Carrozza, S. 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≥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. 2016 Article Flowing in Group Field Theory Space: a Review / S. Carrozza // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 87 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81T15; 81T16; 83D27; 83C45 DOI:10.3842/SIGMA.2016.070 http://dspace.nbuv.gov.ua/handle/123456789/147767 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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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≥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. |
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Carrozza, S. Flowing in Group Field Theory Space: a Review Symmetry, Integrability and Geometry: Methods and Applications |
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Carrozza, S. |
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Carrozza, S. |
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Flowing in Group Field Theory Space: a Review |
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Flowing in Group Field Theory Space: a Review |
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Flowing in Group Field Theory Space: a Review |
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Flowing in Group Field Theory Space: a Review |
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Flowing in Group Field Theory Space: a Review |
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flowing in group field theory space: a review |
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Flowing in Group Field Theory Space: a Review / S. Carrozza // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 87 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 070, 30 pages
Flowing in Group Field Theory Space: a Review?
Sylvain CARROZZA
Université Bordeaux, LaBRI, UMR 5800, 33400 Talence, France
E-mail: sylvain.carrozza@labri.fr
Received March 08, 2016, in final form July 13, 2016; Published online July 16, 2016
http://dx.doi.org/10.3842/SIGMA.2016.070
Abstract. 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 ≥ 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.
Key words: group field theory; quantum gravity; quantum field theory; renormalization
2010 Mathematics Subject Classification: 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 Gd. 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 group1 (both technically and for
?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
1Note 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.
mailto:sylvain.carrozza@labri.fr
http://dx.doi.org/10.3842/SIGMA.2016.070
http://www.emis.de/journals/SIGMA/Tensor_Models.html
2 S. Carrozza
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 amplitudes2 [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 renormaliza-
tion 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 renor-
malizable 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 ϕ 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
g = (g1, . . . , gd) ∈ Gd, dg = dg1 · · · dgd.
2Spin foam amplitudes fall in the class of generalized lattice gauge theories which may be generated by a GFT.
Flowing in Group Field Theory Space: a Review 3
We will also use the short-hand notation:
ϕ1 · ϕ2 =
∫
dgϕ1(g)ϕ2(g),
for any two square-integrable functions ϕ1 and ϕ2 on Gd. The dynamics of the GFT field is
specified by a probability measure
dµCΛ
(ϕ,ϕ) exp (−SΛ[ϕ,ϕ]) (2.1)
or equivalently a generating functional
ZΛ[J, J̄ ] =
∫
dµCΛ
(ϕ,ϕ) exp
(
−SΛ[ϕ,ϕ] + J̄ · ϕ+ ϕ · J
)
. (2.2)
The measure dµCΛ
is the Gaussian measure associated to the covariance CΛ, which is a positive
operator with kernel:
CΛ(g;g′) =
∫
dµCΛ
(ϕ,ϕ)ϕ(g)ϕ(g′).
In other words, CΛ is the propagator of the GFT, or equivalently its free 2-point function. We
explicitly introduced an extra regularization or scale parameter Λ > 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 SΛ[ϕ,ϕ], which we will parametrize as
SΛ[ϕ,ϕ] =
∑
b∈B
tb(Λ)Ib[ϕ,ϕ],
where B is a set of elementary interactions, Ib is the specific monomial in the fields associated to
the interaction b, and tb(Λ) is the running coupling constant associated to b at scale Λ. One main
objective of renormalization is to determine how these coupling constants should be adjusted to
compensate for a change in the cut-off Λ. More precisely, as in ordinary field theory, we will
require that the infrared content of the connected 2k-point functions (or Schwinger functions)
S
(2k)
Λ (g1, . . . ,gk,g
′
1, . . . ,g
′
k) =
(
k∏
i=1
δ
δJ̄(gi)
δ
δJ(g′i)
)
lnZΛ[J, J̄ ]
∣∣∣∣∣
J=J̄=0
,
which also characterize the random measure (2.1), is invariant under a change of Λ. 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 expan-
sion of (for instance) the partition function
ZΛ := ZΛ[0, 0] =
∑
G
∏
b∈B
(−tb(Λ))nb(G)AG(Λ), (2.3)
generates Feynman diagrams G which are in one-to-one correspondence with specific (closed)
2-complexes and are weighted by spin foam amplitudes AG3. We remind the reader that, more
3Note 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.
