Random Tensors and Quantum Gravity

Random Tensors and Quantum Gravity\star\star$\star$This paper is a
contribution to the Special Issue on Tensor Models, Formalism and
Applications. The full collection is available at
http://www.emis.de/journals/SIGMA/Tensor_Models.html

Vincent RIVASSEAU

V. Rivasseau

Laboratoire de Physique Théorique, CNRS UMR 8627, Université Paris XI,
F-91405 Orsay Cedex, France<EMAIL_ADDRESS>http://www.rivasseau.com/

Received March 23, 2016, in final form July 06, 2016; Published online July
15, 2016

We provide an informal introduction to tensor field theories and to their
associated renormalization group. We focus more on the general motivations
coming from quantum gravity than on the technical details. In particular we
discuss how asymptotic freedom of such tensor field theories gives a concrete
example of a natural “quantum relativity” postulate: physics in the deep
ultraviolet regime becomes asymptotically more and more independent of any
particular choice of Hilbert basis in the space of states of the universe.

renormalization; tensor models; quantum gravity

60B20; 81T15; 81T16; 81T17; 82B28

##  1 Introduction

_In physics, a field is a physical quantity that has a value for each point in
space and time._

This current wikipedia definition of a field reminds us how much a preexistent
classical space-time with its associated notion of locality is deeply
ingrained in our common representation of the physical world. However there is
an “emerging” consensus than an _ab initio_ theory of quantum gravity requires
to at least modify and probably abandon altogether the concepts of absolute
locality and absolute classical space-time [75, 89, 90]. It is a task for many
generations of theoretical physicists to come, requiring even more work than
when classical physics had to be abandoned in favor of quantum mechanics.

The tensor track [79, 80, 82] can be broadly described as a step in this
direction and more precisley as a program to explore (Euclidean) quantum
gravity as a (Euclidean) quantum field theory _of_ space-time rather than _on_
space-time, relying on a specific mathematical formalism, namely the modern
theory of random tensors [27, 56, 60, 63] and their associated $1/N$
expansions [24, 25, 57, 58, 61], which generalizes in a natural way the theory
of random vectors and random matrices [40]. In this formalism observables,
interactions, and Feynman graphs are all represented by regular edge-colored
graphs.

It is both simple and natural but also general enough to perform quantum sums
over space-times in any dimension, pondering them with a discrete analog of
the Einstein–Hilbert action [1]. In three dimensions the formalism sums over
all topological manifolds, and in four dimensions over all triangulated
manifolds111Some topological manifolds in four dimensions, such as the $E_{8}$
manifold, cannot be triangulated as a simplicial complex but it is not clear
they are relevant for quantum gravity. with all their different smooth
structures [6, 34, 35, 36, 47, 88].

More precisely the tensor track proposes to explore renormalization group
flows [17, 22, 23, 46, 70] in the associated tensor theory space [81]. In
doing this, it can retain several of the most characteristic aspects of the
successful quantum field theories of the standard model of particle physics,
such as perturbative renormalizability [12, 19, 20, 32, 33, 87] and asymptotic
freedom [9, 10, 21, 83, 85]. Also at least in the simplest cases, random
tensor models and field theories can be contructed non-perturbatively [38, 39,
59, 62, 71, 72].

The tensor track can be considered both as a generalization of random matrix
models, successfully used to quantize two-dimensional gravity [40], and as an
improvement of group field theory [7, 16, 29, 48, 68, 74, 76] motivated by the
desire to make it renormalizable [13, 14, 49, 78]. It proposes another angle
for the study of dynamical triangulations [2, 3, 4]. It also relies on non-
commutative field theory and in particular on the Grosse–Wulkenhaar model [41,
42, 50, 51, 52, 53, 54, 55] as a strong source of inspiration.

This brief paper is intended as an introduction in non-technical terms to this
approach. We warn the reader that it overlaps strongly with previous reviews
and papers of the author such as [79, 80, 81, 82, 83].

##  2 Historical perspective

After a summer course with J. Schwinger and a long road trip through the
United States with R. Feynman, F. Dyson understood how to relate Feynman’s
functional integration (sum over histories) to Schwinger’s equations [45]. The
iterative solution of these “Schwinger–Dyson” equations is then indexed by
Feynman’s graphs. Quantum field theory was born.

From the start it had to struggle with a famous problem: the amplitudes
associated to Feynman graphs contain ultraviolet divergencies. Any quick fix
of this problem through imposing an ultra-violet cutoff violates some of the
most desired properties of the theory such as Euclidean invariance or
Osterwalder–Schrader positivity, the technical property which allows
continuation to real time and unitarity, hence ultimately ensures that quantum
probabilities add up to 1, as they should in any actual experiment.

The full solution of the problem, namely renormalization, took some time to
polish. The textbook example is the one of an interacting theory obtained by
perturbing a non-local free theory by a local interaction with a small
coupling constant. The free theory is represented by a Gaussian measure based
on a non-local propagator, such as $1/(p^{2}+m^{2})$, which becomes however of
shorter and shorter range, hence is _asymptotically local_ in the ultraviolet
limit. In this case the key ingredients allowing renormalization to work are
the notion of scale and the locality principle: high scale (ultraviolet)
connected observables seem more and more local when observed with lower scale
(infrared) propagators. It also requires a power counting tool which allows to
classify observables into relevant, marginal and irrelevant ones. Under all
these conditions, the ultraviolet part of divergent Feynman graphs can be
absorbed and understood as modifying the effective value of the coupling
constant according to the observation scale. This is the standard example of a
renormalization group flow.

The renormalization group of K. Wilson and followers is a powerful
generalization of the previous example. The action takes place in a certain
_theory space_ which is structured by the locality principle: the local
potential approximation corresponds to the purely local powers of the field at
the same point. Such local operators can be dressed with an arbitrary number
of derivatives, creating a hierarchy of “quasi-local” operators. The operator
product expansion expands the average value of any non-local operator, when
observed through infrared probes, into a dominant local part plus correction
terms with more and more derivatives, hence it is an expansion into quasi-
local operators which generalizes the multipolar expansion of close
configurations of charges in classical electrostatics. The (irreversible)
renormalization group flows from bare (ultraviolet) to effective (infrared)
actions by integrating out (quasi-local) fluctuations modes. Fixed points of
the flow need no longer be Gaussian, but interesting ones should still have
only a small number of associated relevant and marginal directions. Indeed it
is under this condition that the corresponding physics can be described in
terms of only a few physical parameters.

