Exact Renormalisation Group Equations and Loop Equations for Tensor Models
In this paper, we review some general formulations of exact renormalisation group equations and loop equations for tensor models and tensorial group field theories. We illustrate the use of these equations in the derivation of the leading order expectation values of observables in tensor models. Fur...
Saved in:
| Date: | 2016 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2016
|
| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147765 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Exact Renormalisation Group Equations and Loop Equations for Tensor Models / T. Krajewski, R. Toriumi // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 43 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-147765 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1477652025-02-09T10:02:13Z Exact Renormalisation Group Equations and Loop Equations for Tensor Models Krajewski, T. Toriumi, R. In this paper, we review some general formulations of exact renormalisation group equations and loop equations for tensor models and tensorial group field theories. We illustrate the use of these equations in the derivation of the leading order expectation values of observables in tensor models. Furthermore, we use the exact renormalisation group equations to establish a suitable scaling dimension for interactions in Abelian tensorial group field theories with a closure constraint. We also present analogues of the loop equations for tensor models. 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. T.K. is partially supported by the grant “ANR JCJC CombPhysMat2Tens”. 2016 Article Exact Renormalisation Group Equations and Loop Equations for Tensor Models / T. Krajewski, R. Toriumi // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 43 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81T17; 81T18 DOI:10.3842/SIGMA.2016.068 https://nasplib.isofts.kiev.ua/handle/123456789/147765 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
In this paper, we review some general formulations of exact renormalisation group equations and loop equations for tensor models and tensorial group field theories. We illustrate the use of these equations in the derivation of the leading order expectation values of observables in tensor models. Furthermore, we use the exact renormalisation group equations to establish a suitable scaling dimension for interactions in Abelian tensorial group field theories with a closure constraint. We also present analogues of the loop equations for tensor models. |
| format |
Article |
| author |
Krajewski, T. Toriumi, R. |
| spellingShingle |
Krajewski, T. Toriumi, R. Exact Renormalisation Group Equations and Loop Equations for Tensor Models Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Krajewski, T. Toriumi, R. |
| author_sort |
Krajewski, T. |
| title |
Exact Renormalisation Group Equations and Loop Equations for Tensor Models |
| title_short |
Exact Renormalisation Group Equations and Loop Equations for Tensor Models |
| title_full |
Exact Renormalisation Group Equations and Loop Equations for Tensor Models |
| title_fullStr |
Exact Renormalisation Group Equations and Loop Equations for Tensor Models |
| title_full_unstemmed |
Exact Renormalisation Group Equations and Loop Equations for Tensor Models |
| title_sort |
exact renormalisation group equations and loop equations for tensor models |
| publisher |
Інститут математики НАН України |
| publishDate |
2016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147765 |
| citation_txt |
Exact Renormalisation Group Equations and Loop Equations for Tensor Models / T. Krajewski, R. Toriumi // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 43 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT krajewskit exactrenormalisationgroupequationsandloopequationsfortensormodels AT toriumir exactrenormalisationgroupequationsandloopequationsfortensormodels |
| first_indexed |
2025-11-25T16:03:06Z |
| last_indexed |
2025-11-25T16:03:06Z |
| _version_ |
1849778860363612160 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 068, 36 pages
Exact Renormalisation Group Equations
and Loop Equations for Tensor Models?
Thomas KRAJEWSKI † and Reiko TORIUMI ‡
† Aix Marseille Université, Université de Toulon, CNRS, CPT,
UMR 7332, 13288 Marseille, France
E-mail: thomas.krajewski@cpt.univ-mrs.fr
‡ Department of Physics, University of California Berkeley, USA
E-mail: torirei@gmail.com
Received February 29, 2016, in final form July 05, 2016; Published online July 14, 2016
http://dx.doi.org/10.3842/SIGMA.2016.068
Abstract. In this paper, we review some general formulations of exact renormalisation
group equations and loop equations for tensor models and tensorial group field theories.
We illustrate the use of these equations in the derivation of the leading order expectation
values of observables in tensor models. Furthermore, we use the exact renormalisation group
equations to establish a suitable scaling dimension for interactions in Abelian tensorial group
field theories with a closure constraint. We also present analogues of the loop equations for
tensor models.
Key words: tensor models; group field theory; large N limit; exact renormalisation equation
2010 Mathematics Subject Classification: 81T17; 81T18
Contents
1 Introduction 2
2 Tensor models 4
2.1 Invariant potentials for random tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Perturbative expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Tensorial group field theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Couplings in the spin network basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Exact renormalisation group equations 11
3.1 Wilson’s ideas in quantum field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Exact renormalisation group equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Polchinski’s equation for invariant tensor models . . . . . . . . . . . . . . . . . . . . . . . 15
3.4 Large N melonic universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5 Group field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.6 Scaling and renormalisation for Abelian models with closure constraint . . . . . . . . . . . 21
3.7 SU(2) models in the spin network basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.8 Wetterich’s equation and nontrivial fixed points . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Loop equations 26
4.1 Reparametrisation invariance in quantum field theory . . . . . . . . . . . . . . . . . . . . 26
4.2 Loop equations for tensor models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 The case of matrix models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4 An application to melonic dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
?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
mailto:thomas.krajewski@cpt.univ-mrs.fr
mailto:torirei@gmail.com
http://dx.doi.org/10.3842/SIGMA.2016.068
http://www.emis.de/journals/SIGMA/Tensor_Models.html
2 T. Krajewski and R. Toriumi
5 Conclusion and outlook 32
A ERGE for low order bubble couplings 32
A.1 Couplings in rank D = 3 tensor models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
A.2 Couplings in rank D = 4 tensor models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
References 35
1 Introduction
Tensor models are higher-dimensional generalisations of matrix models that have been intro-
duced more than twenty five years ago [1]. The aim was to reproduce in dimension D > 2 the
successes of matrix models in providing a theory of random geometries. This is of high interest
in multiple fields of physics, especially in discrete approaches to quantising gravity. The basic
idea is that a suitable integral over a rank D tensor T is the partition function of a theory of
random triangulations in dimension D,∫
rank D
tensors
dµC(T ) exp{−V (T )} =
∑
graph G ⇔ dimension D triangulations
A(G). (1.1)
In this Feynman graph expansion, dµC is a Gaußian measure on tensors, V an interaction
potential and A the resulting weight associated to the graph G. By a certain duality procedure,
these graphs are in one to one correspondence with triangulations of dimension D. Thus random
tensors provide a natural way to construct models of random geometries in dimension D.
However, the combinatorics of tensor models turned out to be much more intricate than
the matrix model one. Therefore, progress in this field remained very slow until a couple of
years ago, when a new impetus arose thanks to the introduction of coloured tensor models by
Gurau [23]. This allowed him to a derive for the first time a 1/N expansion [25], where N is
the size of the tensor and prove Gaußian universality at large N [28]. This breakthrough was
followed by many other works in the field, refining the 1/N expansion or adding new degrees of
freedom that can be interpreted as matter living on the triangulations. Finally, a double scaling
limit has been obtained: when some parameters in V are taken to their critical values together
with N →∞, the series over triangulations reproduces a continuum limit, see [10] and [19]. We
refer to [38] for a recent overview of random tensors.
A second important breakthrough occurred when Ben Geloun and Rivasseau constructed
the first renormalisable group field theory [5]. Group field theories [34] are specific quantum
field theories defined, in dimension D, over D copies of a group often taken to be SU(2) or
several copies of U(1). As far as the combinatorics is concerned, the group field Φ(g1, . . . , gD)
is analogous to the tensor Ti1,...,iD . Thus, a functional integral over Φ generates a sum over
triangulations weighted by some group integrals. When the potential is suitably chosen, these
weights reproduce the spin foam amplitudes in loop quantum gravity. We refer the reader to the
monographs by Rovelli [40] and Thiemann [42] for an account of loop quantum gravity. Then,
group field theory provides a prescription on how to sum these amplitudes. This is crucial in
order to investigate some properties of the theory, like triangulation independence or summation
over topologies. Moreover, if available, a double scaling limit could be interpreted a continuum
limit, including a sum over topologies. This interplay between the various amplitudes also plays
a key role in the renormalisation of group field theories: the renormalised theory is constructed
recursively using the combinatorics of subgraphs, thus the amplitude of smaller triangulations.
Let us note that the model renormalised in [5] is based on a abelian group of the type U(1)d.
This has been followed by the renormalisation of other abelian models. A further decisive step
was made by Carrozza, Oriti and Rivasseau in [16], when a rank three model based on SU(2)
Exact Renormalisation Group Equations and Loop Equations for Tensor Models 3
with closure constraint has been renormalised. Such a model would open the possibility, if
defined nonperturbatively, to construct a quantum theory of gravity in D = 3 that includes
topology changes. Finally, let us mention that the study of D = 4 models with a Barrett–Crane
or Engle–Peirera–Rovelli–Livine/Freidel–Krasnov vertex is currently under scrutiny by various
researchers. These models include the simplicity constraint as well as the closure constraint, the
former being essential in making contacts with general relativity in D = 4.
In this paper, we aim at revisiting these two breakthroughs in light of the exact renorma-
lisation group equation, in Polchinski’s formulation [35]. The latter is a functional differential
equation that allows to implement Wilson’s ideas in a general setting in quantum field theory.
It translates into a flow equation for all the couplings of the theory, hence the name “exact”,
that prescribes how the couplings should be modified in order to take into account the effects
of the high energy modes that have been integrated. As a consequence, the evolution equation
allows to classify the various couplings according to their degree of relevancy. This is what
is done here: In the tensor model case, we identify the dominant couplings in the large N
limit while for group field theories we identify the relevant couplings that govern the high spin
limit.
Moreover, it allows to trade the functional integral formalism for a functional differential
equation, from which many physical consequences can be drawn without having to resort to
Feynman graphs evaluation. This is precisely the point of view adopted here: while the fun-
damental results presented above have been obtained by expanding (1.1) in Feynman graphs,
these results can also be derived within the exact renormalisation group equation. The latter
is an equation for the effective vertices that collects a sum over Feynman graphs. From a geo-
metrical perspective, the effective vertices correspond to boundary triangulations and collect all
triangulations corresponding to a fixed boundary.
However, our approach remains perturbative in nature since no attempt has been made to
define the quantities we manipulate as analytic functions of the parameters of the potential.
Recently, various nonperturbative results have been obtained [27] by summing up the pertur-
bative series. Such an approach is called “constructive field theory” and we refer the reader to
book [37] and the recent review [29] for a detailed account of the field theory case. Alterna-
tively, the exact renormalisation group equation in Wetterich’s formulation can be solved using
a truncation as will be briefly reviewed in Section 3.8. Wetterich’s equation is equivalent at
the perturbative level to Polchinski’s one but more suitable for truncations. In this respect, let
a mention the review by Carrozza [15] in this special issue, dealing with exact renormalisation
group equation, with an emphasis on the Wetterich equation.
As a functional differential equation to be satisfied by the effective theory, exact renormali-
sation group equations bear some common features with the Schwinger–Dyson equations. More
in general, the invariance of the integral (1.1) under change of variables translate into a set
of constraints on the latter that can be expressed as differential operators with respect to the
parameters appearing in the potential. In the matrix model case, these equations are known as
the “loop equations” and play a prominent role, especially in deriving the double scaling limit
and in establishing some connection to integrable hierarchies. For tensor models, analogues of
these equations have been derived in [26] and applied to the large N limit in [8]. The potential
of these equations for tensor models has not yet been completely explored and it is possible that
they lead to significant results, as in the matrix model case.
It is important to emphasize the rationale of our study: Polchinski’s equation and Schwinger–
Dyson equations are exact relations that are satisfied by the effective action and the generating
functions of invariant observables. In our context, we work at the perturbative level only, con-
sidering these quantities as formal series in the couplings. Then, the aforementioned equations
lead to nontrivial relations between these quantities, collecting a sum over Feynman graphs, or
equivalently, a sum over space-time triangulations.
4 T. Krajewski and R. Toriumi
Our paper is organised as follows. In Section 2, we present the general framework of tensor
models and tensorial group field theories. Section 3 is devoted to the exact renormalisation
group equation. First, we introduce Wilson’s ideas and formulate Polchinski’s equation in the
context of quantum field theory. Then, we apply this formalism to tensor models. As a first
application, we derive the scaling properties of the effective couplings which lead to an alternative
proof of Gurau’s universality [28]. These equations are also formulated for group field theories
in a general setting. Then, we apply them to propose a scaling dimension for all the couplings in
the case of abelian group field theories with closure constraints. As a byproduct, we recover the
list of five renormalisable theories first identified in [16]. We close this section by a formulation
of the flow equation for spin networks and a brief discussion of the nonperturbative results
based on Wetterich’s equation. Sections 4 deals with loop equations. We start by expressing
reparametrisation invariance in quantum field theory and then formulate the analogue of the
loop equations for tensor models and stress the analogies between tensors and matrices. Finally,
we illustrate the use of the loop equations in deriving the values of the leading order expectation
values, following [8].