4 S. Carrozza
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 renorma-
lizability of the GFT. At the very least one needs to check that – again formally – the set
of GFT interactions Ib is stable under a shift of the cut-off Λ. 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 re-
course to GFT. They will generate new vacua, supporting new and possibly inequivalent rep-
resentations 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
Flowing in Group Field Theory Space: a Review 5
SO(1, d− 1) or its universal covering. In this review, we will only consider Euclidean groups –
in particular SU(2) in dimension d = 3 – and ignore complications arising from the Lorentzian
signature4. 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:
∀h ∈ G, ϕ(g1h, . . . , gdh) = ϕ(g1, . . . , gd). (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 CΛ is
of the form
CΛ = PC̃ΛP,
where C̃Λ is again a positive operator, and P is the projector on translation invariant GFT fields
with kernel
P(g,g′) =
∫
G
dh
d∏
`=1
δ
(
g`hg
′−1
`
)
.
Hence CΛ 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 SU(2)
BF theory can be interpreted as a theory of Euclidean gravity, and therefore 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 C̃Λ 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 SΛ
comprising a unique type of monomials Ib, which are of order (d + 1) and contract the field
variables pairwise following the pattern of a d-simplex. Such models have very rigid combina-
torial 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 SΛ 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 interac-
tions 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 ten-
sor 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
4The 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.
6 S. Carrozza
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 ≥ 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 {ϕc | c = 1, . . . , d}, with
covariance C̄Λ not necessarily identical to CΛ. The label c is called color, and we conventionally
associate the color 0 to the original field ϕ ≡ ϕ0. The action SΛ is then implicitly defined by
exp
(
−SΛ[ϕ,ϕ]
)
=
∫ [ d∏
c=1
dµC̄Λ
(ϕc, ϕc)
]
exp
(
−Scol
Λ [ϕ,ϕ;ϕc, ϕc]
)
, (2.5)
where the colored GFT action is
Scol
Λ [ϕ0, ϕ0;ϕc, ϕc] = λ(Λ)
∫
∏
0≤`<`′≤d
dg``′
d∏
`=0
ϕ`(g`)
+ λ̄(Λ)
∫
∏
0≤`<`′≤d
dg``′
d∏
`=0
ϕ`(g`) (2.6)
and we have used the convention that g``′ = g`′` together with the notation
g` = (g``−1, . . . , g`0, g`d, . . . , g``+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 pic-
torially represented by white (resp. black) nodes where (d + 1) colored half-edges meet as in
Fig. 1. An half-edge of color ` is associated to a GFT field ϕ` or ϕ`, and is dual to a (d − 1)-
simplex of color `. A pair of edges of colors ` and `′ in turn encodes the integral over the
variable g``′ 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.
Note that half-lines associated to fields ϕ and ϕ (with color 0) are dashed. This is to empha-
size 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 ϕc for
c = 1, . . . , d can therefore be (formally) integrated out. This yields effective interactions Ib
parametrized by d-colored graphs b involving colors c = 1 to d. These graphs are also called
Flowing in Group Field Theory Space: a Review 7
0
1
2
3
1
3
2 0 0 0
11 2 2
3 3
4 4
Figure 1. Colored interaction vertices in dimension d = 3 (left) and d = 4 (right).
01 02 03
0
21 20 23
2
1 3
10
13
12 32
31
30
0
1
2
3 0
12
3
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.
bubbles in the literature (e.g., in [25, 50]; examples in d = 3 are provided in Fig. 3. The coupling
constant tb is moreover proportional to5 (λλ̄)Nb/2, with Nb the number of nodes in the colored
graph b.
The precise form of the effective interaction Ib highly depends on the auxiliary covariance C̄Λ
and is in general quite involved. Let us discuss the special and simple situation in which
CΛ(g;g′) =
d∏
`=1
δΛ
(
g`g
′−1
`
)
, (2.7)
where δΛ is a regularized version of the delta function on G. More precisely, we assume that Λ
is a sharp cut-off in the Fourier expansion of δ6. Under this condition, it can be shown that
the effective Ib are nothing but tensor invariants (up to constant factors and powers of the
cut-off Λ 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 Ib ≡ Trb.
A monomial Trb(ϕ,ϕ) is uniquely determined by its d-colored bubble b, under the following
rules:
5The proportionality factor depends on the combinatorial structure of the bubbles alone, not on the other
ingredients of the model.
6For instance, when G = SU(2), one may define
δΛ(g) :=
∑
j∈ N
2
|j(j+1)≤Λ2
(2j + 1)χj(g),
where χj are the characters of SU(2).
8 S. Carrozza
• a white (resp. black) node of b is associated to a field ϕ (resp. ϕ);
• an edge of color ` represents a convolution of two field variables, both appearing in the
`th copy of the group G.
An example is provided in Fig. 4. In this simplified context, tensor invariant interactions gene-
rate an infinite-dimensional GFT theory space in which colored simplicial models are embedded
as a one-parameter family of models7. 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. 58.
