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Abstract

We show that natural noncommutative gauge theory models on $R^3_\lambda$ can accommodate gauge invariant harmonic terms, thanks to the existence of a relationship between the center of $R^3_\lambda$ and the components of the gauge invariant 1-form canonical connection. This latter object shows up naturally within the present noncommutative differential calculus. Restricting ourselves to positive actions with covariant coordinates as field variables, a suitable gauge-fixing leads to a family of matrix models with quartic interactions and kinetic operators with compact resolvent. Their perturbative behavior is then studied. We first compute the 2-point and 4-point functions at the one-loop order within a subfamily of these matrix models for which the interactions have a symmetric form. We find that the corresponding contributions are finite. We then extend this result to arbitrary order. We find that the amplitudes of the ribbon diagrams for the models of this subfamily are finite to all orders in perturbation. This result extends finally to any of the models of the whole family of matrix models obtained from the above gauge-fixing. The origin of this result is discussed. Finally, the existence of a particular model related to integrable hierarchies is indicated, for which the partition function is expressible as a product of ratios of determinants.
JHEP12(2015)045
Published for SISSA by Springer
Received:August 18, 2015
Accepted:November 13, 2015
Published:December 9, 2015
Noncommutative gauge theories on R3
λ:
perturbatively finite models
Antoine Géré,aTajron Jurićband Jean-Christophe Walletc
aDipartimento di Matematica, Università di Genova,
Via Dodecaneso, 35, I-16146 Genova, Italy
bRuđer Bošković Institute, Theoretical Physics Division,
Bijenička c.54, HR-10002 Zagreb, Croatia
cLaboratoire de Physique Théorique, CNRS, University Paris-Sud, University Paris-Saclay,
Bât. 210, 91405 Orsay, France
E-mail: gere@dima.unige.it,tjuric@irb.hr,
jean-christophe.wallet@th.u-psud.fr
Abstract: We show that natural noncommutative gauge theory models on R3
λcan accom-
modate gauge invariant harmonic terms, thanks to the existence of a relationship between
the center of R3
λand the components of the gauge invariant 1-form canonical connection.
This latter object shows up naturally within the present noncommutative differential cal-
culus. Restricting ourselves to positive actions with covariant coordinates as field variables,
a suitable gauge-fixing leads to a family of matrix models with quartic interactions and
kinetic operators with compact resolvent. Their perturbative behavior is then studied. We
first compute the 2-point and 4-point functions at the one-loop order within a subfamily
of these matrix models for which the interactions have a symmetric form. We find that
the corresponding contributions are finite. We then extend this result to arbitrary order.
We find that the amplitudes of the ribbon diagrams for the models of this subfamily are
finite to all orders in perturbation. This result extends finally to any of the models of the
whole family of matrix models obtained from the above gauge-fixing. The origin of this
result is discussed. Finally, the existence of a particular model related to integrable hier-
archies is indicated, for which the partition function is expressible as a product of ratios of
determinants.
Keywords: Non-Commutative Geometry, Differential and Algebraic Geometry, Matrix
Models, Models of Quantum Gravity
ArXiv ePrint: 1507.08086
Open Access,c
The Authors.
Article funded by SCOAP3.doi:10.1007/JHEP12(2015)045
JHEP12(2015)045
Contents
1 Introduction 1
2 Noncommutative gauge theories on R3
λ3
2.1 Basic properties of R3
λ3
2.2 Differential calculus on R3
λand gauge theory models 5
2.3 A family of gauge invariant classical actions 7
3 Perturbative analysis 10
3.1 Gauge-fixing 10
3.2 Gauge-fixed action at Ω=1 12
3.3 One-loop 2-point and 4-point functions 13
3.4 Finiteness of the diagram amplitudes to all orders 15
4 Discussion 18
A Properties of the kinetic operators 21
B Connected 2-point function at one-loop 23
C Loop summation for the truncated model 24
1 Introduction
Noncommutative Geometry (NCG) [1,2] may provide an appealing way to overcome physi-
cal obstructions to the existence of continuous space-time and commuting coordinates at the
Planck scale [3], triggering a new impulse in the studies on noncommutative field theories
(NCFT). Actually, they appeared in their modern formulation a long time ago within String
Field Theory [4]. This was followed by models on the fuzzy sphere [5,6], gauge theories on
almost commutative geometries [7,8] (for a review on fuzzy sphere and related see e.g [9]).
NCFT on noncommutative Moyal spaces received a lot of attention from the end of the 90’s,
in particular from the viewpoint of perturbative properties and renormalisability [1012].
For reviews, see for instance [1315].
Progresses have been made in the area of NCFT on Moyal spaces Rn
θ,n= 1,2lead-
ing to perturbatively renormalisable scalar fields theories. These encompass the scalar φ4
model with harmonic term on R2
θor R4
θ[1618], this latter being likely non-perturbatively
solvable [19], the translational and rotational invariant related φ4models [20,21] together
with fermionic versions [22,23] and solvable models inherited from the LSZ model [24].
The situation for the gauge theories is not so favorable. Although the construction of gauge
– 1 –
JHEP12(2015)045
invariant classical actions can be easily done from suitable noncommutative differential cal-
culi [2529], the study of quantum properties is rendered difficult by technical complications
stemming mainly from gauge invariance that supplement the UV/IR mixing problem in-
herent in NCFT on Moyal spaces. So far, the construction of a renormalisable gauge theory
on R4
θhas not been achieved. On Moyal spaces, gauge invariant straightforward general-
izations of the above harmonic term do not exist. In this respect, attempts to reconcile the
features of the φ4model with harmonic term with a gauge theoretic framework gave rise to
the gauge invariant model obtained in [30,31]. Interestingly, this action can be interpreted
as (related to) the spectral action of a particular spectral triple [32] whose relationship
to the Moyal geometry has been analysed in [3335]. Unfortunately, its complicated vac-
uum structure explored in [36,37] forbids the use of any standard perturbative treatment.1
Alternative based on the implementation of a IR damping mechanism have been proposed
and studied [3942]. Although this damping mechanism is appealing, it is not known if it
can produce a renormalisable gauge theory on R4
θ. Besides, interpreting the action within
the framework of some noncommutative differential geometry is unclear. Another appealing
approach is the matrix model formulation of noncommutative gauge theory, initiated a long
ago in [43]. For recent reviews, see [4446]. This approach may in some cases allow one
to go beyond the perturbative approach [47,48]. One interesting outcome is that it may
provide a interpretation for the UV/IR mixing for some noncommutative gauge theories in
terms of an induced gravity action. See e.g [49,50].