Scales and locality play the fundamental role in this standard picture of a
renormalization group analysis. Locality is also at the core of the
mathematically rigorous formulation of quantum field theory. It is a key
Wightman axiom [91] and in algebraic quantum field theory [67] the fundamental
structures are the algebras of _local observables_. It is therefore not
surprising that to generalize quantum field theory and the locality principle
to an abstract framework independent of any preexisting space-time takes some
time.

Such a development seems however required for a full-fledged _ab initio_
theory of quantum gravity. Near the Planck scale, space-time should fluctuate
so violently that the ordinary notion of locality may no longer be the
relevant concept. Among the many arguments one can list pointing into this
direction are the Doplicher–Fredenhagen–Roberts remark that to distinguish two
objects closer than the Planck scale would require to concentrate so much
energy in such a little volume that it would create a black hole, preventing
the observation [44]. String theory, which (in the case of closed strings)
contains a gravitational sector, is another powerful reason to abandon strict
locality. Indeed strings are one-dimensional _extended_ objects, whose
interaction cannot be localized at any space time point. Moreover, closed
strings moving in compactified background may not distinguish between small
and large such backgrounds because of dualities that exchange their
translational and “wrapping around” degrees of freedom. Another important
remark is that two and three-dimensional pure quantum gravity are topological
theories. In such theories observables, being functions of the topology only,
cannot be localized in a particular region of space-time.

Remark however that four-dimensional gravity is certainly not a purely
topological theory, as confirmed by the recent observation of gravitational
waves. It is nevertheless an educated guess that pure (Euclidean) four-
dimensional quantum gravity should not completely loose any topological
character. In other words modifications of the topology of space-time should
_not_ be strictly forbidden. Indeed space-times of different (complicated)
topologies can interpolate between fixed boundaries, even when the latter have
simple topology. Suppressing some of them because they do not have a simple
topology would be similar to suppressing arbitrarily instantons from Feynman’s
sum over histories, and this is known to lead to quantum probabilities no
longer adding up to 1.

Space-time dimension four is the first in which gravity has local degrees of
freedom but it is also the first dimension in which exotic smooth structures
can appear on a fixed topological background manifold [88]. It would be
desirable to establish a strong link between these two properties of dimension
four, in particular because smooth structures in four dimensions are
intimately related to gauge theories [43], which describe the other physical
interactions, electroweak and strong, in the standard model. Staying within
the scope of this small review, let us simply remark again that the basic
building blocks proposed by the tensor track, namely tensor invariants, are
dual to piecewise linear manifolds with boundaries. Therefore they precisely
distinguish in four dimension all smooth structures associated to a given
topological structure [34, 35, 36].

##  3 The quantum relativity principle

Let us try to forget for a minute the enormous amount of technical work spent
on various theories of quantum gravity, and consider with a fresh eye a very
general, even naive question: what are the most basic mathematical tools and
physical principles at work in general relativity and quantum mechanics? How
could we most naturally join them together?

The very name of general relativity suggests how Einstein derived it: as a
consequence of the independence of the laws of physics under any particular
choice of space-time coordinates. In a sense our coordinates systems are man-
made: the Earth does not come equipped with meridians or parallels drawn on
the ground. So it is natural to expect the general laws of physics not to
depend on such man-made coordinates. The corresponding symmetry group is the
group of diffeomorphisms of space-time. But since this group depends on the
underlying space-time manifold, in particular of its topology, the classical
general relativity principle is not fully background invariant.

In quantum mechanics, instead of space-time trajectories, the basic objects
are states, elements of a Hilbert space and the observables are operators
acting on them. An important observation is that although there are many
different manifolds in a given dimension, each with its own different
diffeomorphism group, there is only a _single_ Hilbert space of any given
dimension $N$, denoted ${\mathcal{H}}_{N}$ (up to isomorphism, which here
means up to change of orthonormal basis).

There is also a single _separable_ infinite-dimensional Hilbert space, namely
_the_ Hilbert space ${\mathcal{H}}$. Separable means here that ${\mathcal{H}}$
admits a countable orthonormal basis. ${\mathcal{H}}$ can be identified with
the space of square integrable series $\ell_{2}(\mathbb{N})$, or with
$\ell_{2}(\mathbb{Z})=L^{2}({\rm U}(1))$ or with the $L^{2}$ space of square
integrable functions on _any_ Riemannian manifold of _any_ finite dimension.
It is therefore a truly background-independent mathematical structure.

Our universe certainly contains a huge number of degrees of freedom, both of
geometric (gravitational waves) and matter type. Hence
${\mathcal{H}}=\lim\limits_{N\to\infty}{\mathcal{H}}_{N}$ seems the right
mathematical starting point for a background independent quantum theory of
gravity. That this was not emphasized more in the early days of quantum
mechanics probably only means that absolute space-time was still a deeply
ingrained notion in the minds of the theoretical physicists at that time.

It is then tempting to extend the general relativity principle into a quantum
relativity principle by postulating that the laws of physics should be
invariant under the symmetry group of _that unique_ Hilbert space
${\mathcal{H}}$, namely independent of any preferred choice of an orthonormal
basis in ${\mathcal{H}}$. Just like for coordinates systems, an orthonormal
basis seems to be observer-made.

In a very large universe with $N$ degrees of freedom it would amount to
postulate the ${\rm U}(N)$ invariance of physics. However such a postulate
quickly appears a bit too extreme. The only (polynomial) ${\rm U}(N)$
invariants in a finite-dimensional Hilbert space ${\mathcal{H}}_{N}$ are
polynomials in the scalar product. We know that quantum states, being rays in
Hilbert space, can be normalized to 1, hence there is no fully ${\rm
U}(N)$-invariant physical observables which can distinguish between two states
hence interesting observables need to break this full symmetry.