Note however that bubbles may also be dual to topologically singular9 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.
ϕ
1
1
2
2
3 3b =
Trb(ϕ,ϕ) =
∫
[dgi]
6 ϕ(g1, g2, g3)ϕ(g1, g2, g4)
×ϕ(g5, g6, g3)ϕ(g5, g6, g4)
ϕ ϕ
ϕ
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. The first possibility is to choose the covariances CΛ and C̄Λ equal. Both are in particular
degenerate, and the effective interactions Ib become quite complicated and hard to mani-
pulate in concrete calculations. From the point of view of known spin foam models, which
7Colored simplicial models with covariance (2.7) generate coupling constants tb which are functions of λλ̄,
hence the one-dimensional character of this subspace of theories.
8Note 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].
9A 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.
Flowing in Group Field Theory Space: a Review 9
are derived from simplicial discretizations of formal quantum gravity path-integrals, this
is however the most natural construction.
2. The second possibility is to assume, as we have done before, that the auxiliary colored
fields ϕc have trivial covariance C̄Λ. One may argue in this case that imposing suitable
spin foam constraints on the remaining dynamical field ϕ 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 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 = SU(2). The propagator with cut-off may
be defined as
CN (g1, g2, g3; g′1, g
′
2, g
′
3) =
∫
SU(2)
dh
3∏
`=1
K1/N2
(
g`hg
′−1
`
)
−→
N→+∞
P(g1, g2, g3; g′1, g
′
2, g
′
3),
where Kα is the heat-kernel on SU(2) at time α10, and we denote the cut-off by N instead
of Λ in reference to the original literature [51, 52, 54]. The auxiliary covariances appearing in
formula (2.5) are furthermore taken equal to the propagator: C̄N = CN . In [51] Gurau showed
that, under the assumption that the coupling constant asymptotically behaves like
λ(N) ∼
N→+∞
λ0
N3/2
for some fixed λ0, the colored Boulatov model admits a 1/N -expansion. In particular, the
partition function can be expanded as
ZN = N6Z0(λ0λ̄0) +N3Z1(λ0λ̄0) +O(1), (2.8)
where Z0 and Z1 sum over specific infinite families of Feynman diagrams representing spherical
manifolds11. 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 con-
tributions 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 methods12. Sin-
10This defines a regularization of the delta function in which high spin representations are smoothly cut-off:
Kα(g) =
∑
j∈ N
2
e−αj(j+1)(2j + 1)χj(g).
11The family summed over by Z0 – 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 Z1 has as far as we know not been
studied in great details, but it is nonetheless known that it sums spherical manifolds [52].
12In particular, tighter bounds were derived, showing that singular topologies are suppressed in at least N3(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].
10 S. Carrozza
gular topologies having no natural space-time interpretation at this point13, 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 Z0 and Z1. 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 SN of the form
I2(ϕ,ϕ) =
∫
dgdg′ ϕ(g)ϕ(g′)
∫
[dhi]
3K1/N2
(
h1h
−1
2
)
K1/N2
(
h1h
−1
3
)
K1/N2
(
h2h
−1
3
)
×
3∏
i=1
K1/N2
(
gihig
′−1
i
)
.
Using the gauge invariant condition (2.4), one is free to translate the gi variables by (say) h−1
3 .
This reduces the hi dependence of the last line to a dependence in h−1
3 h1 and h−1
3 h2. Performing
the change of variables h1 → h3h1 and h1 → h3h2, we hence obtain an integral which is
completely independent of h3. By normalization of the Haar measure, the new expression of I2
is thus
I2(ϕ,ϕ) =
∫
dgdg′ ϕ(g)ϕ(g′)
∫
dh1dh2K1/N2
(
h1h
−1
2
)
K1/N2(h1)K1/N2(h2)
×K1/N2
(
g1h1g
′−1
1
)
K1/N2
(
g2h2g
′−1
2
)
K1/N2
(
g3g
′−1
3
)
. (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 h1 = h2 = 1l; together
with the second line, this implies that the integrand is also sharply peaked around gi = g′i. This
simple fact entitles us to expand ϕ(g′) in Taylor expansion around g:
ϕ(g′) = ϕ(g) +
d
dt
∣∣∣∣
t=0
ϕ(g(t)) +
1
2
d2
dt2
∣∣∣∣
t=0
ϕ(g(t)) + · · · ,
where g′(t) is an affinely parametrized geodesic from g to g′ in SU(2)3. This reduces I2 to
an infinite sum over tensor invariant contractions of the fields and their derivatives. More
precisely, this procedure can only generate SU(2)-invariant differential operators acting on each
copy of SU(2) and one therefore obtains:
I2(ϕ,ϕ) = a(Λ)
∫
dgϕ(g)ϕ(g) + b(Λ)
∫
dgϕ(g)
(
−
3∑
`=1
∆`
)
ϕ(g) + · · · , (2.10)
where ∆` is the Laplace operator acting on the `th copy of SU(2), and a(Λ), b(Λ) 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 Ib 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 B ⊃ B
labelling generalized trace invariants Trb(ϕ,ϕ). The original bubble interactions Trb(ϕ,ϕ) (with
13While metric singularities are generic in general relativity, topological singularities are completely absent of
standard models of classical space-time.