Recently, scalar field theories on the noncommutative space R3
λ, a deformation of R3
preserving rotation invariance, have been studied in [51]. These appear to have a mild
perturbative behavior and are (very likely) free of ultraviolet/infrared (UV/IR) mixing.
In this respect, one may expect a more favorable situation for the gauge theories on R3
λ
than for those on R4
θ. The space R3
λ, which may by viewed as a subalgebra of R4
θ, has
been first introduced in [52] and generalized in [53]. The use of the canonical matrix
base introduced in [51] (see also [54]) renders the computation tractable, avoiding the
complexity of a direct calculation in coordinates space. A first exploration of gauge theories
on R3
λhas been performed in [55], focused on a particular class of theories for which the
gauge-fixed propagator can be explicitly computed rendering possible a one-loop analysis.
The impact of the expected mild perturbative behavior of the loop diagrams was however
tempered by the occurrence of a nonzero one-loop tadpole signaling quantum instability
of the chosen vacuum. While further study of this quantum instability may reveal new
interesting properties, it seemed desirable to undertake a more systematic investigation
around the construction of other families of gauge theories on R3
λwith stable vacuum and
non trivial dynamics. Reconciling these two features seems to be out of reach in the case
of Moyal spaces but can be achieved when dealing with R3
λ.
In this paper, we show that natural noncommutative gauge theory models on R3
λcan
support gauge invariant harmonic terms, unlike the case of Moyal spaces. This stems
from the existence of a relationship between the center of R3
λand the components of the
gauge invariant 1-form canonical connection which arises in the derivation-based differential
1This technical obstruction can be circumvented on R2
θfor particular vacuum configurations [38].
– 2 –
JHEP12(2015)045
calculus underlying our construction. We focus our analysis on a family of (positive) gauge
invariant actions whose field variables are assumed to be the covariant coordinates, i.e. the
natural objects related to the canonical connection. Then, a suitable BRST gauge-fixing
in the spirit of [38,56,57] gives rise to a family of matrix models with quartic interactions
and kinetic operators (having compact resolvent). Their perturbative behavior is then
examined. We first consider a subfamily of these matrix models for which interactions and
kinetic operators leads to slight technical simplifications and compute the corresponding 2-
point and 4-point functions at the one-loop order. We find that the respective contributions
are finite. We then extend this result to arbitrary order and find that the amplitudes of
the ribbon diagrams for the models pertaining to this subfamily are finite to all orders in
perturbation. It appears that this perturbative finiteness results from the conjunction a
sufficient rapid decay for the propagator, the role played by the radius of the fuzzy sphere
components of R3
λacting as a kind of cut-off together with the existence of an upper bound
for the (positive) propagator depending only of the cut-off. We then extend this result to
any of the matrix models of the whole family obtained from the above gauge-fixing. Finally,
we point out the existence of a particular model related to integrable (2-d Toda) hierarchies
and give the expression of the partition function as a product of ratios of determinants.
The paper is organized as follows. In section 2, we present and discuss the construction
of the relevant family of gauge invariant models. Useful properties on the (derivation
based) noncommutative differential calculus together with the notion of noncommutative
connection inherited from the (commutative) notion of Koszul connection are also recalled.
Section 3is devoted to the gauge-fixing and the perturbative analysis with the one-loop
computations collected in the subsection 3.3 while subsection 3.4 deals with the finiteness
to arbitrary orders. In section 4, we discuss the results and finally consider also a particular
model for which the partition function can be related to ratios of determinants signaling a
relation to integrable hierarchies.
2 Noncommutative gauge theories on R3
λ
2.1 Basic properties of R3
λ
The algebra R3
λhas been first introduced in [52] and further considered in various works [51,
53,55]. Besides, a characterization of a natural basis has been given in [51]. We refer to
these references for more details. Here,2it will be convenient to view R3
λas [51,55]
R3
λ=C[x1, x2, x3, x0]/I[R1,R2],(2.1)
where C[x1, x2, x3, x0]is the free algebra generated by the 4 (hermitean) elements (coordi-
nates) {xµ=1,2,3, x0}and I[R1,R2]is the two-sided ideal generated by the relations
R1: [xµ, xν] = iλεµνρ xρ,R2:x2
0+λx0=
3
X
µ=1
x2
µ,µ, ν, ρ = 1,2,3(2.2)
2To simplify the notations, the associative ?-product for R3
λis understood everywhere in any product
of elements of the algebra. Besides, summation over repeated indices is understood everywhere, unless
explicitly stated.
– 3 –
JHEP12(2015)045
with λ6= 0.R3
λis a unital -algebra, with complex conjugation as involution and cen-
ter Z(R3
λ)generated by x0and satisfying the following strict inclusion R3
λ)U(su(2)),
where U(su(2)) is the universal enveloping algebra of the Lie algebra su(2). Alternative
(equivalent) presentations can be found in e.g [51,53,55].