Beauty has sometimes been defined as a “slightly broken symmetry” and this may
also be a good definition for physics. Just like ordinary quantum field theory
is not exactly local but only asymptotically local in the ultraviolet regime,
the giant ${\rm U}(N)$ invariance of ${\mathcal{H}}_{N}$ at very large $N$
could be asymptotic in a certain regime which by analogy we should still call
the ultraviolet regime. In practice it has to be broken in any actual
experiment. If for no other reason, it should be at least because we are
finite size observers in an effective geometry of space-time. Hence we are led
to consider an attenuated version of invariance of physics under change of
basis, namely

Quantum relativity principle. _In the extreme “ultraviolet” regime
$N\to\infty$ the laws of physics should become _asymptotically_ independent of
any preferred choice of basis in the quantum Hilbert space describing the
universe._

Of course we have to explain in more detail the meaning of the words
“asymptotically independent in the extreme ultraviolet regime $N\to\infty$”.
It has to be understood in a renormalization group sense. To define an
abstract (space-time independent) renormalization group and its corresponding
asymptotic ultraviolet regime is the main goal of the tensor track, and it
requires several ingredients. We need first an initial device to break the
${\rm U}(N)$-symmetry group and allow to label and regroup the degrees of
freedom in a way suited for a renormalization group analysis. In particular it
should allow to group together the degrees of freedom into renormalization
group _slices_. The ones with many degrees of freedom will be called the
ultraviolet slices, and the ones with less degrees of freedom the infrared
slices. In tensor field theories this device is a ${\rm U}(N)$-breaking
propagator for the free theory, whose spectrum allows to label and regroup the
degrees of freedom of ${\mathcal{H}}_{N}$, exactly like Laplacian or Dirac-
based propagators do in ordinary quantum field theory222Even if one does not
like the idea of a non-invariant propagator, to set up a renormalization group
analysis can still be done by breaking ${\rm U}(N)$ invariance at the level of
the cutoffs which separate degrees of freedom into fluctuations and
background..

Once such a device is in place, the renormalization group always then means a
decimation, to find the effective infrared theory after integrating
ultraviolet slices of the theory. Asymptotic invariance in the ultraviolet
regime means that the bare action should be closer and closer to an exactly
${\rm U}(N)$-invariant action. If we agree on the “quantum relativity
principle” above and on this general strategy, the next step is to search for
the most natural symmetry breaking pattern that could occur on the path from
the extreme ultraviolet regime to effective symmetry-broken infrared actions.

##  4 Random tensors

We know that the Hilbert space of a composed physical system is not the direct
sum but the tensor product of the Hilbert spaces of its constituents, and this
aspect is critical to entanglement, a basic feature of quantum mechanics.
Hence the most natural pattern for symmetry breaking of ${\rm U}(N)$
invariance is to change vectors into tensors of a certain rank $d$. This
breaks the ${\rm U}(N)$ group to a smaller group. More precisely if $N$
factors as a product of two integers333This is generically possible at large
$N$ because of the rarefaction of prime numbers. $N=N_{1}.N_{2}$, we can break
${\rm U}(N)$ into ${\rm U}(N_{1})\otimes{\rm U}(N_{2})$, the natural symmetry
group of random rectangular $N_{1}$ by $N_{2}$ Wishart matrices [92] in the
tensor space ${\mathcal{H}}_{N_{1}}\otimes{\mathcal{H}}_{N_{2}}$. Such Wishart
matrices are not just rectangular arrays of random numbers. Their polynomial
interactions and observables, which are product of traces, have the reduced
${\rm U}(N_{1})\otimes{\rm U}(N_{2})$ symmetry rather than a full ${\rm
U}(N_{1}N_{2})$ vector symmetry.

Continuing along this line, if $N=N_{1}.N_{2}\dots N_{d}$, we can break ${\rm
U}(N)$ into ${\rm U}(N_{1})\otimes\cdots\otimes{\rm U}(N_{d})$, the natural
symmetry group of tensors of rank $d$ in a tensor space
${\mathcal{H}}_{N_{1}}\otimes\cdots\otimes{\mathcal{H}}_{N_{d}}$. The reduced
symmetry corresponds to invariance under independent change of coordinates in
each of the tensor factor, namely in each space
${\mathcal{H}}_{N_{1}},\dots,{\mathcal{H}}_{N_{d}}$. The invariant connected
polynomials associated to this symmetry are exactly connected bipartite
$d$-regular edge-colored graphs. Such connected bipartite $d$-regular edge-
colored graphs on $n$ vertices can be enumerated [18] and their number
$Z_{d}(n)$, for $d\geq 3$ grows rapidly with $n$:

$\displaystyle Z_{1}(n)=1,0,0,0,0,\dots,\qquad$ $\displaystyle
Z_{3}(n)=1,3,7,26,97,624,\dots,$ $\displaystyle
Z_{2}(n)=1,1,1,1,1,1,1,\dots,\qquad$ $\displaystyle
Z_{4}(n)=1,7,41,604,13753,\dots.$

Tensor models with canonical trivial propagator and polynomial invariant
interactions have a perturbative expansion indexed by Feynman graphs which,
under expansion of the internal structure of their bipartite $d$-regular edge-
colored vertices, become bipartite $(d+1)$-regular edge-colored graphs. They
typically admit an associated 1/N expansion whose leading graphs, called
melons [26], are in one-to one correspondence with $(d+1)$-ary trees444This is
true when the interactions of these models are themselves melonic; for other
cases, see [28]..

This observation has deep geometric consequences. These graphs precisely
encode all piecewise linear orientable555Non-orientable manifolds can be
included by considering only real $O(N)$ invariance, hence loosing
bipartiteness but not the _colors_. manifolds (with boundaries) in dimension
$d$ [47]. We could therefore consider the reduced symmetry ${\rm
U}(N_{1})\otimes\cdots\otimes{\rm U}(N_{d})$ as a quantum precursor of the
coordinate invariance of general relativity and the invariant connected
polynomials associated to this symmetry as precursors of observables. More
precisely, both interactions and observables of the theory can be associated
with (colored triangulations of) $d$-dimensional spaces and the Feynman graphs
of the theory are associated with (colored triangulations of)
$(d+1)$-dimensional space-time [27].