2 Tensor models
Matrix models turn out to be ubiquitous in mathematics and physics. For instance, they appear
in combinatorics of maps and statistical physics on random triangulated surfaces. Tensor models
aim at providing a higher-dimensional generalisation of matrix models, with a particular view
towards generation of random triangulations of spaces of dimension D > 2. Thus, the basic
variables we consider are rank D complex tensors Ti1,...,iD and its complex conjugate T i1,...,iD ,
with indices ik ∈ {1, . . . , N}, the integer N being the size of the tensor. Let us stress that
the models considered here involve a tensor and its complex conjugate, treated as independent
variables. Furthermore, these tensors are not required to fulfil any symmetry relations upon
permutations of their indices. For instance, in the D = 2 case, they correspond to nonhermitian
matrices.
There is a natural transformation of a rank D complex tensor under the direct product of D
copies of the group U(N)
Ti1,...,iD →
∑
j1,...,jD
U1
i1j1 · · ·UDiDjDTj1,...,jD , T i1,...,iD →
∑
j1,...,jD
U
1
i1j1 · · ·U
D
iDjD
T j1,...,jD , (2.1)
with (U1, . . . , UD) ∈ U(N) × · · · × U(N). This large transformation group is available because
we do not impose any symmetry under permutations of indices, otherwise two indices that can
be permuted must transform in the same way under the unitary group, so that the symmetry
is reduced from U(N)× U(N) down to U(N). Finally, let us also note that it is possible to let
the various indices vary in different ranges, ik ∈ {1, . . . , Nk}. Then, unitary transformations
correspond to the group U(N1) × · · · × U(Nk). Although useful from a combinatorial point of
view, we do not consider this generalisation here and stick to the case N1 = · · · = Nk = N .
2.1 Invariant potentials for random tensors
In analogy with random matrices, we construct a theory of random tensors, we have to define
a certain probability measure on the space of tensors. The latter is of the form
dµC(T, T ) exp{−V (T, T )}
Z
,
Exact Renormalisation Group Equations and Loop Equations for Tensor Models 5
where dµC(C, T ) is a Gaußian measure of covariance C, V (T, T ) the interaction potential and Z
the partition function defined as
Z =
∫
dµC(T, T ) exp{−V (T, T )}.
Then, the expectation value of an observable, taken to be any functionO(T, T ) of the components
of the tensors, is defined as
〈O〉 =
∫
dµC(T, T )O(T, T ) exp{−V (T, T )}
Z
.
Unless otherwise stated, we restrict our attention to invariant random tensors. This means that
dµC(T, T ), O(T, T ) and V (T, T ) have to be invariant under the action of U(N) × · · · × U(N),
see (2.1).
Polynomial invariants of the components of the tensors are constructed using particular
graphs which are called D-bubbles. A D-bubble is a bipartite D-coloured graph. This means
that:
• There are two types of vertices, black ones • and white ones ◦.
• An edge can connect only a black vertex to a white vertex.
• At any vertex there are exactly D incident edges.
• Each edge is decorated by a colour in {1, . . . , D} in such a way that the colours of the D
edges incident to any vertex are all different.
The invariant associated to a D-bubble is defined by assigning a tensor T to a white vertex,
a tensor T to a black vertex. We identify the indices ik in T and ik in T whenever they are
connected by a line of colour k and summing over all tensor indices. In analogy with the trace
invariants of matrix models, such an invariant is written as
TrB(T, T ) =
∑
ie
edge indices
∏
v
black vertices
T iv,1,...,iv,D
∏
v
white vertices
Tiv,1,...,iv,D
∏
e
edges
δie,iv(e),c(e)
δie,iv(e),c(e)
,
where e is an edge between a white vertex v(e) and a black vertex v(e) and c(e) its colour. For
example, the invariants associated to two 3-bubbles are given in Fig. 1.
Let us note that we do not restrict ourselves to connected bubbles, though the invariant of
a disconnected bubble factorizes over its connected components. In the particular case D = 2,
connected bubble invariants are defined by a single even integer, the number n of edges, and
are just traces of power of the matrix MM †, TrB(M,M) = Tr(MM †)n/2. In this case, non
connected bubbles correspond to product of traces, also known as multitrace operators.
A polynomial invariant potential for a rank D tensor is defined by summing over a finite set
of D-bubbles, each weighted by a complex number λB, called its coupling constant,
V (T, T ) =
∑
B
λB
CB
TrB(T , T ). (2.2)
CB is a combinatorial factor introduced for later convenience. It is the cardinal of the group of
transformation of edges and vertices that preserve the bubble. In this paper, we mostly stay
at a formal level, i.e., we expand the functional integral (1.1) in power series of the couplings
using Gaußian integration. In this case, the couplings can be any complex numbers but if we
require the partition function and the expectation values to be real, then we have to take the
couplings to be real. At a nonperturbative level, the main difficulty lies in the fact that the trace
6 T. Krajewski and R. Toriumi
1
2
3
∑
ia
T i1,i2,i3Ti1i2i3
(a) Dipole graph (Gaußian measure).
1
23
2
1
2
3
1
2
2
2
3
∑
ia, jb, kc
T i1i2i3T j1j2j3T k1k2k3Ti1k2j3Tj1i2k3Tk1j2i3
(b) Degree 6 interaction.
Figure 1. Some bubble invariants.
invariants TrB(T , T ) are not necessarily positive, except for a few special cases, like the quartic
models studied in [27]. When these invariant are positive, positive couplings lead to a well
defined integral. In the general case, it may be useful to consider purely imaginary couplings in
such a way that the exponential of the interaction term remain bounded.
There is a single quadratic trace invariant up to a factor. It is associated to the dipole,
a bubble made of two vertices connected by D edges, see Fig. 1. Its trace invariant reads
Trdipole(T, T ) =
∑
i1···iD
i1,...,iD
T i1,...,iDTi1···iD = T · T.
It is used to construct the normalised Gaußian measure
dµC(T, T ) = N exp
{
−
(
t−1T · T
)}
dTdT ,
where N is a normalisation factor and dTdT the standard Lebesgue measure on the space of
components of the tensors T and T . Then, the covariance of the Gaußian measure is simply
Ci1,...,iD|i1,...,iD = tδi1i1 · · · δiDiD , which means that∫
dµC(T, T )Ti1...iDT i1...iD = t δi1i1 · · · δiDiD (2.3)
for positive t.
By the same token, invariant observables are also labelled by D-bubbles,
O(T, T ) =
∑
B
OB
CB
TrB(T , T ).
Thus, we reduce the computation of the expectation value of an observable to that of an in-
variant bubble. Moreover, bubble interactions may be used to generate expectations of bubble
observables,
〈TrB(T , T )〉 =
1
Z
∂
∂JB
∫
dµC(T, T ) exp
{
−V (T, T ) +
∑
B′
JB′ TrB′(T , T )
}∣∣∣∣∣
JB′=0
=
∂
∂JB
logZ
∣∣
JB′=0
. (2.4)
Exact Renormalisation Group Equations and Loop Equations for Tensor Models 7
Alternatively, any bubble observable can be expanded in powers of the components of the tensors.
Thus, the basic object to compute is the expectation value of a product of components〈
Ti1,1...i1,D · · ·Tin,1...in,D T i1,1...i1,D · · ·T in,1...in,D
〉
. (2.5)
In analogy with quantum field theory, these expectation values are referred to as correlation
functions. Because of U(1) invariance, correlation functions vanish if there is not an equal
number of T and T .
Although we restrict ourselves to invariant tensor models, in the more general case of nonin-
variant tensors, the sum over bubbles invariants in (2.2) may be replaced by
V (T, T ) =
∑
B
∏
edges
∑
ie, ie
edge indices
λB
{
ie, ie
}
CB
×
∏
v
black vertices
T iv,1,...,iv,D
∏
v
white vertices
Tiv,1,...,iv,D
∏
e
edges
δie,iv(e),c(e)
δie,iv(e),c(e)
. (2.6)
The invariant potential (2.2) is recovered for
λB
{
ie, ie
}
= λB
∏
e edges
δie,ie .
The motivation behind the choice (2.6) lies in the fact that it is covariant under the group
U(N) × · · · × U(N): it is invariant under simultaneous transformations of λB
{
ie, ie
}
and the
tensors T and T . Moreover, this general construction renders the transition to group field theory
more transparent.
2.2 Perturbative expansion
Let us now give the basic features of the expansion of an arbitrary bubble observable as a formal
power series in the bubble couplings λB. Using (2.4), we reduce such a computation to that of
the partition function, considered as a function of the bubble couplings,
Z =
∫
dµC(T, T ) exp
{
−
∑
B
λB
CB
TrB(T , T )
}
.
To proceed, we expand the exponential and integrate over the tensors using Wick’s theorem
with the covariance (2.3). The latter can be formulated as the computation of the Gaußian
expectation value of the product of n pairs of a tensor and its complex conjugate∫
dµC(T, T ) Ti1,1...i1,D · · ·Tin,1...in,D T i1,1...i1,D · · ·T in,1...in,D
= t
∑
σ∈Sn
∏
1≤k≤D
δi1,k,iσ(1),k
· · · δin,k,iσ(n),k
.
The sum over all permutations Sn is nothing but the familiar sum over all pairings, with the
constraint that a tensor is paired with its complex conjugate. The Gaußian expectation value
of a product of n tensors T with n tensors T vanishes when n 6= n.
Applying Wick’s theorem to any monomial obtained by expanding the potential amounts to
summing over all contractions of T with T . Any such contraction corresponds to the contraction
of a pair of vertices of different colours in a bubble, defined as follows. Consider a bubble B,
which may be disconnected, as is the case if it arises from two different bubble couplings or from
8 T. Krajewski and R. Toriumi
0
1
2
3
1
2
3
→
1
2
3
(a) Generic vertices.
0
3
1
2
1
2
→
1
2
= N
(b) Connected vertices.
Figure 2. Contraction of a pair of vertices.
a single bubble coupling associated with a disconnected bubble. Choose a white vertex v and
black vertex v in B and remove them, leaving a set of coloured half edges. Then, reattach the
half edges respecting the colours. The resulting bubble is denoted by B/vv. It is convenient to
materialise the choice of the pair of vertices in B before contraction by an extra colour 0 edge,
represented by a dashed line, see Fig. 2. Note that if the pairs of vertices are already connected
by k edges in B, then the pair contraction yields k circles, so that we have an extra factor of Nk
resulting from the summation over the k indices.
Accordingly, the perturbative expansion of the partition function can be represented as a sum
over (D+1)-bubbles (see Fig. 3 for an example of such a bubble)
Z =
∫
dµC(T, T ) exp
{
−
∑
B
λB
CB
TrB(T , T )
}
=
∑
G
(D+1)-bubbles
NF (G)tE(G)
CG
∏
B⊂G
(−λB).
This expression is obtained by contracting all pairs of vertices, thus yielding (D+1)-bubbles
once we represent the pair contractions by a colour 0 line. The pair contractions can be used, in
any order, to successively reduce the bubbles to a disjoint union of circles, each of them yielding
a factor of N . On the bubble G, it can be viewed as the total number of faces with colour 0.
Recall that a face of colours 0 and k is a connected subgraph of G obtained by removing all
colours except 0 and k. E(G) is the number of colour 0 edges of G. The bubbles B ⊂ G are
simply the D-bubbles we started with. On the example of Fig. 3, the 3-bubble in the potentials
have been drawn on a shaded disc, there is one disconnected 3-bubble (two dipoles) considered
as a single interaction in the potential, whose contribution to the potential is
1
2
λ 1
2
3
1
2
3
(
T · T
)2
.
The 4-bubble of Fig. 3 is 0-connected but not connected.
In many cases, it is useful to consider the logarithm of Z. The latter is expanded over
0-connected bubbles,
logZ =
∑
G
0-connected (D+1)-bubbles
NF (G)tE(G)
CG
∏
B⊂G
(−λB), (2.7)
Exact Renormalisation Group Equations and Loop Equations for Tensor Models 9
32
1
1
32 00 1 321 32
0
0
Figure 3. A 0-connected bubble with color 0 edges and two 3-bubbles.
⇔
Figure 4. Triangulation of a 3d sphere with 2 tetrahedra.
where a bubble is said to be 0-connected if the graph (with only colour 0 lines) obtained by
contracting all the bubbles B ⊂ G is connected. A bubble may be 0-connected without being
connected if some of the bubbles B ⊂ G are disconnected.
This expansion can be given a geometrical interpretation that may be used in discretised ap-
proaches to quantum gravity and random geometry. Each bubble G with D+1 colors represents
a coloured triangulation of a space of dimension D. Its simplices of dimension D are the vertices
of G attached along their D − 1 faces following the edges of G. Thus, the expansion of the
logarithm of the partition function (2.7) provides a sum over coloured triangulations. If all the
bubbles in the potential are connected, then logZ provides us with a sum over connected trian-
gulations. For instance, the dipole graph represents a triangulation of a sphere by 2 simplices,
see Fig. 4.
Triangulations with boundaries are represented by correlation functions, see (2.5). (D+1)-
bubbles contributing to a correlation with n tensors T and T have 2n colour 0 external legs.