As shown in [51], any element φR3
λhas the following blockwise expansion
φ=X
jN
2X
jm,nNj
φj
mn vj
mn ,(2.3)
where φj
mn C, and the family {vj
mn , j N
2,jm, n j}is the natural orthogonal
basis of R3
λintroduced in [51], stemming from the direct sum decomposition
R3
λ=M
jN
2
M2j+1(C).(2.4)
For fixed j, the corresponding subfamily is simply related to the canonical basis of the
matrix algebra M2j+1(C). The following fusion relation and conjugation hold true
vj1
mnvj2
qp =δj1j2δnq vj1
mp ,(vj
mn)=vj
nm ,jN
2,jm, n, q, p j . (2.5)
The orthogonality among the vj
mn’s is taken with respect to the usual scalar product ha, bi:=
Tr(ab), for any a, b R3
λ. Here, the trace functional Tr can be defined [55] for any
Φ,ΨR3
λas
Tr(ΦΨ) := 8πλ3X
jN
2
w(j)trjjΨj)(2.6)
with w(j)is a center-valued weight factor to be discussed below, trjdenotes the canonical
trace of M2j+1(C), and Φj(resp. Ψj) an element of M2j+1(C)is simply defined from
the expansion (2.3) of Φby the (2j+ 1) ×(2j+ 1) matrix Φj:= (φj
mn)jm,nj(resp.
Ψj:= (ψj
qp)jq,pj). Therefore we have
Tr(ΦΨ) = 8πλ3X
jN
2
w(j)
X
jm,nj
φj
mnψj
nm
,(2.7)
and
trj(vj
mn) = δmn ,hvj1
mn, vj2
pqi= 8πλ3X
j1N
2
w(j1)δj1j2δmpδnq .(2.8)
Eq. (2.6) defines a family of traces depending on the weight factor w(j). Recall that the
particular choice
w(j) = j+ 1 (2.9)
leads to a trace that reproduces the expected behavior3for the usual integral on R3once
the (formal) commutative limit is applied [55]. For a general discussion on this point based
on a noncommutative generalization of the Kustaanheimo-Stiefel map [58], see [59].
3For instance, observe that one easily obtains from (2.7) the expected volume of a sphere of radius λN
with Φj= Ψj=Ijand summing up to j=N
2. Namely, one obtains 8πλ3N
P
k=0
k
2(k+ 1) '4
3π(λN)3.
– 4 –
JHEP12(2015)045
We define x±:= x1±ix2. Other useful relations [51] that will be needed for computa-
tions in the ensuing analysis are
x+vj
mn =λF(j, m)vj
m+1,n vj
mn x+=λF(j, n)vj
m,n1
xvj
mn =λF(j, m)vj
m1,n vj
mn x=λF(j, n)vj
m,n+1
x3vj
mn =λ m vj
mn vj
mn x3=λ n vj
mn
x0vj
mn =λ j vj
mn vj
mn x0=λ j vj
mn ,(2.10)
where
F(j, m) := p(j+m+ 1)(jm).(2.11)
2.2 Differential calculus on R3
λand gauge theory models
The construction of noncommutative gauge models can be conveniently achieved by using
the general framework of the noncommutative differential calculus based on the derivations
of an algebra which has been introduced a long ago [25,26]. The general framework can
actually be viewed as a noncommutative generalization the Koszul approach of differential
geometry [60]. Mathematical details and some related applications to NCFT can be found
in [2729].
In the present paper, we consider as in [55] the differential calculus generated by the
Lie algebra of real inner derivations of R3
λ
G:= Dµ:= Adθµ=i[θµ,·], θµ:= xµ
λ2,µ= 1,2,3,(2.12)
where the inner derivation Dµsatisfy the following commutation relation
[Dµ, Dν] = 1
λµνρ Dρ,µ, ν, ρ = 1,2,3.(2.13)
Denoting, for any nN, by n
Gthe space of n(Z(R3
λ))-linear) antisymmetric maps
ω:GnR3
λ, the corresponding N-graded differential algebra is (Ω
G=nNn
G, d, ×),
with nilpotent differential d: Ωn
Gn+1
Gand product ×on
Gdefined for any ωp
G
and ρq
Gby
(X1, . . . , Xp+1) =
p+1
X
k=1
(1)k+1Xkω(X1,...,k, . . . , Xp+1 )
+X
1k<lp+1
(1)k+lω([Xk, Xl],...,k,...,l, . . . , Xp+1),(2.14)
ω×ρ(X1, . . . , Xp+q) = 1
p!q!X
σSp+q|σ|ω(Xσ(1),...,Xσ(p))ρ(Xσ(p+1),...,Xσ(p+q)),(2.15)
where the Xi’s are elements of Gand |σ|is the signature of the permutation σSp+q.
Let Mdenotes a right-module over R3
λ. Recall that a connection on Mcan be defined
as a linear map :G × MMwith
X(ma) = X(m)a+mXa , zX (a) = zX(a),X+Y(a) = X(a) + Y(a),
for any aR3
λ, any mM,z∈ Z(R3
λ)and any X, Y ∈ G.
– 5 –
JHEP12(2015)045
As we are interested by noncommutative versions of U(1) gauge theories, we assume
from now on M=CR3
λwhich can be viewed as a noncommutative analog of the complex
line bundle relevant for abelian (U(1)) commutative gauge theories. We further restrict
ourself to hermitean connections4for the canonical hermitean structure given by h(a1, a2) =
a
1a2,a1, a2R3
λ.