The single [26] and double [37, 65, 66] scaling limits of such tensor models
for rank $d\geq 3$ have now been identified and correspond to continuous
random trees [64]. This seems at first sight a step backwards. Indeed the
$1/N$ expansion of vector models is also dominated by trees, whereas the $1/N$
expansion of matrix models is dominated by the more complicated planar maps.
Branched polymers, like planar maps are certainly not a good approximation to
our current macroscopic universe, but planar maps seem closer, since they have
higher Hausdorff and spectral dimension. However tensor invariant interactions
and the sub-leading structure of the $1/N$ tensorial expansion are much richer
than their vector and matrix counterparts. This is not too surprising, given
that tensor models of rank $d$ can effectuate a statistical sum over all
manifolds (and many pseudo-manifolds) in dimension $d$.

To go beyond single and double scaling of the simplest models and to find more
interesting infrared limits and phase transitions probably will require to
combine analytic and numerical methods. Tensor group field theory (TGFT),
which we now describe, adds a Laplacian to the propagator of the tensor models
to equip them with a full-fledged notion of renormalization group. It
generalizes in a natural way the matrix renormalization group of the
Grosse–Wulkenhaar model [41, 42, 50, 51, 52, 53, 54, 55]. Such tensor theories
become then suited for a functional renormalization group analysis [17, 22,
23, 46, 70].

Asymptotic ${\rm U}(N^{d})$ invariance and the “quantum relativity principle”
will then be recovered in the deep ultraviolet regime if the theory is
_asymptotically free_. This is due to the combination of two facts. First the
interaction, which broke ${\rm U}(N^{d})$ to ${\rm U}(N)^{\otimes d}$
asymptotically vanishes in the ultraviolet regime because of this asymptotic
invariance. Second the propagator itself, because it is of the inverse
Laplacian type, has a smaller and smaller relative variation in the
ultraviolet regime between different frequencies:

$\displaystyle\frac{\delta}{\delta
p}\big{(}p^{2}+m^{2}\big{)}^{-1}\simeq\frac{p}{(p^{2}+m^{2})^{2}}\ll_{p\to\infty}\big{(}p^{2}+m^{2}\big{)}^{-1}.$

Full ${\rm U}(N^{d})$ asymptotic invariance is then in fact recovered in the
ultraviolet limit precisely in the same asymptotic sense than locality is
asymptotically recovered in the ultraviolet regime of ordinary (Euclidean)
quantum field theory: high energy subgraphs look almost local when observed
through infrared probes with low resolution. We could also say that in such a
scenario the degrees of freedom of the universe behave more and more as the
molecules of a perfect gas in the extreme ultraviolet, high-temperature
regime.

##  5 Tensor group field theories

As already explained, the key difference between random matrix models and non
commutative field theory on Moyal space lies in the modification of the
propagator. Breaking the ${\rm U}(N)$ invariance of matrix models at the
propagator level can be done in many ways, but the simplest way was found in
the Grosse–Wulkenhaar (GW) model [51] on ${\mathbb{R}}^{4}$ equipped with the
Moyal product. It is a matrix model with quartic coupling and a propagator
which can be interpreted as the sum of the Laplacian and an harmonic potential
on ${\mathbb{R}}^{4}$. It is very natural in the matrix base which expresses
the Moyal star product as a matrix product. Such a propagator has a non-
trivial spectrum, hence breaks the ${\rm U}(N)$ invariance of the theory. It
allows to distinguish the infrared (small values of the base index, large
eigenvalues of the propagator) from the ultraviolet regime (large values of
the base index, small eigenvalues of the propagator). There is an associated
renormalization group flow between these two regimes.

Power counting in such models is entirely governed by the underlying $1/N$
expansion (divergent graphs are the planar 2 and 4 point graphs with a single
external face), hence the corresponding matrix renormalization group [46] can
be considered a kind of continuous version of the $1/N$ expansion. It is
remarkable that this model is asymptotically scale invariant, that is in the
ultraviolet regime the beta function tends to zero [41, 42, 50]. Its planar
sector has also recently been beautifully solved [52, 53], showing that four-
dimensional integrability, like four-dimensional asymptotic safety, is not
limited to the famous example of supersymmetric Yang–Mills $N=4$ QFT but is
more general and does not _require_ supersymmetry. Also in a rather surprising
way the model at negative coupling very probably obeys all Wightman axioms
except clustering [54, 55].

In a completely analogous manner, one can define TGFT’s with tensorial
interactions and a soft breaking of the tensorial invariance of their
propagator. Renormalization is again a continuous version of the $1/N$
expansion and the divergent graphs reduce to the melonic sector instead of the
planar sector. The group structure allows to interpret the fields either as
rank $d$ tensors or as ordinary fields defined on $G^{d}$ where $G$ is the Lie
group. The propagator is the inverse of the sum of the Laplacians on each
factor in $G^{d}$. In the simplest case, namely $G={\rm U}(1)$, one can
identify at any rank which are the super-renormalizable and just
renormalizable interactions. Surprisingly perhaps, at rank/dimension 4 the
just renormalizable interactions are of order 6, not 4 [19, 20, 21]. The
quartically interacting model is just renormalizable at rank/dimension 5. It
is also just renormalizable at rank/dimension 3 if the propagator has Dirac-
type rather than Laplace-type power counting. It is remarkable that the
corresponding tensorial RG flow displays the generic property of asymptotic
freedom, at least for quartic interactions [9, 10, 21]; see however [31] for
the subtle issue of interactions of order 6.

The bridge between matrix and tensor renormalizable models has been further
reduced in [15], in which new families of renormalizable models are
identified. It is shown that the rank 3 tensor model defined in [21], and the
Grosse–Wulkenhaar models in two and four dimensions generate three different
classes of renormalizable models by modifying the power of the propagator. A
review on this class of models extends and generalizes this analysis [12]. It
classifies the models of matrix or tensor type with a propagator which is the
inverse of a sum of momenta of the form $p^{2a}$, $a\in]0,1]$. Infinite towers
of (super- and just-) renormalizable matrix models are found.