Such a bubble B gives rise to the boundary bubble ∂B. Vertices of ∂B are external legs of B.
They are connected by an edge of colour k if and only if the two external legs belong to a face
of colour 0k in B. Recall that in a coloured tensor models a face of colours i1 . . . ik in a bubble
is a connected component of the subgraph made of edges of colours i1 . . . ik in B.
2.3 Tensorial group field theories
Tensorial group field theories are particular quantum field theories defined on D copies of
a group, usually SU(2), U(1)d, Rd or SL(2,C). They are constructed in analogy with tensor
models, we replace tensors by functions of D copies of the group,
Ti1,...,iD → Φ(g1, . . . , gD), T i1,...,iD → Φ(g1, . . . , gD).
Note that g is not the complex conjugate of g but an independent group element that is an
argument of the complex conjugate field Φ. In quantum gravity applications, D denotes the
space-time dimension and the group is the Lorentz group, the euclidian group or their universal
10 T. Krajewski and R. Toriumi
g1
g3
g2
g4
Figure 5. The group field as a tetrahedron.
g1
g2
g3
g4
g1
g2
g3
g4
Figure 6. Propagation of a tetrahedron.
covers. The group field Φ(g1, . . . , gD) represents a D − 1 simplex and the group variables
g1, . . . , gD are associated with its D (D−1)-faces, see Fig. 5.
In many models, the group elements are related to the normals of the tetrahedron. Then,
it is assumed that they obey a closure constraint. The closure condition is implemented as an
invariance of the field under left translation,
Φ(g1, . . . , gD) = Φ(hg1, . . . , hgD), Φ(g1, . . . , gD) = Φ(hg1, . . . , hgD). (2.8)
for every group elements h and h. The covariance is defined by the Gaußian integration∫
dµC(Φ,Φ) Φ(g1, . . . , gD)Φ(g1, . . . , gD) = C
(
g1g
−1
1 , . . . , gDg
−1
D
)
,
and the integration of Φ Φ and Φ Φ vanishes. Because of the closure constraint (2.8), the cova-
riance obeys
C
({
geg
−1
e
})
= C
({
hgeg
−1
e h
−1})
.
Note that it is also invariant under a global translation for every colour, since it only depends
on products geg
−1
e . The covariance propagates a tetrahedron, in analogy with the quantum field
theory propagation of a particle, see Fig. 6.
A suitable covariance may be constructed using the heat kernel on the group Hα,
CΛ,Λ0
({
geg
−1
e
})
=
∫ 1
Λ2
1
Λ2
0
dα
∫
dhdh
∏
1≤i≤D
Hα
(
hgig
−1
i h
−1)
. (2.9)
Λ and Λ0 are cut-offs that have to be introduced to avoid divergences. Group integrations are
performed using the Haar measure and implement the required invariances.
As for the tensor models, the general interaction potential can also be expanded over D-
bubbles. However, in this case there is no reason to impose unitary invariance. Therefore, we
use the general form of the bubble couplings (2.6), replacing tensor indices by group elements,
V (Φ,Φ) =
∑
B
1
CB
∫ ∏
e
dhe
∏
v,i
dgv,i
∏
v,i
dgv,iλB({he})
∏
e
δ
(
h−1
e gv(e),c(e)(gv(e),c(e))
−1
)
×
∏
v
black vertices
Φ(gv,1, . . . , gv,D)
∏
v
white vertices
Φ(gv,1, . . . , gv,D), (2.10)
Exact Renormalisation Group Equations and Loop Equations for Tensor Models 11
where δ is the Dirac distribution on the group, defined by
∫
dgδ(g)f(g) = f(1). Note that
we have imposed translation invariance, so that the couplings only depend on the products
gv(e),c(e)(gv(e),c(e))
−1
Then, the perturbative expansion yields a sum over triangulations of dimension D, weighted
by certain graph amplitudes. By a suitable choice of the bubble couplings λB({he}), one can
show that this amplitude can be made equal to a spin foam amplitude. Indeed, a spin foam is
2-complex and can be constructed from the (D+1)-bubbles by adding a 2-cell to any colour 0k
face, for every k.
2.4 Couplings in the spin network basis
The relation to spin foam models can be made more precise if we formulate the theory in terms
of spins j (labelling representations of SU(2)) and intertwiners ι (invariant tensors) in the case
of SU(2). Then, we expand the bubble couplings using the Peter–Weyl theorem,
λB({he}) =
∑
je spins
iv intertwiners
λB({je, ιv})
∏
e
dje
×
∏
e edges
{ ∑
mv,k,mv,k
magnetic indices
(ιv(e))mv(e),1...mv(e),D
Djemv(e),c(e),mv(e),c(e)
(he)(ιv(e))mv(e),1...mv(e),D
}
.
Djmm(h) are the Wigner matrices with dj = 2j+1 the dimension of the spin j representation and
the intertwiner appear because of the closure constraint (2.8). The intertwiners are invariant
tensors in the tensor product of representation of edges incident to a vertex, following the
ordering provided by the colours. They are chosen so that they define a orthonormal basis in
this tensor product. The new couplings λB({je, ιv}) depend on the choice of a spin for every
edge and of an intertwiner for every vertex. Such a representation theoretic data defines a spin
nertwork s, so that the couplings can be simply indexed by a spin network. For example, for
the dipole graph, the spin network formulation yields
λ 1
2
3
(h1, h2, h3) =
∑
j1, j2, j3, ι
λ 1
2
3
(j1, j2, j3, ι)
×
∑
m1,m2,m3
m1,m2,m3
ιm1m2m3D
j1
m1m1
(h1)Dj2
m2m2
(h2)Dj3
m3m3
(h3)ιm1m2m3 .
This can be considered as a non-Abelian Fourier transform,
position space: he ↔ momentum space: je, ιv.
Moreover, it is important to realise that the existence of the intertwiner between representations
labelled by spins puts some constraints on these spins. For instance, for three spins j1, j2, j3,
there exists an intertwiner (and it is then unique up to normalisation) if and only if the triangular
inequality |j1 − j2| ≤ j3 ≤ j1 + j2 is satisfied.
3 Exact renormalisation group equations
3.1 Wilson’s ideas in quantum field theory
Let us start with a heuristic overview of Wilson’s idea in quantum field theory, deferring a more
precise formulation till the next section. Consider a quantum field theory with an action S0
12 T. Krajewski and R. Toriumi
and a UV cut-off Λ0. The basic object of interest is the generating functional of correlation
functions,
Z[J ] =
∫
[Dφ]Λ0 exp{−S0[φ] + Jφ},
where [Dφ]Λ0 is a measure that integrates only over fields with momenta . Λ0.
Wilson’s idea amounts to performing this integration not in a single step but in a series
of partial integrations. The cutoff Λ0 is then reduced to a lower cutoff Λ by integrating the
modes between Λ and Λ0. The low energy physics, for instance correlation functions for fields at
momenta . Λ is then left unchanged. Denoting by φ0 the original field with modes less than Λ0,
φ a field with modes less than Λ and by χ the remaining degrees of freedom between Λ and Λ0,
φ0 = φ+ χ.
This separation of modes translates into the following equation for the generating functional
Z[J ] =
∫
[Dφ0]Λ0 exp{−S0[φ0] + Jφ0}
=
∫
[Dφ]Λ[Dχ]Λ,Λ0 exp{−S0[φ+ χ] + Jφ+ Jχ}
=
∫
[Dφ]Λ exp{−SΛ[φ] + Jφ},
where we have used Jχ = 0 since we are only interested in the low energy correlation functions.
SΛ is the Wilsonian effective action at scale Λ defined by
SΛ[φ] = − log
∫
[Dχ]Λ,Λ0 exp{−S0[φ+ χ]}.
This procedure leads to a flow equation for the effective action
Λ
dSΛ
dΛ
= β(Λ, SΛ), (3.1)
with the initial condition SΛ0 = S0.
The physics of this equation is best captured by expanding the effective action in local
operators, which are monomials of the fields and their derivatives. Writing symbolicallyO[∂d, φn]
such a monomial with n occurrences of the field and d derivatives, the action reads, in a space-
time of dimension D,
SΛ[φ] =
∑
i
local operators
∫
dDx gi(Λ)O
[
∂di , φni
]
.
The flow equation for the effective action (3.1) translates into a system of equations for the
couplings,
Λ
dgi
dΛ
= βi(Λ, {gj}).
It is convenient to write this equation in terms of dimensionless couplings ui(Λ), related to gi(Λ)
by
gi(Λ) = Λδiui(Λ),
Exact Renormalisation Group Equations and Loop Equations for Tensor Models 13
where δi is the canonical dimension of the local operator i. Assuming translational invariance,
this dimension is
δi = D − ni
(
D
2
− 1
)
− di.
Dimensional analysis allows us to write the flow equation for dimensionless couplings as
Λ
dui
dΛ
= −δiui + βi({uj}).
Couplings are classified according to the sign of their dimension:
• relevant couplings with δi > 0,
• irrelevant couplings with δi < 0,
• marginal couplings with δi = 0.
In the vicinity of the Gaußian fixed point u∗i = 0 (i.e., in perturbation theory), the physical
consequences of this equation are the following:
• Universality: Whatever the irrelevant couplings at scale Λ0 are (provided ui(Λ0) are
bounded, so that gi(Λ0) ∼ Λδi0 goes to zero for large Λ0), when Λ0 → ∞, the couplings
at scale Λ are attracted towards a manifold that can be parametrized by the relevant and
marginal couplings only.
• Renormalisability: When Λ0 → ∞ with Λ fixed, the relevant and marginal couplings at
scale Λ diverge. Perturbative renormalisation amounts to imposing their values at Λ by
a measurement. Then, all the other operators can be computed at scale Λ in terms of the
relevant and marginal operators at scale Λ, in the limit Λ0 →∞.
This provides a rationale for renormalisation: When renormalisable couplings (i.e., couplings
with δi ≥ 0) are available, the latter dominate the low energy behaviour in the limit Λ0 → ∞.
Low energy physics can be formulated in terms of low energy parameters. Of course, when no
such couplings exist, for instance if they are forbidden by symmetries, the limit Λ0 → cannot
be taken.
These statements can be rigorously proved in perturbation theory, see [35]. Obviously, a non-
perturbative study is required, especially when marginal couplings are involved or when no
renormalisable couplings are available. It may also be that the theory is attracted towards
a nontrivial fixed point. Finally, let us mention that these ideas also play a prominent role in
constructive physics. The latter aims at a nonperturbative construction of the theory based on
a multiscale analysis of the Feynman graphs [37].
3.2 Exact renormalisation group equations
In order to understand how the previous ideas can be implemented in tensor models and tensorial
group field theories, a more precise formulation is required. Within this section, we remain in
the general setting of a quantum field theory. In order to keep the notations as simple and
universal as possible, we write fields as φi, where i may be a discrete index, a space-time index
or a momentum. We separate the action into a quadratic part and an interaction potential V0[φ].
The quadratic part is included into a Gaußian measure of covariance C(t, t0) which we assume
to be written as an integral
C(t, t0) =
∫ t
t0
dsK(s). (3.2)
14 T. Krajewski and R. Toriumi
∂
∂t
= -
1
2
+
1
2
Figure 7. A graphical interpretation of the Polchinski’s equation.
The interacting part of effective action is defined as
V (t, φ) = − log
∫
dµC(t,t0)(χ) exp{−V0(t, φ+ χ}, (3.3)
with the initial condition V (t0, φ) = V0[φ] since C(t0, t0) = 0 so that there is no integration.
The effective interaction V (t, φ) obey the semi-group relation, see [43],
Z =
∫
dµC(t,t0)[φ] exp{−V0(t, φ)}
=
∫
dµC(t,t1)[φ]
∫
dµC(t1,t0)(χ) exp{−V0(φ+ χ)}
=
∫
dµC(t,t1)[φ] exp{−V (t1, φ)}.
Then, the integration over an infinitesimal shell with t1 = t−dt leads to the following differential
equation, known as Polchinski’s exact renormalisation group equation,
∂V
∂t
=
1
2
∑
i,j
Kij(t)
(
− ∂V
∂φi
∂V
∂φj
+
∂2V
∂φi∂φj
)
. (3.4)
At the perturbative level, the effective potential (3.3) can be expanded over Feynman graphs
using the background field technique. φ dependent vertices are introduced by expanding
V0(φ+ χ) in powers of χ. Then, Feynman graphs are generated by integrating over χ. Then,
the flow equation provides a recursive construction of these graphs by successively adding edges.
The first term corresponds to a tree edge (between two connected components) and the second
one to a loop edge (within the same connected component), see Fig. 7.
While Polchinski’s equation is a powerful tool in the perturbative case, in nonperturbative
investigations (for instance search for nontrivial fixed points using truncations), it is often more
convenient to sum the contributions of the trees. This is achieved by introducing the Legendre
transform Γ of V with respect to some sources. Alternatively Γ can be defined in analogy
with (3.3) by
Γ(t, φ) = − log
∫
dµC(t,t0)(χ) exp
{
−V0(t, φ+ χ) +
∂Γ
∂φ
· χ
}
.