Flowing in Group Field Theory Space: a Review 11
b ∈ 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 Spin(4) in
dimension 4. It remains however to understand how this heuristic calculation may be genera-
lized 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 ϕ over d ≥ 3 copies of a compact Lie group G.
Its free 2-point function is assumed to be of the form
CΛ(g;g′) =
∫ +∞
1/Λ2
dα
∫
G
dh
d∏
`=1
Kα
(
g`hg
′−1
`
)
, (3.1)
which is a regulated version of the formal operator C = P
(
−
d∑
`=1
∆`
)−1
P, Kα and ∆` being
respectively heat-kernels and Laplace operators on G. Equation (3.1) is known as the Schwinger
representation of the propagator, and α is accordingly called a Schwinger parameter. As for the
interaction action SΛ, we assume that it is generated by tensor invariants14:
SΛ[ϕ,ϕ] =
∑
b∈B
tb(Λ) Trb[ϕ,ϕ]. (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 tb 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 SΛ
a local potential15. 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 ∆`.
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
14Note 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.
15Even though we cannot define a potential function, due to the combinatorially non-trivial nature of tensorial
locality.
12 S. Carrozza
Figure 6. A bubble with two connected components.
tensor invariant:
(∫
dgϕ(g)ϕ(g)
)
·
(∫
dgϕ(g)ϕ(g)
)
,
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.
In order to legitimize the class of TGFTs thus defined as a perfectly honest arena for renorma-
lization, we need to comment a bit more on the notion of scale in this context. For definiteness,
let us specialize to G = 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
p2 =
d∑
`=1
j`(j` + 1),
where {j`} are the spin labels associated to the harmonic expansion of the fields. Modes associa-
ted to p ≥ Λ 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-off16. 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 cons-
tants tb(Λ), 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 pro-
gramme 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 per-
spective 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 interest-
ing 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 com-
prising 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
16We 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.
Flowing in Group Field Theory Space: a Review 13
g1
g2
g3
=
∫
dg1 dg2 dg3 . . .
= δ(gg̃−1)
g g̃
g1 g̃1
g2
g3 g̃3
g̃2 = CΛ(g1, g2, g3; g̃1, g̃2, g̃3)
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.
will necessitate a non-perturbative treatment of the GFT renormalization group17. 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 classif ication 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 G, we will denote by L(G), V (G) and N(G) its
set of (internal) dashed lines, bubble vertices and external legs18. Accordingly, the amplitude
associated to a graph G is determined by the following Feynman rules: each node is associated
to an integration over Gd; each colored line internal to a bubble represents a delta function on G;
and finally, dashed lines must be replaced by kernels of CΛ. We provide a 3d example in Fig. 7.
The multiscale analysis19 relies on a discrete slicing of the propagator:
CΛ = CMρ =
∑
i∈N|i≤ρ
Ci, (3.3)
where
C0 :=
∫ +∞
1
dα
∫
G
dh
d∏
`=1
Kα
(
g`hg
′−1
`
)
,
∀ i ≥ 1, Ci :=
∫ M2(i−1)
M−2i
dα
∫
G
dh
d∏
`=1
Kα
(
g`hg
′−1
`
)
.
M > 1 is a fixed but arbitrary slicing parameter and we have assumed that the UV cut-off
is of the form Λ = Mρ with ρ ∈ N. Each covariance Ci is then essentially responsible for the
17Another 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.
18As is customary in the literature, we will allow ourselves to use the same notation for the cardinals of these
sets.
19In 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.
14 S. Carrozza
propagation of the modes with M i−1 . p . M i. This procedure induces a decomposition of
Feynman amplitudes
AG =
∑
µ
AG,µ, (3.4)
according to scale attributions µ := {il ∈ N, l ∈ L(G)}. In this formula, AG,µ is an amplitude
constructed from the sliced propagators {Ci} rather than the full propagator CΛ: that is, to each
line l, we now associate the propagator Cil . The multiscale strategy then consists in looking
for estimates of each of the amplitudes AG,µ separately, rather than of the full amplitude AG .