The emerging picture is a simple classification of quite a large class of
Bosonic renormalizable non-local QFTs of the matrix or tensor type. The
theories of scalar and vector type are neither asymptotically free nor safe
since they have no wave function renormalization at one loop. The theories of
matrix type are generically asymptotically safe (specially for instance all
$\operatorname{Tr}[M^{4}]$ models), since the wave function renormalization
exactly compensates the coupling constant renormalization in the ultraviolet
regime. The theories of tensor type are generically asymptotically free since
the wave function renormalization wins over the coupling constant
renormalization. As discussed in Section 6, which we essentially reproduce
from [83] for self-content, this is a robust fact since it is rooted in the
combinatorial structure of the quartic vertex, which is different for vector,
matrix and tensor models.

The combinatorics of renormalization is neatly encoded in a Connes–Kreimer
combinatorial Hopf algebra. See [69] for a version of this algebra adapted to
multi-scale analysis. This algebra has been explicited in [77] for the case of
the model defined in [19, 20]. It differs significantly from previous
Connes–Kreimer algebras, due to the involved combinatorial and topological
properties of the tensorial Feynman graphs.

###  5.1 Laplacian with gauge projector

TGFTs which include a “Boulatov-type” projector in their propagator [29]
deserve a category of their own as they are both more difficult to renormalize
and closer to the desired geometry in the continuum. Their characteristic
feature is to include an average over a new Lie group variable acting
simultaneously over all tensor threads in the propagator. This average
implements in three dimensions the constraints of the $BF$ theory, which is
classically equivalent to three-dimensional general relativity. In four
dimensions one may similarly use projectors in order to implement the
Plebanski simplicity constraints of four-dimensional gravity [84]. In a
Feynman graph, once the vertex variables have been integrated out, the Feynman
amplitude appears as an integral over one variable for each edge of a product
of delta functions for each face. Each such delta function fixes to the
identity the ordered product of the edge variables around the face. Therefore
it can be interpreted as a theory of simplicial geometry supplemented with a
discrete gauge connection at the level of the Feynman amplitudes of the theory
which has trivial holonomy around each face of the Feynman graph, hence around
each $d-2$ cell of the dual triangulation.

To introduce renormalizable models of this class is again done in two steps.
First one replaces the usual interaction of the Boulatov model by tensor
invariant interactions to allow for a simple power-counting of the $1/N$ type.
Second, one adds to the usual GFT Gaussian measure a Laplacian term to allow
for scale analysis. It is however truly non-trivial that renormalization can
still work for models of this type. Indeed their propagator
$C(g_{1},\dots,g_{d};g^{\prime}_{1},\dots,g^{\prime}_{d})$ is definitely _not_
asymptotic in the ultraviolet regime to a product of delta functions
$\delta\big{(}g_{1}(g^{\prime}_{1})^{-1}\big{)}\cdots\delta\big{(}g_{d}(g^{\prime}_{d})^{-1}\big{)}$
but rather to an average:

$\displaystyle
C(g_{1},\dots,g_{d};g^{\prime}_{1},\dots,g^{\prime}_{d})\simeq_{uv}\int
dh\,\delta\big{(}g_{1}h(g^{\prime}_{1})^{-1}\big{)}\cdots\delta\big{(}g_{d}h(g^{\prime}_{d})^{-1}\big{)}.$

Nevertheless an extension of the locality principle still holds for
_divergent_ graphs, meaning that they can be renormalized again by tensor
invariant counterterms. This is possible because the divergent graphs to
renormalize, which are melonic graphs, precisely have a sufficiently
particular structure for such an extended locality property to hold, although
it does not hold at all for general graphs.

The set of results include the study of a family of Abelian TGFT models in 4d
which was shown to be super-renormalizable for any polynomial interaction
[33], so is similar in power counting to the family of $P(\phi)_{2}$ models in
ordinary QFT. Abelian just renormalizable models also exist in 5 and 6
dimensions and were studied in [87]. They are also asymptotically free [85],
hence in particular obey the “quantum relativity principle” of Section 3 but
in which the Hilbert space is the physical one, restricted by the gauge
condition. Finally an even more interesting non-Abelian model with group
manifold ${\rm SU}(2)$ was successfully renormalized in [32]. For an excellent
review of this subject we refer to [30], and for an extension to homogenous
spaces rather than Lie groups, see [73].

Let us recall that the above studies concern _uncolored_ TGFTs. For the
colored Boulatov model, Ward–Takahashi identities were derived and interpreted
in [8]. In [11] the ultraviolet behavior of the same colored theory but with a
Laplacian added to the propagator is studied. It is shown that all orders in
perturbation theory in the case of the ${\rm U}(1)$ group in three dimensions
are convergent; moreover it was convincingly argued that this finiteness
should also hold for the same model over ${\rm SU}(2)$. Recall however that
colored Bosonic models are a priori non-perturbatively unstable, and that
colored Fermionic models are therefore more interesting, as they should better
behave in that respect. They also have an interesting additional symmetry of
rotation between colors [56].

##  6 Asymptotic freedom

This section is essentially reproduced from [83]. Consider the familiar
Laplacian-based normalized Gaussian measure for $d$-dimensional Bosonic fields
on ${\rm U}(1)^{d}$ with periodic boundary conditions

$\displaystyle
d\mu_{C}(\phi,\bar{\phi})=\left(\prod_{p,\bar{p}\in\mathbb{Z}^{d}}\frac{d\phi_{p}d\bar{\phi}_{\bar{p}}}{2i\pi}\right)\operatorname{Det}(C)^{-1}e^{-\sum_{p,\bar{p}}\phi_{p}C^{-1}_{p\bar{p}}\bar{\phi}_{\bar{p}}},$

where the covariance $C$ is, up to a field strength renormalization, the
inverse of the Laplacian on $S_{1}^{d}$ plus a mass term

$\displaystyle
C_{p,\bar{p}}=\frac{1}{Z}\frac{\delta_{p,\bar{p}}}{p^{2}+m^{2}}.$

Here $p^{2}=\sum\limits_{c=1}^{d}p_{c}^{2}$, $m^{2}$ is the square of the bare
mass, and $Z$ is the so-called wave-function renormalization, which can be
absorbed into a
$(\phi,\bar{\phi})\to\big{(}Z^{-1/2}\phi,Z^{-1/2}\bar{\phi}\big{)}$ field
strength renormalization. If we restrict the indices $p$, which should be
thought as “momenta”, to lie in $[-N,N]^{d}$ rather than in $\mathbb{Z}^{d}$
we have proper (finite-dimensional) fields. We can consider $N$ as the
ultraviolet cutoff, and we are interested in performing the ultraviolet limit
$N\to\infty$.