It obeys the following differential equation, known as Wetterich’s equation, which we write
symbolically as
∂Γ
∂t
=
1
2
Tr
[
C−1KC−1
(
1 + C
∂2Γ
∂φ∂φ
C
)−1
]
, (3.5)
which has to be understood as a geometrical series for the matrix products∑
j,k
Cij
∂2Γ
∂φj∂φk
Ckl.
Exact Renormalisation Group Equations and Loop Equations for Tensor Models 15
Wetterich’s equation is particularly useful in the search for fixed points. A fixed point is an
effective action such β(S∗) = 0. Because Λ dS
dΛ = β(S), a fixed point action describes a scale
invariant theory. These fixed points play an important role in statistical physics. In the case of
the O(N) invariant model in three dimensions, the long distance behavior at a second order phase
transition is governed by a infrared stable fixed point (i.e., with only few divergent directions),
known as the Wilson–Fisher fixed point. In this case the ultraviolet cut-off Λ0 is fixed because
it is nothing but the inverse lattice spacing. The vicinity of the critical point is described
by this fixed point in the long distance limit Λ � Λ0. On the other side, ultraviolet stable
fixed point may allow the large cut-off limit Λ0 → ∞ in theories of fundamental interactions.
Asymptotically free theories provide natural examples with a Gaußian fixed point but more
general fixed points have been found in some simple models, like the D = 3 Gross–Neveu
model, see for instance [11]. Then, physical theories are parametrised by the few coordinates
on the stable manifold. Such a scenario, known as ”asymptotic safety”, has been proposed by
Weinberg and may allow to construct a quantum theory of gravity with only field theoretical
degrees of freedom. We refer the reader to the reviews [3, 36, 39] for more complete overviews
of these renormalisation group equations.
The field theory case presented in the previous section is recovered by taking t = log(Λ/Λref),
where Λref is an immaterial reference scale that disappears from the equations. Then, Ct0,t =
CΛ0,Λ smoothly integrates out modes between Λ and Λ0. In momentum space, it can be written
as
C(Λ0,Λ, p, q) = δ(p+ q)
∫ Λ0
Λ
dΛ′
Λ′
K(Λ′, p).
Sources can be included in V for convenience since we assume that they decouple from the
high energy behaviour. When formulated in momentum space, the flow equation (3.4) for
a translationally invariant field theory takes the form
Λ
dS
dΛ
=
1
2
∫
dDpK(Λ, p)
(
− δV
δφ(p)
δV
δφ(−p) +
δ2V
δφ(p)δφ(−p)
)
.
Expanding the effective interaction in powers of the field
V [Λ, φ] =
∑
n
1
n!
∫
dp1 · · · dpnδ(p1 + · · ·+ pn)Gn(Λ, p1, . . . , pn)φ(p1) · · ·φ(pn)
yields the system of equations
∂Gn
∂Λ
(Λ, p1, . . . , pn) =
1
2
∫
dDpK(Λ, p)
( n∑
i=0
{
−Gi+1(Λ, p, p1, . . . , pi)
×Gn−i+1(Λ,−p, pi+1, . . . , pn) +Gn+2(Λ, p,−p, p1, . . . , pn)
})
. (3.6)
Then, dimensional analysis can be done by further expanding Gn in powers of the momenta.
The next sections are devoted to an implementation of these ideas in the context of tensor
models and tensorial group field theories, following our presentation in [32].
3.3 Polchinski’s equation for invariant tensor models
Let us first consider an analogue of the Polchinski’s exact renormalisation group equation (3.4)
for random tensors. We consider a covariance C(t, t0) such that (3.2). The effective potential is
defined as in (3.3),
V (t, T, T ) = − log
∫
dµC(t,t0)(T
′, T
′
) exp
{
−V0
(
t, T + T ′, T + T
′)}
.
16 T. Krajewski and R. Toriumi
d
dt
= +
Figure 8. Dual interpretation of the flow equation for D = 3.
Since we treat T and T as independent variables, the flow equation for the effective poten-
tial (3.4) takes the form
∂V
∂t
=
∑
i1,...,iD
i1,...,iD
Ki1,...,iD|i1,...,iD(t)
(
− ∂V
∂Ti1,...,iD
∂V
∂T i1,...,iD
+
∂2V
∂Ti1,...,iD∂T i1,...,iD
)
. (3.7)
In order to preserve unitary invariance (2.1), the covariance has to be diagonal, see (2.3). In
this section, we restrict our attention to
Ci1,...,iD|i1,...,iD(t, t0) = K(t− t0)
∏
1≤k≤D
δik,ik ,
with K independent of t and t0. Although simple, such a covariance is not really in accordance
with the spirit of Wilson’s ideas. It would be better to choose a covariance that gradually reduces
the size of the tensors by integrating over some of its components. For simplicity, we restrict
ourselves to an invariant integration. We refer to [20, 21] for an application of a reduction of
the size to matrix models suing Wetterich’s equation.
In the unitary invariant case, we expand the interaction potential over bubbles as in (2.2)
V (t, T, T ) =
∑
B
λB(t)
CB
TrB(T , T ), (3.8)
with couplings that depend on the parameter t controlling the flow.
Polchinski’s exact renormalisation group equation for tensors (3.7) involves the operations of
deriving V with respect to T and T . At the level of bubble couplings, they respectively amount
to removing a white vertex v and a black vertex v. Then, the contraction with the derivative
of the covariance K connects the resulting edges respecting the colours. Therefore, when acting
on bubble couplings in V ,
∑
i1,...,iD
i1,...,iD
Ki1,...,iD|i1,...,iD(t)
(
− ∂V
∂Ti1,...,iD
∂V
∂T i1,...,iD
+
∂2V
∂Ti1,...,iD∂T i1,...,iD
)
is equivalent to a contraction of a pair of vertices, either on two independent couplings (first
term) or on the same coupling (second term). At the dual level of triangulation, the flow
equation can be represented graphically as in Fig. 8.
Equation (3.7) translates into a system of equations for the couplings λB. To explicit this
system, it is useful to introduce the inverse operation of a pair contraction, which is called a cut.
A k-cut c in a bubble B is defined as a subset of edges {e1, . . . , ek} of B with different colours.
The cut bubble Bc is the bubble obtained from B by cutting the k edges {e1, . . . , ek} into half-
edges, attaching to them a new black v and a new white vertex v and joining v and v by D− k
edges carrying the colors not in {e1, . . . , ek}. Then, it follows that B/vv = Bc. The notion of
Exact Renormalisation Group Equations and Loop Equations for Tensor Models 17
1
2
3
→
1
2
3
1
2
3
(a) A 3-cut.
1
2
→ 3
1
2
1
2
(b) A 2-cut.
Figure 9. Examples of cut operations.
k-cut with k 6= D is necessary to deal with the contraction of a pair that is connected by D− k
edges.
This ensures that Bc is a bubble with D colours. In particular, if c is a 0-cut, Bc is just
the disjoint union of B with a dipole. A 1-cut on an edge e is just the insertion on e of a pair
of vertices joined by D − 1 edges carrying the colours different from that of e. Substituting
the bubble expansion (3.8) into the flow equation (3.7), we arrive at the system of differential
equations governing the bubble couplings. Thus, the evolution of the couplings reads
∂λB
∂t
= K
D∑
k=0
∑
k-cut c
ND−kλBc − K
∑
D-cut c
κ(Bc)>κ(B)
∑
B′,B′′
Bc=B′∪B′′, v∈B′, v∈B′′
λB′λB′′ ,
In this equation, the sum runs over k-cuts c for 0 ≤ k ≤ D. The second term involves a sum-
mation over D-cut that increases the number of connected components κ(B) of B and over ways
of writing Bc as a disjoint union of a bubble B′ containing v and a bubble B′′ containing v.
Finally, let us emphasize that even if the initial potential V0 does not contain nonconnected
bubbles, the latter are generated by the loop-like term in the flow equation. Furthermore, they
play a key role in the large N limit.
3.4 Large N melonic universality
In order to study the large N limit, we make a change of variables in the couplings,
λB(t) = N δ(B)uB(t).
δ(B) has to be determined in such a way that the rescaled variables uB are of order unity
in N . The latter play the role of the dimensionless variables in the Wilsonian approach to
quantum field theory, with δ analogous to the canonical dimension of a coupling. This amounts
to parametrising the potential as
V (t, T, T ) =
∑
B
N δ(B)uB(t)
CB
TrB(T , T ).
18 T. Krajewski and R. Toriumi
This change of variables factorizes the leading order behaviour of the bubble couplings in the
large N limit.
Since the covariance involves the dipole and the latter could as well be considered as an
interaction, invariance under shifts of part of the covariance into the interaction requires that
the constant K be written as K = N δ(
1
2
3 ). In a more general discussion, a multiplicative factor
that admits a finite limit when N → ∞ could be included. For our purposes, this is irrelevant
and we choose this constant to be 1. In the renormalisation group language, this number can be
considered as a finite, cut-off independent wave function renormalisation and can be removed
by a constant rescaling of the fields.
The couplings uB are finite at large N if the obey a renormalisation group equation involving
only negative powers of N ,
∂uB
∂t
= β0({uB′}) +
1
N
β1({uB′}) +
1
N2
β2({uB′}) + · · · ,
so that in the large N limit the couplings uB are finite and solely determined by β0. To deter-
mine δ(B) such that this is the case, let us assume that it is of the form
δ(B) = α+ βκ(B) + γv(B),
where κ(B) is the number of connected components of the bubble B and v(B) its total number
of vertices. It is not necessary to introduce the number of edges since e(B) = D
2 v(B), but
more refined numbers could be used, like the number of faces. Then, the renormalisation group
equation reads
∂uB
∂t
=
D∑
k=0
∑
k-cut c
N δ(Bc)+D−k−δ(B)−δ(
1
2
3 )uBc
−
∑
D-cut c
κ(Bc)>κ(B)
∑
B′,B′′
Bc=B′∪B′′, v∈B′, v∈B′′
N δ(B′)+δ(B′′)−δ(B)−δ(
1
2
3 )uB′uB′′ ,
Note that for the dipole δ(
1
2
3 ) = α+β+2γ. Then, the tree-like term (second term in the previous
equation) is independent of N because v(B) = v(B′) + v(B′′)− 2 and κ(B) = κ(B′) + κ(B′′)− 1.
The power of N in the loop-like term (first term in the previous equation) can be written as
δ(Bc) +D − k − δ(B)− δ( 1
2
3 ) = D − α− k − βκ(B, c), (3.9)
where κ(B, c) is defined as the number of connected components of B containing edges of the
cut, except for a D-cut that increases the number of connected components of B, in which case
κ(B, c) = 0. This implies that κ(Bc) = κ(B)−κ(B, c)+1. This exponent is negative if we choose
α = D and β = −1 with γ arbitrary1, but other choices are possible, see the discussion in [9].
Then, the renormalisation group equation is
∂uB
∂t
=
D∑
k=0
∑
k-cut c
Nκ(B,c)−kuBc −
∑
D-cut c
κ(Bc)>κ(B)
∑
B′,B′′
Bc=B′∪B′′, v∈B′, v∈B′′
uB′uB′′ . (3.10)
We always have k ≥ κ(B, c) and k = κ(B, c) if and only if all edges of the cut belong to
different connected components of B. In particular, this holds for any 0-cut and any 1-cut. The
evolution equations for low order bubble couplings are given in Appendix A.1 for D = 3 and in
Appendix A.2 for D = 4.
1We thank Nicolas Dub for pointing this to us.
Exact Renormalisation Group Equations and Loop Equations for Tensor Models 19
(a) Melonic bubbles. (b) Non melonic bubbles.
Figure 10. Melonic and nonmeloinc bubbles for D = 3 and D = 4.
Since the exponent of N in (3.10) is negative, the limit N →∞ can be taken. For matrices
(D = 2), the large N limit is equivalent to the equation derived in [22].
In this regime, only D-cuts that disconnect the graph or k-cuts with each edge in distinct con-
nected components (including the 0-cuts) survive. Staring with the empty bubble coupling u∅,
that corresponds to logZ, only melonic bubbles are generated when solving iteratively (3.10)
in powers of t − t0. Recall that a bubble M is said to be melonic if, for every white vertex v,
there is a black vertex v such that the removal of v and v increases the number of connected
components by D − 1.
Consequently, there is a kind of melonic universality: Whatever the initial couplings uB(t0)
are, the large N partition function only depends on the melonic couplings uM(t0). Thus, as far
as the partition function is concerned, nonmelonic bubbles are irrelevant. The partition function
is all we need to compute the expectation value of a bubble since〈
trB(T, T )〉 = − 1
ND−κ(B)
∂
∂uB(t0)
logZ = Nκ(B) ∂u∅(t)
∂uB(t0)
,
since logZ = −NDu∅(t). If B is melonic, the derivative ∂u∅(t)
∂uB(t0) is of order unity. If B is not
melonic, then it is suppressed by negative powers N . Taking the second derivative of logZ
shows that the expectation value of a product of melonic bubbles factorizes at leading order.