A mere application of the above definition yields
Dµ(a) := µ(a) = Dµa+Aµa ,
Aµ:= µ(I),with A
µ=Aµ,(2.16)
for aR3
λand µ= 1,2,3. The definition of the curvature
F(X, Y ) := [X,Y]− ∇[X,Y ],X, Y ∈ G ,
yields
F(Dµ, Dν) := Fµν = [µ,ν]− ∇[Dµ,Dν]=DµAνDνAµ+ [Aµ, Aν] + 1
λµνρ Aρ.(2.17)
The group of gauge transformations, defined as the group of automorphisms of the module
compatible with both hermitean and right-module structures, is easily found to be the
group of unitary elements of R3
λ,U(R3
λ), with left action of R3
λ. For any g∈ U(R3
λ)and
φR3
λ, one has gg=gg=I,φg=. From the definition of the gauge transformations
of the connection given by g
µ=gµg, for any g∈ U(R3
λ), one infers
Ag
µ=gAµg+gDµg , and Fg
µν =gFµν g . (2.18)
The existence of a canonical gauge invariant connection, denoted hereafter by inv, stems
from the existence of inner derivations in the Lie algebra of derivations that generates the
differential calculus. See [25,26] for a general analysis. In the present case, one finds
inv
µ(a) = Dµaµa=iaθµ,aR3
λ,(2.19)
with curvature Finv
µν = 0. A natural gauge covariant tensor 1-form is then obtained by
forming the difference between inv
µand any arbitrary connection. The corresponding
components, sometimes called covariant coordinates, are given by
Aµ:= µ− ∇inv
µ=Aµ+µ,i= 1,2,3,(2.20)
and one has A
µ=−Aµ,µ= 1,2,3(A
µ=Aµ). By using (2.17), one obtains
Fµν = [Aµ,Aν] + 1
λµνρ Aρ.(2.21)
One easily verifies that for any aR3
λ, and g∈ U(R3
λ), the following gauge transformations
hold true
(inv
µ(a))g=inv
µ(a),Ag
µ=gAµg , µ= 1,2,3.(2.22)
4Given a hermitean structure, says h:M×MR3
λ,is hermitean if Xh(m1, m2) = h(X(m1), m2) +
h(m1,X(m2)), for any X∈ G,m1, m2M.
– 6 –
JHEP12(2015)045
Define the real invariant 1-form Θ1
Gby
Θ1
G: Θ(Dµ) = Θ(Adθµ) = θµ.(2.23)
By making use of (2.14) and (2.15), one easily check that
d(iΘ) + (iΘ)2= 0 ,(2.24)
reflecting Finv
µν = 0.
The form Θrelated to the 1-form invariant canonical connection supports an interesting
interpretation. Recall [25,26] that a natural noncommutative analog of a symplectic form
is defined as a real closed 2-form ωsuch that for any element ain the algebra, there exists a
derivation Ham(a)(the analog of Hamiltonian vector field) verifying ω(X, Ham(a)) = X(a)
for any derivation X. One then observes that ω:= dΘ2
Gcan be viewed as the natural
symplectic form on the algebra R3
λin the setting of [25,26] with Ham(a) = Adia for any
aR3
θas the noncommutative analog of Hamiltonian vector field and
{a, b}:= ω(Ham(a),Ham(b)) = i[a, b](2.25)
the related (real) Poisson bracket.
2.3 A family of gauge invariant classical actions
Families of gauge-invariant functional (classical) actions can be easily obtained from the
trace of any gauge-covariant polynomial functional in the covariant coordinates Aµ, namely
Sinv(Aµ) = Tr(P(Aµ)). Here, we will assume that the relevant field variable is Aµ, akin to
a matrix model formulation of gauge theories on R3
λ, thus proceeding in the spirit of [38].
Natural requirement for the gauge-invariant functional are:
i) P(Aµ)is at most quartic in Aµ,
ii) P(Aµ)does not involve linear term in Aµ(not tadpole at the classical order),
iii) the kinetic operator is positive.
Set from now on
x2:=
3
X
µ=1
xµxµ.
We observe that gauge theories on R3
λcan accommodate a gauge-invariant harmonic term
Tr(x2AµAµ). This property simply stems from the fact that x2∈ Z(R3
λ)combined with
the gauge-invariance of the 1-form canonical connection whose components in the module
are given by
inv(I)µ:= Ainv
µ=µ(2.26)
as it can be readily obtained from (2.16) and (2.19). One easily checks that
(Ainv
µ)g= (µ)g=µ,(2.27)
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JHEP12(2015)045
as a mere combination of (2.12) and (2.18). Now, the relation R2(2.1) and (2.12) imply
3
X
µ=1
(µ)(µ) = 1
λ4x2=1
λ4(x2
0+λx0),(2.28)
in which the l.h.s. is obviously gauge-invariant since (2.27) holds true while the r.h.s. belongs
to Z(R3
λ)as a polynomial in x0. Hence, the gauge-invariant object P3
µ=1(µ)2belongs to
the center of R3
λ. Therefore, by using the cyclicity of the trace, one can write (summation
over repeated αindice understood)
Tr
3
X
µ=1
(µ)g(µ)g(Ag
αAg
α)
= Tr
g
3
X
µ=1
(µ)(µ)g(AαAα)
= Tr
3
X
µ=1
(µ)(µ)(AαAα)
(2.29)
where we used Pµ(µ)(µ)∈ Z(R3
λ)to obtain the last equality. Note that such a
gauge-invariant harmonic term cannot be built in the case of gauge theories on the Moyal
space R4
θ[30,31] simply because, says x2
ν=1,2,3,4, while still related to a gauge invariant
object (a canonical gauge-invariant connection still exists, see e.g [2729]), does not belong
to the center of R4
θ.
It is convenient to work with hermitean fields. Thus, we set from now on
Aµ=iΦµ
so that Φ
µ= Φµfor any µ= 1,2,3. The above observation, combined with the requirements
i) and ii) given above points towards the following general expression for a gauge-invariant
action
S(Φ) = 1
g2Tr κΦµΦνΦνΦµ+ηΦµΦνΦµΦν+µνρΦµΦνΦρ+ (M+µx2µΦµ
=1
g2Tr ηκ
4µ,Φν]2+η+κ
4{Φµ,Φν}2+µνρΦµΦνΦρ
+ (M+µx2µΦµ,(2.30)
where from now on Einstein summation convention is used, the trace is still given by (2.7)
and g2,κ,η,ζ,Mand µare real parameters. The corresponding mass dimensions are
[κ]=[η]=0,[g2]=[ζ] = 1,[M]=2,[µ] = 4 (2.31)
so that the action (2.30) is dimensionless, assuming that the “engineering” dimension 3of
the noncommutative space is the relevant dimension.