The generating function for the moments of the model is

$\displaystyle{\mathcal{Z}}(g,J,\bar{J})=\frac{1}{{\mathcal{Z}}}\int
e^{\bar{J}\cdot\phi+J\cdot\bar{\phi}}e^{-\frac{g}{2}V(\phi,\bar{\phi})}d\mu_{C}(\phi,\bar{\phi}),$
(6.1)

where ${\mathcal{Z}}={\mathcal{Z}}(g,J,\bar{J})|_{J=\bar{J}=0}$ is the
normalization, $g$ is the coupling constant, and the sources $J$ and $\bar{J}$
are dual respectively to $\bar{\phi}$ and $\phi$. The generating function for
the connected moments is $W=\log{\mathcal{Z}}(g,J,\bar{J})$.

The simplest quartic vector, matrix and tensor field theories correspond to
this choice of the propagator and only differ in the _combinatorial_ way in
which the momenta indices of the four fields branch at the quartic interaction
vertex $V(\phi,\bar{\phi})$.

The _vector_ interaction is the square of the quadratic (mass) term, hence it
is _disconnected_ , that is it factorizes into two disjoint pairs

$\displaystyle
V_{V}=\langle\bar{\phi},\phi\rangle^{2}=\sum_{p,q}(\phi_{p}\bar{\phi}_{p})(\phi_{q}\bar{\phi}_{q}).$
(6.2)

It is just renormalizable for $d=4$.

A _matrix_ quartic (connected) interaction is easily defined only for $d=2r$
even. It is obtained by splitting the initial index as a pair $(p,q)$ with
$p=(p_{1},\dots,p_{r})$, $q=(q_{1},\dots,q_{r})$, hence splitting the space
${\mathcal{H}}_{d}={\mathcal{H}}_{r}\otimes{\mathcal{H}}_{r}$. The field
$\phi$ is then interpreted as the matrix $\phi_{pq}$, with conjugate matrix
$\phi^{\star}=^{t}\bar{\phi}$ and the vertex $V_{M}$ is an invariant trace

$\displaystyle
V_{M}=\operatorname{Tr}\phi\phi^{\star}\phi\phi^{\star}=\sum_{p,q,p^{\prime},q^{\prime}}\phi_{pq}\bar{\phi}_{p^{\prime}q}\phi_{p^{\prime}q^{\prime}}\bar{\phi}_{pq^{\prime}}.$
(6.3)

It it is just renormalizable for $r=4$, hence $d=8$.

Finally the simplest _tensor_ interaction $V_{T}$ is the color-symmetric sum
of _melonic_ [26] quartic interactions

$\displaystyle V_{T}=\sum_{c}V_{c},$ $\displaystyle
V_{c}(\phi,\bar{\phi})=\operatorname{Tr}_{c}(\operatorname{Tr}_{\hat{c}}\phi\bar{\phi})^{2}=\sum_{p,\bar{p},q,\bar{q}}\left[\phi_{p}\bar{\phi}_{\bar{p}}\prod_{c^{\prime}\neq
c}\delta_{p_{c^{\prime}}\bar{p}_{c^{\prime}}}\right]\delta_{p_{c}\bar{q}_{c}}\delta_{q_{c}\bar{p}_{c}}\left[\phi_{q}\bar{\phi}_{\bar{q}}\prod_{c^{\prime}\neq
c}\delta_{q_{c^{\prime}}\bar{q}_{c^{\prime}}}\right],$ (6.4)

where $\operatorname{Tr}_{\hat{c}}\phi\bar{\phi}$ means partial trace in
${\mathcal{H}}_{d}=\otimes_{c=1}^{d}{\mathcal{H}}_{c}$ over all colors except
$c$, and $\operatorname{Tr}_{c}$ means trace over color $c$. The corresponding
model is just renormalizable for $d=5$ [86], with a corresponding
Connes–Kreimer algebra [5].

These three different combinatorial models have just renormalizable power
counting, like the ordinary scalar $\phi^{4}_{4}$ theory. But the class of
divergent graphs is more restricted in the combinatorial case. Remark that in
all cases the interactions $V$ are positive for $g>0$. Hence the models are
stable for this sign of the coupling constant, which we now assume.

We want to compare the one-loop beta function computation for these just
renormalizable quartic vector, matrix and tensor field theories. This
computation is most conveniently performed in the intermediate field
representation.

An intermediate field $\sigma$ splits the quartic vertex in two halves, as
pictured in Fig. 1, through the simple integral representation

$\displaystyle e^{-\frac{g}{2}V(\phi,\bar{\phi})}=\int
d\nu(\sigma)e^{i\sqrt{g}\bar{\phi}\phi\cdot\sigma}.$ (6.5)

In this formula $d\nu$ is a Gaussian measure with covariance 1 on the
intermediate field $\sigma$. The three cases (6.2)–(6.4) lead to $\sigma$
fields of different nature and to different combinatorial rules for the dot in
(6.5). In the vector case the $\sigma$ field is a _scalar_ , reflecting the
already factorized nature of (6.2). In the matrix and tensor case (6.3), (6.4)
$\sigma$ is a _matrix_. More precisely, in the matrix case (6.3), it is a
single matrix with its two arguments in ${\mathbb{Z}}^{4}$. In the tensor case
(6.4), it is the sum of five different colored matrices $\sigma_{c}$ with
their two arguments in ${\mathbb{Z}}$, one for each color $c$, and should be
properly written as

$\displaystyle{\sigma}=\sum_{c}\sigma_{c}\otimes\mathbb{I}_{\hat{c}}.$

Figure 1: The vertex is cut in two by the intermediate field representation.
From top to bottom: the vector, matrix and tensor case. Incoming and outgoing
arrows distinguish $\phi$ and $\bar{\phi}$.