The actual computation of this expectation value is performed in Section 4.4. Note that the
melonic universality holds for any D, including the matrix model case D = 2, for which all
interactions are melonic. However, the Gaußian ansatz presented in Section 4.4 does not hold
for matrices. These techniques also allow to show the large N Gaußian nature of the correlation
functions for random tensors, see [32].
Finally, let us emphasize that our treatment is perturbative: The partition and the expecta-
tion values of observables are considered as formal power series in the initial couplings uB(t0)
and the large N limits holds order by order. It is not excluded that small 1/N contributions
add up and alter the scaling exponents at a nonperturbative level. This could happen, for in-
stance, by resuming contributions to the dipole graph, thus leading to a nontrivial wave function
renormalisation.
3.5 Group field theory
We consider now tensorial group field theories, analogous to tensor models, provided we replace
rank D tensors by functions over D copies of a group G,
Ti1,...,iD → Φ(g1, . . . , gD), T i1,...,iD → Φ(g1, . . . , gD).
As already emphasized, g is not the complex conjugate of g but an independent group element
that is an argument of the complex conjugate field Φ. The covariance includes both a UV cut
off Λ0 and an IR one Λ (see (2.9)), and depends on the D products of group elements gig
−1
i ,
C
(
Λ,Λ0,
{
gig
−1
i
})
=
∫ 1
Λ2
1
Λ2
0
dαK
(
α,
{
gig
−1
i
})
,
20 T. Krajewski and R. Toriumi
so that
Λ
∂C
∂Λ
(
Λ,Λ0,
{
gig
−1
i
})
= − 2
Λ2
K
(
1
Λ2
,
{
gig
−1
i
})
.
In the following general discussion, K(α, {gig−1
i }) is left unspecified. It is often convenient to
take it as a product of D heat kernels averaged over the group. The general interaction can be
expanded in terms of bubble couplings (see (2.10)),
V (Λ,Φ,Φ) =
∑
B
1
CB
∫ ∏
e
dgv(e),c(e)dgv(e),c(e)
× λB
(
Λ,
{
gv(e),c(e)(gv(e),c(e))
−1
}) ∏
v
black vertices
Φ(gv,1, . . . , gv,D)
∏
v
white vertices
Φ(gv,1, . . . , gv,D),
where the bubble couplings only depend on gv(e),c(e)g
−1
v(e),c(e) because of the global translation
invariance.
For group field theories, Polchinski’s exact renormalisation group equation reads (see the
tensor model analogue (3.7))
Λ
∂V
∂Λ
= − 2
Λ2
∫ ∏
1≤i≤D
dgidgi
×K
(
1
Λ2
,
{
gig
−1
i
})(
− δV
δΦ({gi})
δV
δΦ({gi})
+
δ2V
δΦ({gi})δΦ({gi})
)
. (3.11)
Let us determine the system of equations satisfied by the bubble couplings λB. These involve
k-cuts c on the edges of B. Recall that a k-cut is a subset of k edges of distinct colours, with
i1, . . . , iD−k the colours of the edge not in the cut. We cut these edges, attach two new vertices vc
and vc and complete the graph by connecting these vertices by D− k edges ei1 , . . . , eik carrying
the colours not in the cut. We denote by Bc the resulting bubble and, if k = D and if Bc has
one more connected component than B, B′c and B′′c are any bubbles such that Bc = B′c ∪B
′′
c with
vc ∈ B′c and vc ∈ B′′c .
Then, inserting the expansion of V in terms of bubble couplings (2.10) in the exact renor-
malisation group equation (3.11) and identifying the contribution of a bubble on both sides lead
to
Λ
∂λ
∂Λ
({
gv(e),c(e)(gv(e),c(e))
−1
})
= − 2
Λ2
∫ ∏
1≤i≤D
dgidgiK
(
1
Λ2
,
{
gig
−1
i
})
×
{ ∑
0≤k≤D
∑
c
λBc
(
Λ,
{
gv(e),c(e)(gv(e),c(e))
−1
}
e∈B, gi, gi
)
−
∑
cD-cut
disconnecting B
∑
B′c∪B′′c
λB′c
(
Λ,
{
gv(e),c(e)(gv(e),c(e))
−1
}
e∈B′c
, gi
)
× λB′′c
(
Λ,
{
gv(e),c(e)(gv(e),c(e))
−1
}
e∈B′′c
, gi
)}
. (3.12)
This expression is not very illuminating, but interesting consequences can be drawn from it in
specific models. In the next two sections, we investigate the case of Abelian models with closure
constraints and SU(2) models in the spin network basis.
Exact Renormalisation Group Equations and Loop Equations for Tensor Models 21
3.6 Scaling and renormalisation for Abelian models with closure constraint
In this section, we consider Abelian models based on the group U(1)d. Elements of U(1) are
written as g = exp 2iπ θL with θ ∈ [0, L], with L being a length. The covariance is of heat kernel
type
C(Λ,Λ0, {θi − θi}) =
∫ 1
Λ2
1
Λ2
0
dα
∑
{pi}∈ 1
L
ZdD
exp
−
α ∑
1≤i≤D
p2
i + i
∑
pi(θi − θi)
δ∑ pi,0,
where we have enforced the condition
∑
i pi = 0 so that the closure constraint is fulfilled.
The exact renormalisation group equation written in terms of bubble couplings (3.12) is
Λ
∂λB({pe})
∂Λ
= − 2
Λ2
D∑
k=0
∑
k-cut c
∑
{pl}l/∈c
exp
(
− 1
Λ2
{
D∑
i=1
p2
i
})
λBc({pe}e∈B, {pl}l /∈c)δ D∑
i=1
pi,0
+
2
Λ2
∑
D-cut c
κ(Bc)>κ(B)
∑
B′,B′′
Bc=B′∪B′′,
v∈B′, v∈B′′
exp
(
− 1
Λ2
{
D∑
i=1
p2
i
})
λB′({pe}e∈B′)λB′′({pe}e∈B′′)δ D∑
i=1
pi,0
. (3.13)
This equation is the group field theory analogue of (3.6) in the framework of quantum field
theory. Note that because of the closure constraint, the bubble couplings are only defined on
subspaces of momentum space such that the sum of momenta incident to any vertex is 0. We
further assume L to be large enough (1/L� Λ) so that momenta can be treated as continuous
variables. Then we can trade sums for integrals∑
p
→ Ld
∫
dp and δ∑ pi,0 → L−dδ
(∑
pi
)
.
Otherwise, a more precise but cumbersome analysis can be performed using the Poisson resum-
mation. Finally, we also allow derivatives of the bubble couplings with respect to momenta and
impose rotational invariance in momentum space. This last hypothesis is fulfilled if we assume
that the initial couplings are invariant under rotation since the propagator is.
To understand the large Λ behaviour, we repeat the steps followed in Section 3.4. Let us
introduce the analogue of dimensionless bubble couplings uB defined by
λB({pe},Λ) = Λδ(B)uB({qe},Λ) with pe = Λqe.
δ(B) is not the dimension as would be derived by rescaling all lengths, L included. It is merely
a scaling dimension for large Λ that encodes the dominant behaviour of λB.
In terms of these new couplings, the exact renormalisation group equation (3.13) is
Λ
∂uB({qe})
∂Λ
= −δ(B)uB({qe}) +
∑
e
qe
∂uB({qe})
∂qe
− 2
D∑
k=0
∑
0≤k≤D
ck-cut
Λδ(Bc)−δ(B)−2+d(D−k)−d
Ld−d(D−k)
×
∫ ∏
l /∈c
dql exp
{
−
(∑
i
q2
i
)}
uBc({qe}e∈B, {ql}l /∈c)δ
(∑
i
qi
)
+ 2
∑
D-cut c
κ(Bc)>κ(B)
∑
Bc=B′∪B′′,
v∈B′, v∈B′′
Λδ(B
′)+δ(B′′)−δ(B)−2
Ld
exp
{
−
(∑
i
q2
i
)}
uB′({qe}e∈B′)uB′′({qe}e∈B′′).
22 T. Krajewski and R. Toriumi
Suppose that this equation can be written with only negative powers of Λ,
Λ
∂uB
∂Λ
= β0({u′B}) +
1
Λ
β1({u′B}) +
1
Λ2
β2({u′B}) + · · · .
Then, the large Λ behaviour is entirely governed (at least in perturbation theory) by β0. To this
aim, let us search for a suitable scaling dimension for the bubble interactions in the form
δ(B) = α+ βκ(B) + γv(B).
The exponent in the tree-like term is
δ(B′) + δ(B′′)− δ(B)− 2 = α+ β + 2γ − 2,
while for the loop like term it is
δ(Bc)− δ(B)− 2 + d(D − k)− d = −dk − βκ(B, c) + β + 2γ − 2 + d(D − 1).
We recall that κ(B, c) is defined as the number of connected components of B containing edges
of the cut, except for a D-cut that increases the number of connected components of B, in which
case κ(B, c) = 0. It obeys κ(Bc) = κ(B)− κ(B, c) + 1.
Setting α = d(D − 1), β = −d, and 2γ = −d(D − 2) + 2, the exponent of the tree-like
term vanishes while the loop-like one it is d(κ(B, c) − k), thus always negative. Therefore, the
couplings uB are governed by a flow equation that only involves negative powers of Λ. This
leads to a scaling dimension
δ(B) = d(D − 1)− dκ(B)− (d(D − 2)− 2)
v(B)
2
.
For connected bubbles, this is the scaling dimension found in [16] by Carrozza, Oriti and Ri-
vasseau using multiscale analysis.
Assuming that the general picture presented in Section 3.1 for quantum field theory remains
valid, perturbatively renormalisable interactions correspond to relevant and marginal interac-
tions, or equivalently δ(B) ≥ 0. This leads us to only five values of D and d for which such
interactions can be found. Indeed, if we also take into account the term associated with rescaling
of momenta,∑
e
qe
∂uB({qe})
∂qe
that is diagonalised by degree n homogenous polynomials in qe, renormalisable terms are char-
acterised by
δ(B, n) = (D − 1)d− κ(B)d− [(D − 2)d− 2]
v(B)
2
− n ≥ 0. (3.14)
In particular, among all the renormalisable interactions, we always have the mass term δ 1
2
3 ,0 = 2
and for the kinetic term δ 1
2
3 ,2 = 0.
Let us end by listing all renormalisable interactions in Abelian models with a closure con-
straint. The condition δ(B) ≥ 0 implies v(B) ≤ 2 + 4
d(D−2)−2 . Non trivial interactions involve
bubbles with at most four vertices, so that we are left with only two possibilities: d(D− 2) = 4
and d(D − 2) = 3, which leads to the following 5 renormalisable theories. This list, with con-
nected interactions only, was first obtained in [16] using the multiscale analysis of Feynman
graphs.
Exact Renormalisation Group Equations and Loop Equations for Tensor Models 23
• D = 3 and d = 4 so that δ = 8 − 4κ − v. The renormalisable interactions is quartic and
melonic
(δ = 0).
The fixed point structure of a non-Abelian version of this model has been studied in [14].
• D = 4 and d = 2 so that δ = 6− 2κ− v. The renormalisable interactions are quartic
(δ = 0), (δ = 0).
Note that the second interaction is not melonic (called necklace in [9]).
• D = 6 and d = 1 so that δ = 5− κ− v. The renormalisable interactions are quartic
(δ = 0), (δ = 0), (δ = 0).
The last two interactions are not melonic. This model was shown to be renormalisable
in [41, 33] and its fixed point structure was further investigated in [7].
• D = 3 and d = 3 so that δ = 6 − 3κ − v/2. The renormalisable interactions are quartic
and sextic
(δ = 1), (δ = 0), (δ = 0), (δ = 0).
The first term is in fact superrenormalisable (δ > 0) and the last one is not melonic.
Its non-Abelian counterpart is the tensorial SU(2) group field theory in D = 3 which
corresponds to three-dimensional Euclidian quantum gravity without the cosmological
constant, first shown to be renormalisable in [16].
• D = 5 and d = 1 so that δ = 4− κ− v/2. The renormalisable interactions are quartic and
sextic. We first have the melonic ones
(δ = 1), (δ = 0), (δ = 0), (δ = 0).
The last interaction is not connected (κ = 2). This model, with the nonconnected in-
teraction, was shown to be renormalizable in [41]. Besides, there are also nonmelonic
renormalisable interactions.
Note that all these bubble couplings with v ≥ 4 have dimension 0 or 1, so that there is no
possibility of adding a derivative coupling. Indeed, the latter corresponds to a term with n > 0
in (3.14), which would lead to a negative dimension, the case n = 1 being forbidden by rotational
symmetry. Of course, we could always add quadratic terms (mass and kinetic terms) to the list
of interaction. Then, the kinetic term can play the role of a derivative coupling.
It is also worthwhile to notice that some of the renormalisable interactions are not neces-
sary melonic. As for unitarily invariant random tensors, nonmelonic interactions appear with
a negative power of Λ in the flow of renormalisable melonic interactions. Therefore, they do not
require renormalisation if the bare theory only contains melonic interactions. However, if the
bare theory contains nonmelonic renormalisable interactions, the latter requires renormalisation.