We will mainly focus on sub-families involving positive actions obtained from (2.30).
In order to make contact with some notations of refs. [30,31], we set
κ= 2(Ω + 1), η = 2(Ω 1),(2.32)
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JHEP12(2015)045
where the real parameter is dimensionless, thus fixing for convenience the overall nor-
malization of the term µ,Φν]2in (2.30). This latter action can be rewritten as
S(Φ) = 1
g2Tr Fµν i
λµνρ ΦρFµν i
λµνρ Φρ+ Ω {Φµ,Φν}2+µνρΦµΦνΦρ
+ (M+µx2µΦµ
=1
g2Tr F
µν Fµν + Ω {Φµ,Φν}2+ 0µνρΦµΦνΦρ+M0+µx2ΦµΦµ,(2.33)
with
ζ=ζ0+4
λ;M=M0+2
λ2.(2.34)
We note that the first two terms in the gauge-invariant action S(Φ) (2.33) are formally
similar to those occurring in the so-called induced gauge theory on R4
θ[30,31].
S(Φ) is positive when
0, µ > 0, ζ = 0, M > 0(2.35)
or
0, µ > 0, ζ =4
λ, M > 2
λ2,(2.36)
as it can be realized respectively from the 1st and 2nd equality in (2.33) (see also section 3
and the appendix for the positivity of the kinetic operator).
In the rest of this paper, we will focus on the family of actions fulfilling the first
condition (2.35), namely
S=1
g2Tr Fµν i
λµνρ ΦρFµν i
λµνρ Φρ+{Φµ,Φν}2+(M+µx2µΦµ.(2.37)
The equation of motion for (2.37) given by
4(Ω + 1)(ΦρΦµΦµ+ ΦµΦµΦρ) + 8(Ω 1)ΦµΦρΦµ+ 2(M+µx2ρ= 0,(2.38)
one infers that Φρ= 0 is the absolute minimum of (2.37).5
In the section 3, we will show that one class of gauge-invariant models pertaining to the
families (2.37), (2.33) yields after gauge-fixing to a finite theory at all orders in perturbation.
This stems from the conjunction of the gauge-invariant harmonic term in (2.30)µx2ΦµΦµ,
the orthogonal sum structure of R3
λ(2.4) and the existence of a bound on the (absolute
value of) the propagator for Φµ. This will be discussed at the end of the paper. Notice that
in the Moyal case only the term Mis allowed by gauge invariance.
5There are also other nontrivial solutions of the equation of motion related to (2.30). Namely, there
is one more solution belonging to the center Z(R3
λ)given by ΦµΦµ=M+µx2
2(κ+η). We found also solution
outside the center given by Φi=f xi, where f=ηλ±qη2λ232[x2(κ+η)ηλ2](M+µx2)
8[x2(κ+η)ηλ2]. The corresponding
quantum field theories are still under investigation.
– 9 –
JHEP12(2015)045
3 Perturbative analysis
3.1 Gauge-fixing
We set
Φµ=X
j,m,n
(φµ)j
mnvj
mn ,µ= 1,2,3.(3.1)
The kinetic term of the classical action (2.37)Sis given by
SKin(Φ) = 1
g2Tr(Φµ(M+µx2µ)(3.2)
=8πλ3
g2X
j,m,n
w(j)(M+λ2µj(j+ 1))|(φµ)j
mn|2(3.3)
where w(j)is the center-valued weight introduced in (2.7)) and we used (2.5), (2.10), (2.7)
and
x0=λX
j,m
j vj
mm, x2=λ2X
j,m
j(j+ 1) vj
mm,(3.4)
stemming from (2.10) and (2.3). Recall that we have assumed that the condition (2.35)
holds true. We assume for the moment that w(j)is a polynomial function of j, thus insuring
a suitable decay of the related propagators at large indices. We will specialize to the cases
w(j)=1and w(j) = j+ 1 in a while.
Now, defining the kinetic operator by
SKin(Φ) = X
j,m,n,k,l
(φµ)j1
mnGj1j2
mn;kl(φµ)j2
kl ,
one can write
Gj1j2
mn;kl =8πλ3
g2w(j1)M+λ2µj1(j1+ 1)δj1j2δnkδml.(3.5)
The relation (3.5) defines a positive self-adjoint operator. The corresponding details are
collected in the appendix A.
The gauge-invariance of S(2.37) can be translated into invariance under a nilpotent
BRST operation δ0defined by the following structure equations [38]
δ0Φµ=i[C, Φµ], δ0C=iCC (3.6)
where Cis the ghost field. Recall that δ0acts as an antiderivation with respect to the
grading given by (the sum of) the ghost number (and degree of forms), modulo 2. C(resp.