The advantage of this representation is that the $\bar{\phi}$ and $\phi$
functional integral becomes quadratic, hence can be performed explicitly,
yielding

$\displaystyle{\mathcal{Z}}(g,J,\bar{J})=\frac{1}{{\mathcal{Z}}}\int
d\nu(\sigma)\int
d\mu_{C}(\phi,\bar{\phi})e^{i\sqrt{g}\bar{\phi}\phi\cdot\sigma}e^{\bar{J}\cdot\phi+J\cdot\bar{\phi}}$
$\displaystyle\hphantom{{\mathcal{Z}}(g,J,\bar{J})}{}=\frac{1}{{\mathcal{Z}}}\int
d\nu(\sigma)e^{\langle\bar{J},C^{1/2}R(\sigma)C^{1/2}J\rangle-\operatorname{Tr}\log\left[\mathbb{I}-i\sqrt{g}C^{1/2}{\sigma}C^{1/2}\right]},$

where $R$ is the symmetric _resolvent_ operator

$\displaystyle R(\sigma)\equiv\frac{1}{\mathbb{I}-i\sqrt{g}C^{1/2}\sigma
C^{1/2}}.$

In all cases the one-loop beta function boils down to the same computation, up
to subtle differences of a purely combinatorial nature. Let us call
$\Gamma_{2p}$ the $2p$-point vertex function, hence the sum of one-particle
irreducible amputated Feynman amplitudes with $2p$ external legs. The
renormalized BPHZ prescriptions are defined by momentum space subtractions at
zero momentum, which we can restrict to _divergent_ graphs (e.g., planar in
the matrix case, melonic in the tensor case)

$\displaystyle\frac{g_{r}}{2}=-\Gamma_{4}(0),\qquad
Z-1=\left[\frac{\partial}{\partial p^{2}}\Gamma_{2}\right](0),$ (6.6)

where $g_{r}$ is the renormalized coupling. Performing the field strength
renormalization we can rescale to 1 the wave function renormalization at high
ultraviolet cutoff at the cost of using a _rescaled_ bare coupling
$g^{\prime}_{b}=Z^{-2}g_{b}$. The one-loop $\beta_{2}$ coefficient shows how
this rescaled bare coupling evolves at fixed $g_{r}$ when $N\to\infty$. It
writes

$\displaystyle g^{\prime}_{b}=g_{r}\big{[}1+\beta_{2}g_{r}(\log N+{\rm
f\/inite})+O\big{(}g_{r}^{2}\big{)}\big{]},$ (6.7)

where $N$ is the ultraviolet cutoff, and “finite” means bounded as
$N\to\infty$. As well-known $\beta_{2}>0$ corresponds to a coupling constant
which flows out of the perturbative regime in the ultraviolet. $\beta_{2}<0$
corresponds to asymptotic freedom: the (rescaled) bare coupling flows to zero
as $N\to\infty$. $\beta_{2}=0$ is inconclusive as the analysis of the
renormalization group flow must be pushed further, but, if reproduced at
higher orders, indicates a fully scale invariant theory.

It is easier to compute the bare perturbation theory, as it does not involve
any subtraction. Starting from (6.6), we find that $\Gamma_{4}$ and $Z-1$
always involve the _same_ logarithmically divergent sum, namely

$\displaystyle\sum_{q\in[-N,N]^{4}}\frac{1}{\big{(}q^{2}+m_{r}^{2}\big{)}^{2}}=2\pi^{2}\log
N+{\rm f\/inite},$ (6.8)

where $m_{r}^{2}=Zm^{2}-\Gamma_{2}(0)$ is the renormalized mass. However this
sum arises with various combinatoric coefficients. More precisely

$\displaystyle\Gamma_{4}(0)=-\frac{g_{b}}{2}\left[1-ag_{b}\sum_{q\in[-N,N]^{4}}\frac{1}{\big{(}q^{2}+m_{r}^{2}\big{)}^{2}}+O\big{(}g_{b}^{2}\big{)}\right],$
(6.9) $\displaystyle Z=1+\frac{\partial\Gamma_{2}}{\partial
p^{2}}\Big{|}_{p=0}=1+bg_{b}\sum_{q\in[-N,N]^{4}}\frac{1}{\big{(}q^{2}+m_{r}^{2}\big{)}^{2}}+O\big{(}g_{b}^{2}\big{)},$
(6.10)

where $a$ and $b$ are combinatoric coefficients that depend on the particular
case (vector, matrix or tensor) considered.

Since $g^{\prime}_{b}=Z^{-2}g_{b}$, multiplying (6.7) by $Z^{2}$ and taking
into account (6.6)–(6.10), which imply $g_{r}=g_{b}+O(g_{b}^{2})$ and
$Z=1+O(g_{b})$, we find

$\displaystyle Z^{2}\Gamma_{4}(0)=-\frac{g_{b}}{2}\big{[}1-\beta_{2}g_{b}(\log
N+{\rm f\/inite})+O\big{(}g_{b}^{2}\big{)}\big{]},$

hence in all cases

$\displaystyle\beta_{2}=(a-2b)2\pi^{2}.$

We are left with the simple problem of computing the coefficients $a$ and $b$
of the one loop leading diagrams for $\Gamma_{4}$ and $Z$.

Figure 2: The (single) one loop melonic graph in the tensor case for
$\Gamma_{4}$ and $\Gamma_{2}$ are trees for the intermediate field (dashed)
lines.