24 T. Krajewski and R. Toriumi
3.7 SU(2) models in the spin network basis
In the SU(2) case which is involved in many spin-foam models, it is possible to expand all the
bubble couplings on spin networks, see [31], rather than just bubbles. For the group SU(2), spin
network observables provide a nice basis. We work with the heat kernel
H
(
α, gg−1
)
=
∑
j
(2j + 1) exp
{
−αj(j + 1)
} ∑
−j≤m≤j
Djmm
(
gg−1
)
.
The covariance is obtained by averaging over the group
C
(
Λ,Λ0,
{
gig
−1
i
})
=
∑
j1...jD spins
ι intertwiner
( ∑
m1,...,mD
m1,...,mD
ιm1,...,mDDj1m1m1
(
g1g
−1
1
)
· · ·
× DjDmDmD
(
gDg
−1
D
)
ιm1,...,mDdj1 · · · djD exp
(
−α
{
j1(j1 + 1) + · · ·+ jD(jD + 1)
}))
,
where we performed the group averaging thanks to the relation∫
dhDj1
m1m′1
(h) · · · DjD
mDm
′
D
(h) =
∑
ι
ιm1...mD ιm′1...m′D .
Then, the spin network formulation of (3.12) is
Λ
∂λB({je, ιv})
∂Λ
= − 2
Λ2
∑
0≤k≤D
ck-cut
∑
ι
∑
{jl}l/∈c
∏
l
djl exp
(
− 1
Λ2
{
D∑
i=1
ji(ji + 1)
})
× λBc({je}e∈B, {jl}l /∈c, {iv}v∈B, ι) +
2
Λ2
∑
D-cut c
κ(Bc)>κ(B)
∑
ι
∑
B′,B′′
Bc=B′∪B′′,
v∈B′, v∈B′′
exp
(
− 1
Λ2
{
D∑
i=1
ji(ji + 1)
})
× λB′({je′′}e′′∈B′{ιv′}v′∈B′ , ι)λB′′({je′′}e′′∈B′′{ιv′′}v′′∈B′′ , ι).
Let us emphasize the analogy with (3.13), the spin playing the role of the momenta. The
constraint is implemented by the insertion of the interwiner ι at the two vertices created by the
cut. Consequently, it acts as a constraint since we can cut some edges only if their spins admit
an intertwiner. For D = 3 this constraint is the triangle inequality. It is also interesting to note
the similarity with the tensor model case, with the factor ND−k replaced by the product over
the D − k edges not in the cut of the dimensions djl .
3.8 Wetterich’s equation and nontrivial fixed points
The search for nontrivial fixed points in group field theories have been investigated by several
authors. We give here a very brief account of the rationale behind these works since they also
make use of exact renormalisation group equations. We refer the reader to the original literature
[4, 6, 7, 14] for the precise calculations and the phase diagrams.
These works use Wetterich’s equation (3.5) expanded as
∂Γ
∂t
= Tr
[
C−1KC−1
(
1 + C
∂2Γ
δΦδΦ
C
)−1
]
=
∞∑
n=0
(−1)n Tr
C−1KC−1C
∂2Γ
δΦδΦ
· · ·C ∂2Γ
δΦδΦ︸ ︷︷ ︸
n times
. (3.15)
Exact Renormalisation Group Equations and Loop Equations for Tensor Models 25
B1 B2
32
1
1
32
1
1
3232
0
0
v2
v2
v1
v1
Figure 11. A contribution to the evolution of a quartic melon.
In this framework, a heat kernel type of covariance is used and the effective average action Γ is
expanded over bubble couplings as in (2.10). Then, (3.15) is expanded a series of colour 0 one
loop graphs whose vertices are the D-bubbles in the expansion of Γ.
To obtain the contribution to ∂λB
∂t , one has to look for all possibilities of writing B as
B =
(
B1 ∪ · · · ∪ Bn
)/(
v1v2, v2v3, . . . , vn−1vn, vnv1
)
,
where vi and vi are respectively a white and a black vertex in Bi. For instance, ∂
∂tλ involves
a term quadratic in λ , see Fig. 11. In order to write down the evolution equation of a coupling
pertaining to a given bubble B in terms of all the bubbles in the chain B1, . . . ,Bn, it is therefore
necessary to find all such possible chains of bubbles. This is solved by the following algorithm,
involving a sequence of n cuts since they are the inverse of inverse operation of a contraction of
a line of color 0.
1. The first cut c1 is arbitrary and results in Bc1, with two new vertices v1 and v1. This cut
determines the insertion of the derivative of the covariance K, materialised by a cross in
Fig. 11.
2. The next cuts c2, . . . , cn must always increase the number of connected components (there-
fore they involve all D colors) with the constraint that each new connected component
contains 1 or 2 new vertices vi and vj . If the first cut has not increased the number of
connected components, all the cuts have to be performed on the connected component
of Bc1 containing v1 and v1; otherwise they may be performed anywhere.
3. Partition the set of connected components of Bc1...cn into n sets, each of which containing
one of the white vertices and one of the black vertices previously constructed in such a way
that joining corresponding vertices leads to a cycle of colour 0 lines.
4. The n sets of the partition define the n bubbles B1, . . . ,Bn
As is usual, for a given bubble there is a maximal value of n, if we exclude the trivial cuts
involving all the D edges incident to a given vertex. Except for the first cut, this kind of cut
simply amounts to an insertion of a chain of dipoles which may be removed by a change in the
propagator.
To search for fixed points, one usually resorts to truncations. It means that we select a few
couplings, usually including the mass m2Φ·Φ and the kinetic term (Z−1)Φ·∆Φ, both associated
to the dipole graph, as well as some interactions, encoded in the bubble couplings, expanded
26 T. Krajewski and R. Toriumi
Figure 12. The quartic melon in D = 6.
around zero momentum (in Abelian theories). For instance, in the model considered by Benedetti
and Lahoche in [7] in D = 6 with the group U(1), only the interaction associated to the quartic
melon given in Fig. 12 has been retained. We focus now on this model and review briefly their
work. In this case, the flow equations reduce to the system
∂m2
∂t
= βm2
(
Z,m2, λ, t
)
,
∂Z
∂t
= βZ
(
Z,m2, λ, t
)
,
∂λ
∂t
= β
(
Z,m2, λ, t
)
.
The wave function renormalisation is not a coupling and one has to rescale the fields as Φ →
Z−1/2Φ and Φ→ Z−1/2Φ in order to have fixed points. Second, it is also necessary to rescale m
and λ according to their scaling dimension for large t,
λ = et(6−D)Z2λ, m2 = e2tZm2.
Note that 6−D is the scaling dimension of the bubble found in equation (3.14), with Λ ∝ et.
Finally, determining Z in terms of m and λ leads to a system of equations
∂m2
∂t
= βm2
(
m2, λ, t
)
,
∂λ
∂t
= β
(
m2, λ, t
)
.
Although nonautonomous (i.e., still involving t) because of the finite size of the group, there are
fixed points in the limit t→∞. Besides the Gaußian one with m = 0 and λ = 0, there are two
others fixed points. One of them cannot be connected to the Gaußian one by a trajectory because
of the presence of a singularity. The other one has one relevant and one irrelevant direction. It
corresponds to a negative value of m2, possibly indicating a phase transition. Similar results
have been obtained by the other authors mentioned at the beginning of this section.
4 Loop equations
4.1 Reparametrisation invariance in quantum field theory
To begin with, let us consider again a general quantum field theory whose fields are collectively
denoted by φi. The basic object of interest is the generating function for correlation functions,
which we write symbolically as
Z[J ] =
∫
[Dφ] exp{−S[φ] + Jφ}.
In this section, we consider formal algebraic relations satisfied by the generating function, treat-
ing the field as a finite-dimensional variable for simplicity. This is enough for the heuristic
purpose of this following presentation.
Because it is an integral over all fields, the generating functional Z[J ] must be invariant under
any change of variables in the space of fields, a property known as reparametrisation invariance.
Consider an infinitesimal change of variable, which we write as
φi → φi + εXi[φ].
Exact Renormalisation Group Equations and Loop Equations for Tensor Models 27
Invariance of Z[J ] under such a change of variable translates into the equation∫
[Dφ]
∑
i
(
∂Xi
∂φi
−Xi ∂S
∂φi
+XiJi
)
exp{−S[φ] + Jφ} = 0.
The first term is the Jacobian of the change of variable and the two others arise from the
variation of the argument of the exponential. Since every function of the field can be obtained
as a derivative acting on the source, we arrive at∑
i
{
∂Xi
∂φi
(
∂
∂J
)
−Xi
( ∂
∂J
)
∂S
∂φi
(
∂
∂J
)
+ JiX
i
(
∂
∂J
)}
Z[J ] = 0. (4.1)
Introducing the differential operator
LX =
∑
i
{
∂Xi
∂φi
(
∂
∂J
)
−Xi
(
∂
∂J
)
∂S
∂φi
(
∂
∂J
)
+ JiX
i
(
∂
∂J
)}
.
Invariance under reparametrisation takes the following simple form
LXZ[J ] = 0.
Although relatively complicated, these differential operators obey a simple commutation relation
[LX ,LY ] = L[X,Y ]
with [X,Y ] the commutator between the two vectors fields X and Y defined on the space of
fields
[X,Y ]i =
∑
j
Xj ∂Y
i
∂φj
− Y j ∂X
i
∂φj
.
[X,Y ] is nothing but the infinitesimal part of the commutator between the change of variables
associated to X and Y . Therefore, the operators L define a representation of the Lie algebra of
infinitesimal change of variables.
When the vector field X is independent of φ, the associated change of variables is simply
a translation in the space of fields, φi → φi + εi. Then, (4.1) takes the form{
∂S
∂φi
(
∂
∂J
)
+ Ji
}
Z[J ] = 0.
These are the Schwinger–Dyson equations. which are the quantum analogue of the equations of
motion. They provide enough information to reconstruct Z[J ] in perturbation theory, see [43].
4.2 Loop equations for tensor models
Let us apply the general framework depicted in the previous section to unitarily invariant random
tensor models. These equations where first derived by Gurau in [24] at leading order and at all
orders in [26]. We follow the presentation given in [30].
We start with an interaction of the form given in (2.2), except that we do not include the
combinatorial factor,
V (T, T ) =
∑
B
λB TrB(T , T ).
28 T. Krajewski and R. Toriumi
The inclusion of the combinatorial factor CT is immaterial since it amounts to trading λB for λB
CB
.
Here, we have included all bubbles, even nonconnected ones and the empty one. The latter is
simply a constant λ∅ that does not couple to the tensors. Then, solving the theory amounts to
computing the partition function as a function of all the bubble couplings,
Z({λB}) =
∫
dµC(T, T ) exp{−V (T, T )}, (4.2)
with a diagonal covariance C = t δi1i1 · · · δiDiD . This has to be considered as a formal power
series in the variables λB, ordered by the number of vertices of the bubbles, in such a way that
at a given order there is only a finite number of terms. The expectation value of a bubble is
then obtained by〈
TrB(T , T )
〉
= − ∂
∂λB
(logZ).
If only a subset of bubble couplings are present in the actual potential, all couplings not present
in the potential have to be set to 0 after the derivation.
For unitarily invariant models, there is a preferred set of change of variables parametrised
by bubbles with a vertex removed. Indeed, if v is a black vertex, then we define TrB\v(T , T )
as TrB(T , T ) with the tensor T at v removed. Treating T and T as independent variables, the
change of variables associated with B\v is
T → T + εB\v TrB\v(T , T ), T → T , (4.3)
with εB\v an infinitesimal parameter. Performing this change of variables in (4.2) leads to the
identity∫
dµC(T, T )
{
−t−1 TrB(T , T ) +
∑
v∈B
white vertex
N l(B,v,v) TrB/vv(T , T )
−
∑
B′
v′∈B′ white vertex
λB′ Tr(B′∪B)/vv′(T , T )
}
exp{−V (T, T )} = 0.
For any bubble B, recall that B/vv is the bubble with the pair of vertices vv contracted (see
Fig. 2) and l(B, v, v) is the number of edges in B that connect v and v. The first two terms arise
from the change of the Gaußian measure, the second one being the Jacobian. The third term
comes from the change of the potential. Note that the contribution of the covariance is of the
same form as that of the dipole, considered as a bubble coupling.
Writing this equation in terms of bubble expectation values, it reads〈
TrB(T, T )
〉
= t
∑
v∈B
white vertex
N l(B,v,v)
〈
TrB/vv(T , T )
〉
− t
∑
B′ v′∈B′
white vertex
λB′
〈
Tr(B′∪B)/vv′(T , T )
〉
. (4.4)
Consider a bubble in the potential as a triangulation of a space of dimension D − 1. Then,
(4.4) has the following interpretation. On a boundary triangulation, consider the simplex of
dimension D− 1 labelled by vertex v. Then, the expectation of an observable constructed using
such a boundary triangulation is a combination of the expectation values for which this triangle
is connected to another triangle on the same boundary triangulation or in a different one, see
Fig. 13. The power of N counts the number of simplices of dimension D − 2 that are shared
by the two simplices that we identify, see Fig. 14. Low powers of N correspond to topology
changes, so that the latter are expected to be subdominant.