Φi) has ghost number +1 (resp. 0). Fixing the gauge symmetry can be conveniently done
by using the gauge condition
Φ3=θ3.(3.7)
This can be implemented into the action by enlarging (3.6) with
δ0¯
C=b , δ0b= 0 (3.8)
– 10 –
JHEP12(2015)045
where ¯
Cand bare respectively the antighost and the Stueckelberg field (with respective
ghost number 1and 0) and by adding to Sa BRST invariant gauge-fixing term given
by (2.37)
Sfix =δ0Tr ¯
C3θ3)= Tr b3θ3)i¯
C[C, Φ3].(3.9)
Integrating over the Stueckelberg field byields the constraint Φ3=θ3into (2.37), while the
ghost part can be easily seen to decouple.6
Now, we define the kinetic operator by
K:= G+ 8ΩL(θ2
3),(3.10)
where G=M+µx2and L(θ2
3)is the left multiplication by θ2
3. The resulting gauge-fixed
action can be written (up to an unessential constant term) as
Sf
=S2+S4,(3.11)
with
S2=1
g2Tr 1,Φ2) Q0
0Q! Φ1
Φ2!!,
Q=K+i4(Ω 1)L(θ3)D3,(3.12)
S4=4
g2Tr Ω(Φ2
1+ Φ2
2)2+ (Ω 1)(Φ1Φ2Φ1Φ2Φ2
1Φ2
2).(3.13)
The gauge-fixed action (3.11) is thus described by a rather simple NCFT with “flavor
diagonal" kinetic term (see (3.12)) and quartic interaction terms. We find also convenient
to introduce the complex fields
Φ = 1
21+iΦ2),Φ=1
21iΦ2),(3.14)
so that the gauge-fixed action Sf
can be expressed alternatively into the form
Sf
=2
g2Tr ΦQΦ+ ΦQΦ+16
g2Tr (Ω + 1)ΦΦΦΦ+ (3Ω 1)ΦΦΦΦ.(3.15)
At this level, some comments are in order.
The action (3.15) bears some similarity with the (matrix model representation of) the
action describing the family of complex LSZ models [24].
For Ω = 1/3, the quartic interaction potential depends only on ΦΦ, so that the
action is formally similar to the action describing an exactly solvable LSZ-type model
investigated in [24]. Only the respective kinetic operators are different. It turns out
that the partition function for Sf
Ω= 1
3
(3.15) can be actually related to τ-functions
6Recall it amounts to consider an “on-shell" formulation for which nilpotency of the BRST operation
(and corresponding BRST-invariance of the gauge-fixed action) is verified modulo the ghost equation of
motion.
– 11 –
JHEP12(2015)045
of integrable hierarchies. More precisely, due to the orthogonal decomposition of
R3
λ(2.4), the partition function can be expressed as a product of factors labelled by
jN
2, each one being expressible as a τ-function for a 2-d Toda hierarchy. Note that
each factor can be actually interpreted as the partition function for the reduction of
the gauge-fixed theory (3.11) on the matrix algebra M2j+1(C). The corresponding
analysis will be presented in a separate publication [64].
For Ω = 1, the kinetic operator in (3.15) simplifies while the interaction term takes
a more symmetric form, as it is apparent e.g from (3.13). We will find that the
corresponding theory is finite to all orders in perturbation.
3.2 Gauge-fixed action at Ω = 1
In this subsection, we will assume Ω=1. The corresponding action is
Sf
Ω=1 =1
g2Tr 1,Φ2) K0
0K! Φ1
Φ2!!+4
g2Tr 2
1+ Φ2
2)2.(3.16)
The kinetic term is expressed as
Sf
2,Ω=1 =8πλ3
g2X
j,m,n
w(j)M+µλ2j(j+ 1) + 8
λ2n2|(φ1µ)mn|2+ (1 2),(3.17)
where we used
x2
3=λ2X
j,m
m2vj
mm.(3.18)
Accordingly, the “matrix elements" of the kinetic operator can be written as
Kj1j2
mn;kl := 8πλ3
g2w(j1)M+µλ2j1(j1+ 1) + 4
λ2(k2+l2)δj1j2δmlδnk .(3.19)
Note that (3.19) verifies
Kj1j2
mn;kl =Kj1j2
lk;nm =Kj1j2
mn;lk (3.20)
reflecting reality of the functional action and the self-adjointness of K(see appendix A;
recall we use the natural Hilbert product ha, bi= Tr(ab)).
The inverse of (3.19) (i.e the matrix elements of the propagator) Pj1j2
mn;kl is then
defined by
X
j2,k,l
Kj1j2
mn;lkPj2j3
kl;rs =δj1j3δms δnr,X
j2,n,m
Pj1j2
rs;mnKj2j3
nm;kl =δj1j3δrl δsk,(3.21)
leading to
Pj1j2
mn;kl =g2
8πλ3
1
w(j1)(M+λ2µj1(j1+ 1) + 4
λ2(k2+l2))δj1j2δmlδnk .(3.22)
We will start the perturbative analysis by computing the 2-point (connected) correlation
function at the first (one-loop) order. To prepare the discussion, we introduce sources
– 12 –
JHEP12(2015)045
variables for the Φα’s, namely Jα=P
j,m,n
(Jα)j
mnvj
mn, for any α= 1,2. Then, a standard
computation yields the free part of the generating functional of the connected correlation
functions W0(J)given (up to an unessential prefactor) by
eW0(J)=Z2
Y
α=1 DΦαe(Sf
2Ω=1+Tr(ΦαJα)) =Z2
Y
α=1 DΦαeP((φα)j1
mnKj1j2
mn;kl(φα)j2
kl +(Jα)j
mn(φα)j
nm)
= exp 1
4X(Jα)j1
mnPj1j2
mn;kl(Jα)j2
kl ,(3.23)
where we have defined for further convenience
(Jα)j:= 8πλ3w(j)(Jα)j,jm, n j(3.24)
for any jN
2. To obtain (3.23), one simply uses the generic field redefinition among the
fields components given by
(φα)j
mn = (φ0
α)j
mn 1
2Pj
nm;kl(Jα)j
kl = (φ0
α)j
mn 1
2(Jα)j
rsPr s;nm.
Correlation functions involving modes (φα)j
mn will be obtained from the successive action of
the corresponding functional derivatives δ
δ(Jα)j
nm
on the full generating functional. We use
eS41,Φ2)eTr(JαΦα)=eS4δ
δJ1,δ
δJ2eP(Jα)j
mn(φα)j
nm (3.25)
where
S4δ
δJ1
,δ
δJ2=X8πλ3
g2w(j)Sj
4δ
δJ(3.26)
in which Sj
4denotes the Trjpart of the interaction term in the action (3.16). We then write
eW(J)=eS4δ
δJ1,δ
δJ2eW0(J)
to obtain
W(J) = W0(J) + ln 1 + eW0(J)eS4δ
δJ1,δ
δJ21eW0(J),(3.27)
where S4is defined by (3.26). The expansion of both the logarithm and eS4then gives rise
to the perturbative expansion.