The scale attributions µ have the considerable advantage that they allow to optimize the naive
bounds one would derive for AG , and as a consequence to more precisely understand the origin
of the divergences. For instance, given a graph G with scale attribution µ, one can realize that
divergences may only be generated by high subgraphs: these are defined as subgraphs H ⊂ 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 bounds20, 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 G is defined as a maximal bicolored
path in G, with the restriction that one of the two colors must be 0. For convenience, we will
simply attribute the color ` to a face consisting of a path with colors 0 and `. We will say that
a line l ∈ L(G) pertains to f (l ∈ 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 G or not. We will denote by F (G)
(resp. Fext(G)) the set of closed (resp. open) faces of 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 εlf , of size L × F and with only 0 or 1 entries: εlf = 1 if
l ∈ f , and εlf = 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
K∑
l∈f
αl
−→∏
l∈f
hl
,
where αl and hl 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
K∑
l∈f
αl
gs(f)
−→∏
l∈f
hl
g−1
t(f)
,
where gs(f) and gt(f) are boundary variables associated to the fields sitting at the source (s(f))
and target (t(f)) ends of f .
20This is due to the fact that the associated 2-complexes are simply connected, see again [27, 28].
Flowing in Group Field Theory Space: a Review 15
f
1 1
1
α1 , h1
α3 , h3 α2 , h2
←→ Kα1+α2+α3
(h1h2h3)
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 G is
ω(G) := −2L(G) +D (F (G)−R(G)) ≥ 0,
where R(G) is the rank of the adjacency matrix εlf of 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., µ = {i` = i, l ∈ L(G)}), one can show thatAG,µ has an exponential
scaling of the form Mω(G)i when i → +∞21. This corresponds to the situation in which 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 de-
gree, 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 that22
ω = D(d− 2)− D(d− 2)− 2
2
N +
∑
k∈N∗
[(D(d− 2)− 2)k −D(d− 2)]n2k +Dρ, (3.5)
where N is the number of external legs and n2k the number of bubbles of valency k. The whole
non-trivial dependence in the rank has been included in the combinatorial quantity
ρ := F −R− (d− 2)(L− V + 1).
The key missing ingredient leading to a general classification of models is a bound on ρ. The
following proposition, which was first derived in [35], serves this purpose.
Proposition 3.2. Let G be a non-vacuum graph. Then
ρ(G) ≤ 0,
and ρ(G) = 0 if and only if G is a melonic graph.
21When ω(G) = 0 the divergences are logarithmic.
22The variable ρ 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).
16 S. Carrozza
ρ = −1 ρ = 0
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.
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.
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, ω 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 vmax of the
renormalizable bubbles. The complete list, established in [35]23, is reported in Table 1. Models
of type A to E are candidate just-renormalizable GFTs, and are in principle the most interes-
ting 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.
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 = SU(2), which was proven
renormalizable at all orders in [35]. Its flow equations were then studied in greater details
23The classification of [35] includes only just-renormalizable models, we have added super-renormalizable and
finite models for completeness.
Flowing in Group Field Theory Space: a Review 17
Table 1. List of power-counting renormalizable melonic models.
Type d D vmax ω Explicit examples
A 3 3 6 3−N/2− 2n2 − n4 + 3ρ G = SU(2) [35, 31]
B 3 4 4 4−N − 2n2 + 4ρ G = SU(2)×U(1) [32]
C 4 2 4 4−N − 2n2 + 2ρ –
D 5 1 6 3−N/2− 2n2 − n4 + ρ G = U(1) [84, 81]
E 6 1 4 4−N − 2n2 + ρ G = U(1) [84, 81, 22]
F 3 2 arbitrary 2− 2V –
G 4 1 arbitrary 2− 2V G = U(1) [36, 63]
H 3 1 arbitrary 1− L− V < 0 G = U(1) [62]
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 renorma-
lizable model of type B based on the group SU(2)×U(1) has been considered in [32]. Examples
of Abelian 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 ϕ6 model of type D requires the inclusion of a non-melonic and non-connected
interaction of the form (ϕ · ϕ)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 bubbles24, remains consistent with only melonic bubbles included25. 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 renor-
malizable 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 ϕ4 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 = 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 = 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 ϕ4 necklace bubbles [23] in this context,
which will generate divergent but non-melonic 2-point graphs (with ρ = −1). See Fig. 10.
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
24Graphs with ρ = −1 or −2 might in principle still lead to divergences.
25Note 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.
18 S. Carrozza
ρ = −1
Figure 10. Non-melonic 2-point graph with ρ = −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 ω = 0 and is therefore logarithmically divergent in this case.