Exact Renormalisation Group Equations and Loop Equations for Tensor Models 29
〈
B
v
〉
= t
∑
v∈B
white vertex
N l(B,v,v)
〈
v v
B
〉
− t
∑
B′
v′∈B′ white vertex
λB′
〈
B B′
v v
〉
Figure 13. Geometrical interpretation of equation (4.4).
(a) Triangle not sharing an edge.
→ N
(b) Triangle sharing an edge.
Figure 14. Contraction of a pair of triangles.
Since a bubble expectation can be generated by derivation with respect to a bubble coupling,
the previous equation can be written as a differential operator acting on the partition function
LB\vZ = 0
with
LB\v = t−1 ∂
∂λB
−
∑
v∈B
white vertex
N l(B,v,v) ∂
∂λB/vv
+
∑
B′
v′∈B′ white vertex
λB′
∂
∂λ(B′∪B)/vv′
. (4.5)
These equations are constraints on the partition function expressed as differential operators with
respect to the couplings. They are tensor analogues of the loop equations for matrix models,
which are recovered when D = 2, see Section 4.3.
The commutation relation between two such differential operators reads[
LB1\v1
,LB2\v2
]
=
∑
v′1 6=v1
white vertex in B1
L(B1∪B2)/v′1v2
−
∑
v′2 6=v2
white vertex in B2
L(B1∪B2)/v′2v1
(4.6)
30 T. Krajewski and R. Toriumi
and simply reproduces the commutation relation between two changes of variables of the form
(4.3). These operators generate a Lie algebra whose structure is related to a Connes–Kreimer
algebra, see [18].
By the same token, removing a white vertex on a bubble defines a change of variables for the
conjugate tensor
T → T, T → T + +εB\v TrB\v(T , T ).
It leads to an analogous equation LB\v Z = 0. The operators LB\v obey similar commutations
relations and commute with LB\v,[
LB\v,LB′\v′
]
= 0,
because we treat T and T as independent variables.
In our derivation of the constraints, we have included all possible bubble couplings in the
action, even for nonconnected bubbles and for the empty one. However, most of the authors do
not consider such couplings. The empty bubble can only appear as B/vv when B is a dipole. If
the empty bubble is not present in the action, we assume that ∂
∂λ∅
is the identity. If we start
with only connected bubble couplings, nonconnected ones can appear only in the Jacobian. The
latter can be generated by higher order derivations on the connected ones. Indeed, let us denote
by (B/vv)c the connected components of B/vv. Then, we simply have to replace the differential
operators by
LB\v = t−1 ∂
∂λB
−
∑
v∈B
white vertex
N l(B,v,v)
∏
c connected
component of B/vv
(
− ∂
∂λ(B/vv)c
)
+
∑
B′
v′∈B′ white vertex
λB′
∂
∂λ(B′∪B)/vv′
.
This is in general a differential operator of order > 1. Nevertheless, the form of the loop
equations and all commutation relations remain unchanged.
4.3 The case of matrix models
When D = 2, coloured tensor models reduce to a theory of complex nonhermitian matrices of
size N . In this case, connected bubble are just indexed by a integer n which is the equal number
of black and white vertices. Then, connected bubble invariants are just traces of the nth power
of M †M and the potential is
V (M,M †) = λ0 +
∞∑
n=1
λn Tr
(
M †M
)n
,
where we have included the empty bubble with n = 0. The change of variable associated to a
bubble with a black vertex removed (see (4.3)) is
M →M + εnM
(
M †M
)n−1
, M † →M †,
for n > 1.
To find the explicit form of the differential operators LB\v (4.5), first consider the case of
a dipole, n = 1. Then, B/vv is empty and we get
L1 = t−1 ∂
∂λ1
−N2 ∂
∂λ0
+
∑
k≥1
kλk
∂
∂λk
.
Exact Renormalisation Group Equations and Loop Equations for Tensor Models 31
For n > 1, B/vv has either one connected component and a free loop (if v and v are adjacent),
or two connected components (if v and v are not adjacent). This leads to
Ln = t−1 ∂
∂λn
− 2N
∂
∂λn−1
−
∑
2≤k≤n−1
∂2
∂λk−1∂λn−k
+
∑
k≥1
kλk
∂
∂λn+k−1
.
In order to compare with the standard form of the loop equations in the literature, it is convenient
to define
λ0 = N2t0, λ1 = N
(
t1 − t−1
)
, λn = Ntn for n > 1,
so that the potential reads
V (M,M †) = N
∞∑
n=0
tn Tr
(
M †M
)n
.
Then, using ∂Z
∂t0
= −N2Z, the differential operators take the form
Ln =
1
N2
∑
1≤k≤n
∂2
∂tk−1∂tn−k
+
∑
k≥1
ktk
∂
∂tn+k−1
.
These operators obey the commutation relation[
Lm,Ln
]
= (m− n)Lm+n−1. (4.7)
This is nothing but the commutation relation of the operators xn ∂
∂x , known as the Witt algebra.
After a shift n→ n+1 and introducing an extra operator L−1, these are the Virasoro constraints,
see [2].
Finally, it may be interesting to note that the Witt algebra also appears as a subalgebra of
the constraint algebra of the tensor models. Indeed, if we define
Ln =
∑
B, n(B)=n
v ∈ B black vertex
LB\v,
with n(B) the number of white vertices of B, then the commutation relation (4.6) implies those
of the Witt algebra (4.7).
4.4 An application to melonic dominance
In Section 3.4, we have shown that only melonic couplings contribute at leading order to the
partition function, provided each bubble coupling is written as λB = ND−1uB, considering only
connected bubbles for simplicity. Then, Schwinger–Dyson equations can be used to evaluate the
expectation value of such a bubble, see [8]. Indeed, with only melonic bubblesM, (4.4) reduces to〈
TrM(T, T )
〉
= t
∑
v∈M
white vertex
N l(B,v,v)
〈
TrM/vv(T , T )
〉
− t
∑
M′, v′∈B′
white vertex
uM′
〈
Tr(M′∪M)/vv′(T , T )
〉
. (4.8)
Let us make the Ansatz that the expectation value is of Gaußian type with a yet to be determined
covariance G(t, {uM′}) depending on all melonic couplings in V ,〈
TrM(T, T )
〉
= NGn(M)
32 T. Krajewski and R. Toriumi
with n(M) the equal number of black or white vertices ofM. We also assume that the integral
for a product of melonic bubbles factorizes in the large N limit. Then (4.8) reduces, for D > 2, to
−t−1Gn(M) +Gn(M)−1 −
∑
M′
n(M′)uMGn(M′)+n(M)−1 = 0.
The assumption D > 2 is crucial here since in this case there is a single v such that the contri-
bution of the second term in (4.8) is dominant, with all others subdominant.
This equation is solved if
G = t+ t
∑
M′
n(M′)uM′Gn(M′) = 0.
If a finite number of bubbles are present in the action, an explicit perturbative expansion of G
in terms of the couplings can be obtained by a recursive solution, starting with G = t at lowest
order.
5 Conclusion and outlook
In this survey, we have presented the exact renormalisation group equations and the loop equa-
tions for random tensors and group field theories. Exact renormalisation group equations are
powerful devices that allow to classify the interactions according to their degree of relevancy in
a specified limit. For unitarily invariant tensor models, the most relevant interactions are the
melonic ones. In the group field theory case in the limit of large cut-off, the theory is governed
by renormalisable interactions. On the other side, loop equations are constraints on the partition
function arising from reparametrisation invariance.
It is fair to say that the applications of these equations to tensor models and group field
theories have not yet been completely explored. Recent works by various groups suggest the
following research directions.
• The group field theories we have presented are based on abelian groups. However, powerful
techniques based on the heat kernel have been developed by Carrozza, Oriti and Rivasseau
in [12, 13, 17] in the context of multiscale analysis. It would certainly be very useful to
adapt these techniques to flow equations. This would also be particularly useful in the
renormalisation of D = 4 group field theories related to general relativity.
• Most of the works presented here is of perturbative nature. However, exact renormali-
sation group equations, especially Wetterich’s one, can be used in the nonperturbative
regime. Ben Geloun, Benedetti, Carrozza, Lahoche and Oriti have already investigated
the existence of fixed points using truncations, see [4, 6, 7, 14] and the review [15].
• Melonic interactions turn out to be the most relevant ones. Nevertheless, the effects of
nonmelonic ones have not yet been understood. This is particularly true in group field
theories where some models exhibit nonmelonic interaction of scaling dimension 0.
• Loop equations play a fundamental role in the theory of random matrices, establishing
the relation with conformal field theory and integrable hierarchies. It would be of high
interest to establish similar connections in the case of random tensors.
A ERGE for low order bubble couplings
In this appendix, a few example of evolutions equation for rescaled couplings are given.
Exact Renormalisation Group Equations and Loop Equations for Tensor Models 33
A.1 Couplings in rank D = 3 tensor models
∂
∂t
u =
[
u
]∣∣
0 cut
+
[
3u
]∣∣
1 cut
−
[
u2
]∣∣
3 cuts
+
1
N
[
3u
]∣∣
2 cuts
+
1
N3
[
u
]∣∣
3 cuts
,
∂
∂t
u =
[
u
]∣∣
0 cut
+
[
4u + 2u
]∣∣
1 cut
−
[
4u u
]∣∣
3 cuts
+
1
N
[
8u + 2u + 2u
]∣∣
2 cuts
+
1
N2
[
4u
]∣∣
3 cuts
+
1
N3
[
4u
]∣∣
3 cuts
,
∂
∂t
u =
[
u
]∣∣
0 cut
+
[
6u
]∣∣
1 cut
+
[
6u
]∣∣
2 cuts
−
[
4u u
]∣∣
3 cuts
+
1
N
{[
6u
]∣∣
2 cuts
+
[
6u
]∣∣
3 cuts
}
+
1
N3
[
2 u
]∣∣
3 cuts
,
∂
∂t
u =
[
u
]∣∣
0 cut
+
[
6u + 3u
]∣∣
1 cut
−
[
6u u + 3u2
]∣∣
3 cuts
+
1
N
[
3u + 6u + 12u + 6u
]∣∣
2 cuts
+
1
N2
[
12u + 6u
]∣∣
3 cuts
+
1
N3
[
6u + 3u
]∣∣
3 cuts
,
∂
∂t
u =
[
u
]∣∣
0 cut
+
[
4u + u + 4u
]∣∣
1 cut
−
[
6u u + 2u2
]∣∣
3 cuts
+
1
N
[
6u + 6u + 3u + 10u + 2u
]∣∣
2 cuts
+
1
N2
[
6u + u + 8u
]∣∣
3 cuts
+
1
N3
[
6u + 2u
]∣∣
3 cuts
,
∂
∂t
u =
[
u
]∣∣
0 cut
+
[
9u
]∣∣
1 cut
+
[
18u
]∣∣
2 cuts
+
[
6u − 6u u − 6u2
]∣∣
3 cuts
+
1
N
{[
9u
]∣∣
2 cuts
+
[
18u
]∣∣
3 cuts
}
+
1
N3
[
3u
]∣∣
3 cuts
,
34 T. Krajewski and R. Toriumi
∂
∂t
u =
[
u
]∣∣
0 cut
+
[
3u + 4u + 2u
]∣∣
1 cut
+
[
8u + 4u
]∣∣
2 cuts
−
[
6u u + 4u u
]∣∣
3 cuts
+
1
N
{[
3u + 8u + 2u + 2u
]∣∣
2 cuts
+
[
2u + 8u + 14u
]∣∣
3 cuts
}
+
1
N2
[
4u
]∣∣
3 cuts
+
1
N3
[
5u
]∣∣
3 cuts
.
A.2 Couplings in rank D = 4 tensor models
∂
∂t
u =
[
u
]∣∣
0 cut
+
[
4u
]∣∣
1 cut
−
[
u2
]∣∣
3 cuts
+
1
N
[
6u
]∣∣
2 cuts
+
1
N2
[
4u
]∣∣
3 cuts
+
1
N4
[
u
]∣∣
4 cuts
,
∂
∂t
u =
[
u
]∣∣
0 cut
+
[
2u
]∣∣
1 cut
−
[
4u u
]∣∣
4 cuts
+
1
N
{[
6u
]∣∣
1 cut
+
[
18u + 6u
]∣∣
2 cuts
}
+
1
N2
[
2u + 12u + 12u + 6u
]∣∣
3 cuts
+
1
N3
[
12u
]∣∣
4 cuts
+
1
N4
[
4 u
]∣∣
4 cuts
,
∂
∂t
u =
[
u
]∣∣
0 cut
+
[
8u
]∣∣
1 cut
−
[
4u u
]∣∣
4 cuts
+
1
N
[
4u + 16u + 4u
]∣∣
2 cuts
+
1
N2
[
16u + 16u
]∣∣
3 cuts
+
1
N3
[
8u + 4u
]∣∣
4 cuts
+
1
N4
[
4u
]∣∣
4 cuts
,
∂
∂t
u =
[
u
]∣∣
0 cut
+
[
8u
]∣∣
1 cut
+
[
12u
]∣∣
2 cuts
−
[
4u u
]∣∣
4 cuts
+
1
N
{[
12u
]∣∣
2 cuts
+
[
24u
]∣∣
3 cuts
}
+
1
N2
{[
8u
]∣∣
3 cuts
+
[
8u + 6u
]∣∣
4 cuts
}
+
1
N4
[
2u
]∣∣
4 cuts
.