3.3 One-loop 2-point and 4-point functions
The computational details of the one-loop contribution to the 2-point function are collected
in the appendix B. From (B.8), it can be realized that the quadratic part of the classical
action receives a 1st order (one-loop) contribution Γ1
2α)given by
Γ1
2α) = 32πλ3
g2X
jN
2
X
jm,n,r,pj
(φα)j
pr w(j)Pj
rm;np(φα)j
mn
+X
jp,r,nj
3(φα)j
pr
j
X
m=j
w(j)Pj
rm;mn
(φα)j
np
,(3.28)
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JHEP12(2015)045
in which the 1st (resp. 2nd) term corresponds to the non-planar (resp. planar) contribution.
Writing generically Γ1
2α) = 32πλ3
g2P(φα)j
mnσj
mn;kl(φα)j
kl, we have explicitly
σNP j
pr;mn =w(j)Pj
pr;mn (3.29)
σP j
pr;nm = 3δmp
j
X
m=j
w(j)Pj
rm;mn.(3.30)
One can easily verify that (3.30) and (3.29) are always finite, even for j= 0 and
j→ ∞ and without any singularity whenever M > 0, which is assumed here. This is
obvious for (3.29). For the planar contribution, one simply observes that the summation
over m, which corresponds to an internal ribbon loop, satisfies the estimate
j
X
m=j
w(j)Prm;mn =δnr
j
X
m=j
g2
8πλ3
1
(M+λ2µj(j+ 1) + 4
λ2(m2+n2))
δnr
g2
8πλ3
2j+ 1
(M+λ2µj(j+ 1)) (3.31)
which is always finite for any jN
2. Note that no dangerous UV/IR mixing shows up in
the computation of the one-loop 2-point function.
Eq. (3.31) reflects simply the existence of an estimate obeyed by the propagator (3.22)
(see (3.32) below). This can be used in the subsection 3.4 to show the finiteness of the
theory to all orders in perturbation. Indeed, we have from (3.22):
0Pj1j2
mn;kl Π(M, j1)
w(j1)δj1j2δmlδnk ,(3.32)
for any j1, j2N
2,j1m, n, k, l j1,, where
Π(M, j ) := g2
8πλ3
1
(M+λ2µj(j+ 1)) .(3.33)
A similar analysis can be carried out for the 1-loop contributions to the 4-point func-
tion showing that those contributions are again finite. For instance, consider the vertex
functional for one specie Φα, written generically as (no sum over α)
Γ1
4α) = X
mi,ni,ri,si
Vm1,m2,n1,n2,r1,r2,s1,s2(φα)j
m1m2(φα)j
n1n2(φα)j
r1r2(φα)j
s1s2.(3.34)
Typical planar contributions to the vertex functional are of the form
ΓP1
4X
X
jp,qj
w2(j)Pj
n1p;qr2Pj
pm2;s1qδm1n2
×δs2r1(φα)j
m1m2(φα)j
n1n2(φα)j
r1r2(φα)j
s1s2,(3.35)
where the factor w2(j)comes from the 2 vertex contributions to the loop. One can easily
check that
X
jp,qj
w2(j)Pj
n1p;qr2Pj
pm2;s1qδn1r2δs1m2(2j+ 1)Π(M, j )2,(3.36)
which is finite for any value of jand decays to 0as j3when j→ ∞.
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JHEP12(2015)045
Other planar 1-loop contributions to the vertex function can be checked to be finite by
using a similar argument.
There are 3 species of non-planar contributions with typical respective contributions
being of the form
Γ1
14 Xw2(j)Pj
m1n2;s1r2Pj
n1m2;r1s2(φα)j
m1m2(φα)j
n1n2(φα)j
r1r2(φα)j
s1s2,(3.37)
Γ1
24 X X
p
w2(j)Pj
m1p;s1r2Pj
pn2;r1s2δm2n1!
×(φα)j
m1m2(φα)j
n1n2(φα)j
r1r2(φα)j
s1s2,(3.38)
Γ1
34 X X
p,q
w2(j)Pj
pm2;qs2Pj
n1p;s1qδm1n2δs2r1!
×(φα)j
m1m2(φα)j
n1n2(φα)j
r1r2(φα)j
s1s2,(3.39)
where obvious summations are not explicitly written. By further performing the sum-
mations over pand qin (3.38)–(3.39) thanks to the delta functions in the propagators
Pj
mn;kl (3.22), we arrive easily at the following estimates:
w2(j)Pj
m1n2;s1r2Pj
n1m2;r1s2Π(M, j )2δm1r2δn2s1δn1s2δm2r1(3.40)
X
p
w2(j)Pj
m1p;s1r2Pj
pn2;r1s2Π(M, j )2δm1r2δr1n2(3.41)
X
p,q
w2(j)Pj
pm2;qs2Pj
n1p;s1qΠ(M, j )2δs1s2δm2n1,(3.42)
leading to finite non-planar contributions to the vertex functional (3.34). A similar conclu-
sion holds true for the other non-planar contribution. Notice, by the way that the r.h.s. of
each of the relations (3.36) and (3.40)–(3.42) decay to zero as j4for j→ ∞.
As for the 2-point function, the diagram amplitudes for the 4-point function are finite,
thanks to the existence of the bound for the propagator (3.32) together with the fact that
loop summation indices are bounded by ±j. Summarizing the above 1-loop analysis, a
simple inspection shows that no singularity can occur for j= 0 within the present model
(recall M > 0) while the only source for divergence might come from the limit j→ ∞. But
such divergences are prevented to occur thanks to the upper bound (3.32) and the decay
of Π(M, j )(3.33) at large j, namely Π(M, j)j2for j→ ∞ so that the model (3.16) is
finite at the one-loop order. In the next subsection, we will show that this property extends
to any order of perturbation.