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 equiv-
alent) definitions of renormalized quantities. One can for example rely on the celebrated
Bogoliubov–Parasiuk–Hepp–Zimmermann (BPHZ) scheme, which amounts to using p2 = 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 divergences26. 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 con-
stants tb,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:
tb,i−1 − tb,i = βi({tb′,j | i ≤ j ≤ ρ}).
The n-point functions may then be expressed as formal multi-series in the tb,i’s, with convergent
coefficients in the limit ρ→ +∞. Moreover, the effective expansion provides a new perspective
on the more standard renormalized BPHZ expansion: at fixed multiscale parameter µ (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, renormaliza-
tion 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 compli-
cated 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 ana-
lysis 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.
26We remind the reader that, given a graph G, two divergent subgraphs H1,H2 ∈ G are said to overlap if neither
of the three relations is verified: H1 ∩H2 = ∅, H1 ⊂ H2, H2 ⊂ H1.
Flowing in Group Field Theory Space: a Review 19
ϕ(g1)
ϕ(g2)
ϕ(g3)
ϕ(g4)
∼
ϕ(g3)
ϕ(g4)
ϕ(g1)
ϕ(g2)
+ · · ·
h1 , α1
h2 , α2
1
23 ν(α1, α2)×
Figure 11. Approximation of a tracial graph as an effective trace invariant contribution plus corrections.
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 α1 and α2, the integrand of its amplitude is:
∫
dh1dh2 [Kα1+α2(h1h2)]2
∫
∏
i<j
dgij
Kα1
(
g11h1g
−1
31
)
Kα2
(
g−1
21 h2g41
)
× δ
(
g12g
−1
22
)
δ
(
g13g
−1
22
)
δ
(
g42g
−1
32
)
δ
(
g43g
−1
33
)
ϕ(g1)ϕ(g2)ϕ(g3)ϕ(g4).
The question is whether this expression can be approximated by an elementary tensor invariant
in the sector α1, α2 → 027. 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:
∫
dh [Kα1+α2(h)]2
∫
∏
i<j
dgij
Kα1
(
g11hg
−1
31
)
Kα2
(
g−1
21 g41
)
× δ
(
g12g
−1
22
)
δ
(
g13g
−1
22
)
δ
(
g42g
−1
32
)
δ
(
g43g
−1
33
)
ϕ(g1)ϕ(g2)ϕ(g3)ϕ(g4).
It is then obvious that, in the large scale limit, h is peaked around the identity and therefore g11
(resp. g21) is identified to g31 (resp. g41). 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
ν(α1, α2).
We say that a graph is contractible if its bulk holonomies can be trivialized in the UV region,
as illustrated in the example before28. 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 SU(2), even though it is allowed
27We remind the reader that the Schwinger parameters should be thought of as inverse squared momenta p−2.
28This is equivalent to simple-connectedness of the 2-complex, i.e., flat connections are trivial up to gauge.
20 S. Carrozza
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 renormali-
zation 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 equa-
tions may only be formulated for dimensionless coupling constants ub(Λ), which are appro-
priate rescalings of tb(Λ) by powers of Λ. This important aspect may be formalized by the
notion of canonical dimension db of tensor invariant bubbles, which has been discussed at length
in [22, 31, 60]. For melonic models (with closure constraint), it is defined as
db := D(d− 2)− (D(d− 2)− 2)
Nb
2
,
where Nb is the valency of the bubble b (i.e., its number of nodes). Hence the dimensionless
coupling constants are defined as
ub(Λ) :=
tb(Λ)
Λdb
,
and b is renormalizable (or, equivalently, perturbatively relevant) if and only if db ≥ 0. In view
of the expression (3.5) for the divergence degree ω, it appears that the most divergent graphs
contributing to the running of tb diverge like Λdb , and yield corrections to the dimensionless
couplings ub 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 dimension29.
29This possibility is actively explored in the context of 4d models with Barrett–Crane simplicity constraints [Car-
rozza S., Lahoche V., Oriti D., work in progress].
Flowing in Group Field Theory Space: a Review 21
UVIR
SΛΓk
Effective average action Bare action
tb(Λ) = ub(Λ) Λdbtb(k) = ub(k) kdb
Effective action
Γ0
Figure 12. The effective average action interpolates between the bare action in the ultraviolet and the
full effective action in the infrared.
In the Wetterich–Morris [67, 86] approach to the functional renormalization group, a one-
parameter family of deformed generating functionals is introduced
ZΛ,k[J, J̄ ] :=
∫
dµCΛ
(ϕ,ϕ) exp
(
−SΛ[ϕ,ϕ]− ϕ ·Rk · ϕ+ J̄ · ϕ+ ϕ · J
)
.