Exact Renormalisation Group Equations and Loop Equations for Tensor Models 35
Acknowledgements
T.K. is partially supported by the grant “ANR JCJC CombPhysMat2Tens”.
References
[1] Ambjørn J., Durhuus B., Jónsson T., Three-dimensional simplicial quantum gravity and generalized matrix
models, Modern Phys. Lett. A 6 (1991), 1133–1146.
[2] Ambjørn J., Jurkiewicz J., Makeenko Y.M., Multiloop correlators for two-dimensional quantum gravity,
Phys. Lett. B 251 (1990), 517–524.
[3] Bagnuls C., Bervillier C., Exact renormalization group equations: an introductory review, Phys. Rep. 348
(2001), 91–157, hep-th/0002034.
[4] Ben Geloun J., Martini R., Oriti D., Functional renormalisation group analysis of tensorial group field
theories on Rd, arXiv:1601.08211.
[5] Ben Geloun J., Rivasseau V., A renormalizable 4-dimensional tensor field theory, Comm. Math. Phys. 318
(2013), 69–109, arXiv:1111.4997.
[6] Benedetti D., Ben Geloun J., Oriti D., Functional renormalisation group approach for tensorial group field
theory: a rank-3 model, J. High Energy Phys. 2015 (2015), no. 3, 084, 40 pages, arXiv:1411.3180.
[7] Benedetti D., Lahoche V., Functional renormalization group approach for tensorial group field theo-
ry: a rank-6 model with closure constraint, Classical Quantum Gravity 33 (2016), 095003, 35 pages,
arXiv:1508.06384.
[8] Bonzom V., Revisiting random tensor models at large N via the Schwinger–Dyson equations, J. High Energy
Phys. 2013 (2013), no. 3, 160, 25 pages, arXiv:1208.6216.
[9] Bonzom V., Delepouve T., Rivasseau V., Enhancing non-melonic triangulations: a tensor model mixing
melonic and planar maps, Nuclear Phys. B 895 (2015), 161–191, arXiv:1502.01365.
[10] Bonzom V., Gurau R., Ryan J.P., Tanasa A., The double scaling limit of random tensor models, J. High
Energy Phys. 2014 (2014), no. 9, 051, 49 pages, arXiv:1404.7517.
[11] Braun J., Gies H., Scherer D.D., Asymptotic safety: a simple example, Phys. Rev. D 83 (2011), 085012,
15 pages, arXiv:1011.1456.
[12] Carrozza S., Tensorial methods and renormalization in group field theories, Springer Theses, Springer,
Cham, 2014, arXiv:1310.3736.
[13] Carrozza S., Discrete renormalization group for SU(2) tensorial group field theory, Ann. Inst. Henri
Poincaré D 2 (2015), 49–112, arXiv:1407.4615.
[14] Carrozza S., Group field theory in dimension 4 − ε, Phys. Rev. D 91 (2015), 065023, 10 pages,
arXiv:1411.5385.
[15] Carrozza S., Flowing in group field theory space: a review, arXiv:1603.01902.
[16] Carrozza S., Oriti D., Rivasseau V., Renormalization of a SU(2) tensorial group field theory in three dimen-
sions, Comm. Math. Phys. 330 (2014), 581–637, arXiv:1303.6772.
[17] Carrozza S., Oriti D., Rivasseau V., Renormalization of tensorial group field theories: Abelian U(1) models
in four dimensions, Comm. Math. Phys. 327 (2014), 603–641, arXiv:1207.6734.
[18] Connes A., Kreimer D., Renormalization in quantum field theory and the Riemann–Hilbert problem. I. The
Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 210 (2000), 249–273, hep-
th/9912092.
[19] Dartois S., Gurau R., Rivasseau V., Double scaling in tensor models with a quartic interaction, J. High
Energy Phys. 2013 (2013), no. 9, 088, 33 pages, arXiv:1307.5281.
[20] Eichhorn A., Koslowski T., Continuum limit in matrix models for quantum gravity from the functional
renormalization group, Phys. Rev. D 88 (2013), 084016, 15 pages, arXiv:1309.1690.
[21] Eichhorn A., Koslowski T., Towards phase transitions between discrete and continuum quantum spacetime
from the renormalization group, Phys. Rev. D 90 (2014), 104039, 14 pages, arXiv:1408.4127.
[22] Gurau R., A diagrammatic equation for oriented planar graphs, Nuclear Phys. B 839 (2010), 580–603,
arXiv:1003.2187.
[23] Gurau R., Colored group field theory, Comm. Math. Phys. 304 (2011), 69–93, arXiv:0907.2582.
http://dx.doi.org/10.1142/S0217732391001184
http://dx.doi.org/10.1016/0370-2693(90)90790-D
http://dx.doi.org/10.1016/S0370-1573(00)00137-X
http://arxiv.org/abs/hep-th/0002034
http://arxiv.org/abs/1601.08211
http://dx.doi.org/10.1007/s00220-012-1549-1
http://arxiv.org/abs/1111.4997
http://dx.doi.org/10.1007/JHEP03(2015)084
http://arxiv.org/abs/1411.3180
http://dx.doi.org/10.1088/0264-9381/33/9/095003
http://arxiv.org/abs/1508.06384
http://dx.doi.org/10.1007/JHEP03(2013)160
http://dx.doi.org/10.1007/JHEP03(2013)160
http://arxiv.org/abs/1208.6216
http://dx.doi.org/10.1016/j.nuclphysb.2015.04.004
http://arxiv.org/abs/1502.01365
http://dx.doi.org/10.1007/JHEP09(2014)05
http://dx.doi.org/10.1007/JHEP09(2014)05
http://arxiv.org/abs/1404.7517
http://dx.doi.org/10.1103/PhysRevD.83.085012
http://arxiv.org/abs/1011.1456
http://dx.doi.org/10.1007/978-3-319-05867-2
http://arxiv.org/abs/1310.3736
http://dx.doi.org/10.4171/AIHPD/15
http://dx.doi.org/10.4171/AIHPD/15
http://arxiv.org/abs/1407.4615
http://dx.doi.org/10.1103/PhysRevD.91.065023
http://arxiv.org/abs/1411.5385
http://arxiv.org/abs/1603.01902
http://dx.doi.org/10.1007/s00220-014-1928-x
http://arxiv.org/abs/1303.6772
http://dx.doi.org/10.1007/s00220-014-1954-8
http://arxiv.org/abs/1207.6734
http://dx.doi.org/10.1007/s002200050779
http://arxiv.org/abs/hep-th/9912092
http://arxiv.org/abs/hep-th/9912092
http://dx.doi.org/10.1007/JHEP09(2013)088
http://dx.doi.org/10.1007/JHEP09(2013)088
http://arxiv.org/abs/1307.5281
http://dx.doi.org/10.1103/PhysRevD.88.084016
http://arxiv.org/abs/1309.1690
http://dx.doi.org/10.1103/PhysRevD.90.104039
http://arxiv.org/abs/1408.4127
http://dx.doi.org/10.1016/j.nuclphysb.2010.06.022
http://arxiv.org/abs/1003.2187
http://dx.doi.org/10.1007/s00220-011-1226-9
http://arxiv.org/abs/0907.2582
36 T. Krajewski and R. Toriumi
[24] Gurau R., A generalization of the Virasoro algebra to arbitrary dimensions, Nuclear Phys. B 852 (2011),
592–614, arXiv:1105.6072.
[25] Gurau R., The complete 1/N expansion of colored tensor models in arbitrary dimension, Ann. Henri
Poincaré 13 (2012), 399–423, arXiv:1102.5759.
[26] Gurau R., The Schwinger Dyson equations and the algebra of constraints of random tensor models at all
orders, Nuclear Phys. B 865 (2012), 133–147, arXiv:1203.4965.
[27] Gurau R., The 1/N expansion of tensor models beyond perturbation theory, Comm. Math. Phys. 330 (2014),
973–1019, arXiv:1304.2666.
[28] Gurau R., Universality for random tensors, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), 1474–1525,
arXiv:1111.0519.
[29] Gurau R., Rivasseau V., Sfondrini A., Renormalization: an advanced overview, arXiv:1401.5003.
[30] Krajewski T., Schwinger–Dyson equations in group field theories of quantum gravity, in Symmetries and
Groups in Contemporary Physics, Nankai Ser. Pure Appl. Math. Theoret. Phys., Vol. 11, World Sci. Publ.,
Hackensack, NJ, 2013, 373–378, arXiv:1211.1244.
[31] Krajewski T., Toriumi R., Polchinski’s equation for group field theory, Fortschr. Phys. 62 (2014), 855–862.
[32] Krajewski T., Toriumi R., Polchinski’s exact renormalisation group for tensorial theories: Gaussian univer-
sality and power counting, arXiv:1511.09084.
[33] Lahoche V., Oriti D., Rivasseau V., Renormalization of an Abelian tensor group field theory: solution at
leading order, J. High Energy Phys. 2015 (2015), no. 4, 095, 41 pages, arXiv:1501.02086.
[34] Oriti D., Group field theory and loop quantum gravity, arXiv:1408.7112.
[35] Polchinski J., Renormalization and effective Lagrangians, Nuclear Phys. B 231 (1984), 269–295.
[36] Polonyi J., Lectures on the functional renormalization group method, Cent. Eur. J. Phys. 1 (2003), 1–71,
hep-th/0110026.
[37] Rivasseau V., From perturbative to constructive renormalization, Princeton Series in Physics, Princeton
University Press, Princeton, NJ, 1991.
[38] Rivasseau V., The tensor track, III, Fortschr. Phys. 62 (2014), 81–107, arXiv:1311.1461.
[39] Rosten O.J., Fundamentals of the exact renormalization group, Phys. Rep. 511 (2012), 177–272,
arXiv:1003.1366.
[40] Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press,
Cambridge, 2004.
[41] Samary D.O., Vignes-Tourneret F., Just renormalizable TGFT’s on U(1)d with gauge invariance, Comm.
Math. Phys. 329 (2014), 545–578, arXiv:1211.2618.
[42] Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical
Physics, Cambridge University Press, Cambridge, 2007.
[43] Zinn-Justin J., Quantum field theory and critical phenomena, Int. Ser. Monogr. Phys., Vol. 113, Oxford
University Press, 2002.
http://dx.doi.org/10.1016/j.nuclphysb.2011.07.009
http://arxiv.org/abs/1105.6072
http://dx.doi.org/10.1007/s00023-011-0118-z
http://dx.doi.org/10.1007/s00023-011-0118-z
http://arxiv.org/abs/1102.5759
http://dx.doi.org/10.1016/j.nuclphysb.2012.07.028
http://arxiv.org/abs/1203.4965
http://dx.doi.org/10.1007/s00220-014-1907-2
http://arxiv.org/abs/1304.2666
http://dx.doi.org/10.1214/13-AIHP567
http://arxiv.org/abs/1111.0519
http://arxiv.org/abs/1401.5003
http://dx.doi.org/10.1142/9789814518550_0050
http://arxiv.org/abs/1211.1244
http://dx.doi.org/10.1002/prop.201400043
http://arxiv.org/abs/1511.09084
http://dx.doi.org/10.1007/JHEP04(2015)095
http://arxiv.org/abs/1501.02086
http://arxiv.org/abs/1408.7112
http://dx.doi.org/10.1016/0550-3213(84)90287-6
http://dx.doi.org/10.2478/BF02475552
http://arxiv.org/abs/hep-th/0110026
http://dx.doi.org/10.1515/9781400862085
http://dx.doi.org/10.1002/prop.201300032
http://arxiv.org/abs/1311.1461
http://dx.doi.org/10.1016/j.physrep.2011.12.003
http://arxiv.org/abs/1003.1366
http://dx.doi.org/10.1017/CBO9780511755804
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://dx.doi.org/10.1093/acprof:oso/9780198509233.001.0001
1 Introduction
2 Tensor models
2.1 Invariant potentials for random tensors
2.2 Perturbative expansion
2.3 Tensorial group field theories
2.4 Couplings in the spin network basis
3 Exact renormalisation group equations
3.1 Wilson's ideas in quantum field theory
3.2 Exact renormalisation group equations
3.3 Polchinski's equation for invariant tensor models
3.4 Large N melonic universality
3.5 Group field theory
3.6 Scaling and renormalisation for Abelian models with closure constraint
3.7 SU(2) models in the spin network basis
3.8 Wetterich's equation and nontrivial fixed points
4 Loop equations
4.1 Reparametrisation invariance in quantum field theory
4.2 Loop equations for tensor models
4.3 The case of matrix models
4.4 An application to melonic dominance
5 Conclusion and outlook
A ERGE for low order bubble couplings
A.1 Couplings in rank D=3 tensor models
A.2 Couplings in rank D=4 tensor models
References
|