  * •

In the vector case, $a=1$. The only divergent graph is the one on the left of
Fig. 2. Resolvents (derivatives with respect to $J$ and $\bar{J}$) come up
with a factor 1 and no symmetry factorial, whether terms from the
$\operatorname{Tr}\log$ expansion come up with a factor $1/n$ for a
$\operatorname{Tr}\big{(}i\sqrt{g}C^{1/2}{\vec{\sigma}}C^{1/2}\big{)}^{n}$
(because of the Taylor series of the logarithm), plus a symmetry factor $1/k!$
if there are $k$ of them (this factor comes from expansion of the
exponential). The combinatoric weight for the tree graph at order $g^{2}$ for
$\Gamma_{4}$ is therefore $1$, which decomposes into a $1/2$ for the single
loop vertex ($n=2$, $k=1$) times a factor 2 for the two Wick contractions. In
this vector case $b=0$ since the one loop tadpole on the right of Fig. 2, the
only contributing graph at order $g$, does not have any external momentum
dependence. Hence

$\displaystyle\left[\frac{\partial}{\partial
p^{2}}\Gamma_{2}\right](0)=O\big{(}g^{2}\big{)}\quad\Rightarrow\quad b=0.$

Hence $\beta_{2}=2\pi^{2}$ and the theory has no UV fixed point, at least in
this approximation.

  * •

In the tensor case, $a=1$ for the same reasons than in the vector case. Indeed
for any of the five melonic interactions there is a single divergent graph of
the corresponding color of the type pictured on the left of Fig. 2. But now we
also have $b=1$. Indeed remark first that $b>0$ because two minus signs
compensate, one in front of $g$ in (6.1) and the other coming from the mass
subtraction, since

$\displaystyle\left[\frac{1}{p_{c}^{2}}\left(\frac{1}{q^{2}+p_{c}^{2}+m_{r}^{2}}-\frac{1}{q^{2}+m_{r}^{2}}\right)\right]_{p_{c}=0}=-\frac{1}{\big{(}q^{2}+m_{r}^{2}\big{)}^{2}}.$

The combinatorics is then 1 because there is a single loop vertex with $n=1$,
$k=1$ and a single Wick contraction to branch it on the external resolvent as
shown for the graph on the right of Fig. 2. Summing over the colors $c$ of
this contraction simply reconstructs $\sum_{c}p_{c}^{2}=p^{2}$. In conclusion
$\beta_{2}=-2\pi^{2}$ and the theory is asymptotically free, in agreement with
[21, 22, 86]. Wave function, or field strength renormalization won over
coupling constant renormalization because of the square power in $Z^{2}$.

  * •

In the matrix case, the vertex crossing symmetry means more terms diverge
logarithmically than in the vector and tensor cases, namely those
corresponding to planar _maps_ in the intermediate representation. The
crossing symmetry is a ${\mathbb{Z}}_{2}$ symmetry, but it acts differently on
$\Gamma_{4}$ and $\Gamma_{2}$. Since the one loop graph for $\Gamma_{4}$ has
_two_ vertices, hence two (dotted) $\sigma$ propagators, the crossing symmetry
acts twice independently and generates an orbit of four planar maps,
represented in the top part of Fig. 3. In contrast the crossing symmetry acts
only once on the orbit of the $\Gamma_{2}$ term, generating only the two
planar maps pictured in the bottom part of Fig. 3. Hence $a=4$ and $b=2$ which
leads to $\beta_{2}=0$! This combinatorial “miracle” persists at all orders:
in fact the logarithmically divergent part of $Z^{2}\Gamma_{4}(0)$ is exactly
0 at all orders in $g$, as can be shown through combining a Ward identity with
the Schwinger–Dyson equations of the theory [41]. The corresponding theory is
asymptotically _safe_.

Figure 3: The dominant one loop graphs in the matrix case for $\Gamma_{4}$ and
$\Gamma_{2}$ are planar in the intermediate field representation.

##  7 Conclusion

In conclusion we would like to recall the main positive mathematical aspects
of the tensor track:

  * •

it proposes to perform a sum both over any topology and any smooth structure
up to dimension/rank 4, following the paradigm of Feynman functional integral
quantization (“sum over histories”),

  * •

this sum is naturally pondered by a discrete version of the Einstein–Hilbert
action,

  * •

it is possible to introduce an abstract notion of scales through Laplacian-
type operators,

  * •

the tensor symmetry is compatible with renormalization and acts as a
substitute for locality,

  * •

it supports a power counting tool, namely the $1/N$ tensor expansion,

  * •

renormalization group flows can be investigated in the tensor theory space,

  * •

the simplest models are asymptotically free, hence obey therefore to the
natural extension of the general relativity principle sketched above: physics
in the extreme ultraviolet limit becomes asymptotically independent of any
preferred basis in the huge Hilbert space of states of the universe,

  * •

constructive control is possible at least in simple (super-renormalizable)
cases.

Let us immediately temper however all these positive points with a lot of
caveats. Of course enormous work lies ahead of the tensor track program, which
is only one among many competing approaches to quantum gravity. Among the
major problems to tackle in our approach we can list finding Euclidean axioms
including the right generalization of the Osterwalder–Schrader positivity
axiom, to allow in particular emergence of Lorentzian time and causality;
constructive treatment of more than quartic interactions; renormalization
group evolution from the arborescent to more realistic macroscopic phases (see
[25] for a step in that direction); consequences of the theory for cosmology
scenarios and for black holes; and addition of the standard model matter
fields to the picture.

Altogether we nevertheless have the impression that the tensor track has
matured enough to be taken seriously and explored further. At the physical
level it suggests an emergent space-time scenario with an initial arborescent
phase of the universe. This result should not be immediately discarded as non-
physical, since the richness of subdominant tensor interactions could lead
this arborescent phase to evolve later into geometries closer to our actual
universe.

### Acknowledgements

We thank R. Avohou, D. Benedetti, J. Ben Geloun, V. Bonzom, S. Carrozza, S.
Dartois, T. Delepouve, O. Samary Dine, R. Gurau, T. Krajewski, V. Lahoche, L.
Lionni, D. Oriti, A. Tanasa, F. Vignes-Tourneret and R. Wulkenhaar for
discussions and contributions on many aspects of the tensor track program.

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