The new operator Rk 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 Γk, which is the Legendre transform of Wk[J, J̄ ] :=
ln(Zk,Λ[J, J̄ ]), appropriately shifted by the 2-point counter-term ϕ ·Rk ·ϕ (see [22] for a detailed
discussion). Interestingly, for suitable choices of Rk, Γk can be shown to interpolate between the
bare action SΛ = ΓΛ in the UV, and the full effective action Γ0 (or in other words the generating
functional of one-particle irreducible graphs) in the infrared, which we illustrate in Fig. 12.
Finally, as derived in detail in [22] for TGFTs with gauge invariant condition, the effective
average action verifies a Wetterich equation
k∂kΓk[ϕ,ϕ] =
∫
dg1dg2dg3k∂kRk(g1;g2)
(
Γ
(2)
k +Rk
)−1
(g2,g3)P(g3;g1), (4.1)
where
Γ
(2)
k [ϕ,ϕ](g;g′) :=
δ2Γk
δϕ(g)δϕ(g′)
[ϕ,ϕ]
is the full interacting propagator at scale k and P is as before the projector onto gauge inva-
riant 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 poten-
tial 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 trun-
cations 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 empir-
ical 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.
22 S. Carrozza
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 renor-
malizable 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 dimen-
sion 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 SU(2)×U(1) as in [32].
This theory is renormalizable up to order 4, therefore a natural ansatz for the effective average
action is30
Γk(ϕ,ϕ) = −Z(k)ϕ ·∆ϕ+ Z(k)u2(k)k2 + Z(k)2u4(k) ,
where for convenience we directly represented tensor interactions by their associated colored
graphs31, and we have introduced a wave-function parameter Z(k). In this group dimension,
the mass term has canonical dimension 2 and the marginal ϕ4 interactions have as they should
dimension 0. It was shown in [32] that the perturbative renormalization group flow reduces in
this truncation to
k
∂u2(k)
∂k
= −2u2(k)− 3πu4(k),
k
∂u4(k)
∂k
= −2πu4(k)2. (4.2)
One notices a major difference with ordinary scalar field theories: the derivative of the ϕ4
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 = SU(2), which is renormalizable up to order
6 interactions. Accordingly, one can choose the following ansatz for the effective average action:
Γk(ϕ,ϕ) = −Z(k)ϕ ·∆ϕ+ Z(k)u2(k)k2 + Z(k)2u4(k)k
+ Z(k)3u6,1(k) + Z(k)3u6,2(k) ,
30We use the short-hand notation ∆ :=
3∑̀
=1
∆`.
31Note 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.
Flowing in Group Field Theory Space: a Review 23
and we note the change of dimensionality of the ϕ4 interactions with respect to the previous
situation. The one-loop perturbative flow can in this case be approximated by32
k
∂u2(k)
∂k
≈ −2u2(k)− au4(k),
k
∂u4(k)
∂k
≈ −u4(k)− b(u6,1(k) + 2u6,2(k)),
k
∂u6,1(k)
∂k
≈ −cu4(k)u6,1(k),
k
∂u6,2(k)
∂k
≈ −du4(k)u6,2(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 u6,1 and u6,2 both converge to 0 in the ultraviolet. As shown in [31]33,
this is actually misleading because, given the form of the flow equations, it can be proven
that u4(k) necessarily reaches negative values for large k, which has the effect of making the ϕ6
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 positive34.
This example shows that the question of asymptotic freedom can be tricky in ϕ6 renormalizable
models, because the ϕ4 super-renormalizable coupling constants have a non-trivial influence on
the marginal ϕ6 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 ε-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 − ε, 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 SU(2) × U(1)D−3 for arbitrary D ≥ 3, and
then analytically continue the parameter D := 4− ε to the interval 3 ≤ D ≤ 4. When ε is small
enough, one may assume that the ϕ4 truncation remains pertinent:
Γk(ϕ,ϕ) = −Z(k)ϕ ·∆ϕ+ Z(k)u2(k)k2 + Z(k)2u4(k)kε .
Note that u4 acquired a small canonical dimension ε. This has the effect of slightly modifying
the flow equations (4.2) to
k
∂u2(k)
∂k
≈ −2u2(k)− 3πu4(k), k
∂u4(k)
∂k
≈ −εu4(k)− 2πu4(k)2. (4.4)
32These flow equations were not explicitly evaluated in [32], but they immediately follow from the computations
reported there.
33This 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 u2
6,1, u2
6,1 and u6,1u6,2) which have
been neglected in (4.3). The main conclusions are however the same.
34The situation is a more subtle when u6,1u6,2 < 0, we do not exclude the possibility that one could define an
asymptotically free theory with positive action in this particular case.
24 S. Carrozza
u2
u4
u∗2
u∗4
Figure 13. Qualitative phase portrait of the renormalization group flow equations (4.4), with arrows
oriented from high to small scales.
Accordingly, one formally finds a new solution to the fixed point equation:
u∗2 ≈
3
4
ε+O
(
ε2
)
, u∗4 ≈ −
1
2π
ε+O
(
ε2
)
.
Qualitatively, the renormalization group flow (4.4) is therefore as represented in Fig. 13. Ex-
trapolating to ε = 1, this suggests the existence of a TGFT analogue of the Wilson–Fisher
fixed point of 3d local scalar field theory in the SU(2) model of [35]. This hypothesis should
of course be taken with a grain of salt, since the first terms in the ε-expansion may only give
a crude idea of what is really going on at ε = 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 ε = 1
[Carrozza S., Lahoche V., work in progress].
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 U(1) (type E) was studied in the ϕ4 melonic
truncation
Γk(ϕ,ϕ) = −Z(k)ϕ ·∆ϕ+ Z(k)u2(k)k2 + Z(k)2u4(k) .
For such Abelian models, it is convenient to use a Litim cut-off operator Rk [65] which, in
momentum space, is defined by the kernel
Rk(p;p) = Z(k)
(
k2 − p2
)
Θ(k2 − p2)
6∏
`=1
δp`p′` ,
Flowing in Group Field Theory Space: a Review 25
where p` ∈ Z label 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 � 1) and IR (k → 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 ε-expansion of 3d models, the coupling constant u4
at the fixed point is positive (and u2 is negative).
A non-compact version of this model, based on the Abelian group R, was investigated in [17].
The renormalization group analysis requires in this case a further regularization of infinite vol-
ume divergences. This was implemented by a compactification of R into U(1), thus resulting
in a regularized theory identical to [22]. The authors could then define an appropriate thermo-
dynamic limit, capturing the properties of the model in the limit in which the volume of U(1)
is scaled to infinity. Once more, a fixed point of the Wilson–Fisher type is found in this trun-
cation, 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 ϕ 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 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
p1+· · ·+p6 = 0, which can be explicitly dealt with. With SU(2) the situation is not so simple: the
gauge invariance condition encodes complicated recoupling relations among the 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 ϕ6 truncation, and compare
them with the features we extrapolated from the formal ε-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
26 S. Carrozza
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 (Tab-
le 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 ϕ4 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 = 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 ne-
cessitates 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 SL(2,R) and SL(2,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. Alter-
native 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 con-
Flowing in Group Field Theory Space: a Review 27
straints. 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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http://dx.doi.org/10.1088/0264-9381/18/1/308
http://dx.doi.org/10.1088/0264-9381/18/1/308
http://arxiv.org/abs/gr-qc/0002095
http://dx.doi.org/10.1103/PhysRevD.88.024011
http://arxiv.org/abs/1302.1781
http://dx.doi.org/10.1515/9781400862085
http://dx.doi.org/10.1063/1.4715396
http://arxiv.org/abs/1112.5104
http://dx.doi.org/10.1209/0295-5075/111/60011
http://arxiv.org/abs/1507.04190
http://dx.doi.org/10.1017/CBO9780511755804
http://dx.doi.org/10.1103/PhysRevD.88.105003
http://arxiv.org/abs/1303.7256
http://dx.doi.org/10.1088/0264-9381/31/18/185005
http://dx.doi.org/10.1088/0264-9381/31/18/185005
http://arxiv.org/abs/1401.2096
http://dx.doi.org/10.1088/0264-9381/32/17/175012
http://arxiv.org/abs/1411.7213
http://dx.doi.org/10.1007/s00220-014-1930-3
http://dx.doi.org/10.1007/s00220-014-1930-3
http://arxiv.org/abs/1211.2618
http://dx.doi.org/10.1017/CBO9780511755682
http://dx.doi.org/10.1017/CBO9780511755682
http://arxiv.org/abs/gr-qc/0110034
http://dx.doi.org/10.1016/0370-2693(93)90726-X
http://dx.doi.org/10.1016/0370-1573(74)90023-4
1 Introduction
2 From simplicial to tensorial GFTs
2.1 Renormalization of GFTs: motivations and basic ingredients
2.2 Simplicial GFT models and tensorial theory space
2.3 Large N expansion and extended tensorial theory space: heuristic derivation
3 Perturbatively renormalizable TGFTs with closure constraint
3.1 A general class of models: local 'potential' approximation
3.2 Power-counting theorem and classification of models
3.3 Renormalization, subtraction schemes and contractible graphs
4 Renormalization group and non-trivial fixed points
4.1 Effective average action and Wetterich equation
4.2 Example: perturbative treatment of rank-3 models
4.3 Non-perturbative aspects and truncations
5 Summary and outlook
References
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