3.4 Finiteness of the diagram amplitudes to all orders
We first observe that (3.33) is related obviously to the propagator for the “truncated" gauge
model obtained by simply dropping the field Φ3in the action (2.37). Notice that this latter
formally may be viewed as resulting from the gauge choice Φ3= 0 in (3.9) instead of
Φ3=θ3. For convenience, we quote here the expression for the propagator of the truncated
– 15 –
JHEP12(2015)045
theory which can be simply read off from the r.h.s. of (3.32) and (3.33):
(G1)j1j2
mn;kl =δj1j2δmnδk l
Π(M, j1)
w(j1)(3.43)
which depends only on a single jN
2, says j1.
The “truncated model" belongs to one particular class of NCFT on R3
λamong those
which have been investigated in [51] where it was shown that the models in this class are
finite to all orders in perturbation. We first discuss useful property of this model.
The key observation is that the amplitude of any ribbon diagram depends only on one
jN
2. Indeed, observe e.g the δj1j2in the propagator (3.43) plus its j-dependence and the
delta functions in any quartic vertex. These δjmjk’s all boil down to a single one in the
computation of any amplitude.
Since the propagator (3.43) depends on the bounded indices m, n, . . . only through
Kronecker delta’s, the summations over the indices of any loop can be exactly carried out
so that any ribbon loop contributes to a factor
(2j+ 1)ε, ε 2(3.44)
to a given amplitude. This can be understood from a simple inspection of the Kronecker
delta’s and the summations over the indices for a ribbon loop built from any N-point sub-
diagram Am1,n1,...,mN,nNand a propagator (3.43) that can be taken to be (Q1)j
m1n1;m2n2
without loss of generality. Namely, one has
Am3,n3,...,mN,nN=X
jm1,n1,m2,n2jAm1,n1,...,mN,nN(Q1)j
m1n1;m2n2.(3.45)
There are 4 summed (internal) indices related to the product of N delta’s coming from the
N-point sub-diagram by the 2 delta’s of the propagator depending only on internal indices.
Two summations can be trivially performed leading to N remaining delta functions. There
are a priori 3 possibilities depending how the 2 remaining summed indices are distributed
among the delta’s: either a single delta depends only on one internal index, or one get
a product of two such deltas, one of each internal index, or the 2 summations combine 2
deltas among the None leading to N2remaining deltas. The details are given in the
appendix C. Notice that the value ε= 2 is obtained from purely algebraic and combinatorial
arguments and represents actually the maximal power of the factor 2j+ 1 any loop can
contribute. A refinement of this analysis by taking into account indices conservation may
well lower the maximal value of this exponent by one unit. Nevertheless, it turns out that
the use of this somewhat crude maximal value in the ensuing analysis is sufficient to prove
the finiteness of arbitrary amplitudes. Summarizing the above discussion, it appears that
the loop summations decouple from the related propagators in the computation of diagram
amplitudes for the truncated model, so that any loop simply contribute by a power of
(2j+ 1) given by (3.44). This leads to a major simplification in the analysis of amplitudes
of arbitrary order, as it will be shown in a while.
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JHEP12(2015)045
To end up with perturbative considerations within the truncated model, consider now
a general ribbon diagram Drelated to this model.7Any ribbon diagram built from the
quartic vertices is characterized by a set of positive integer (V, I , F, B).Vis the number
of vertices, Ithe number of internal ribbons. Fis the number of faces. Recall that Fis
obtained by closing the external lines of a diagram and counting the number of closed single
lines. Finally, Bis the number of boundaries which is equal to the number of closed lines
with external legs. The number of ribbon loops if given by
L=FB. (3.46)
Let gNbe the genus of the Riemann surface on which Dcan be drawn. Recall that gis
determined by the following relation
22g=VI+F. (3.47)
Now consider the amplitude ADfor a diagram characterized by the parameters (V, I, F, B).
It is a (positive) function of j, obviously finite and non singular for j= 0, built from
the product of Vvertex factors, each vertex contributing to w(j)up to unessential finite
factor, Ipropagators (3.43) with summations over indices corresponding to FBloops
which, by the decoupling argument discussed above, give a net overall factor bounded by
(2j+ 1)2(FB). Therefore, we can write
ADKw(j)VIΠ(M, j )I(2j+ 1)2(FB)=K0w(j)VI(2j+ 1)2(FB)
(j2+ρ2)I(3.48)
where Kand K0are finite constants and ρ2=M
λµ2and we have isolated the factor w(j).
Recall that the choice w(j) = j+ 1 as given in (2.9) leads to a trace reproducing at the
formal commutative limit the expected behavior for the usual integral on R3. The natural
choice w(j) = 1 is related to a functional trace built from all the canonical traces of the
components M2j+1(C)occurring in the decomposition of R3
λ, (2.4). To study both cases
when taking the j→ ∞ of the r.h.s. of (3.48), we will set conveniently
w(j)jα, α = 0,1,for j→ ∞.(3.49)
The r.h.s. of (3.48) is always finite for j= 0 while it is also finite for j→ ∞ provided
ω(D) = αI + 2B+ 2(2g2) + V(2 α)0,(3.50)
where we used (3.47) and one has still α= 0,1. For g1, one has ω(D)>0. The case
g= 0, for which the finiteness condition (3.50) becomes ω(D) = αI + 2B+V(2 α)40
requires a closer analysis. In fact, when V= 2 a simple inspection shows that (3.50)
7Recall that any ribbon in such a diagram is made of two lines each carrying 2 bounded indices, says
m, n ∈ {−j, . . . , j}. Thus, a ribbon carries 4 bounded indices (as the propagator (3.43)). Notice that
there is a conservation of the indices along each line, as it can be seen by observing the delta function
in the expression of the propagator (3.43), each delta defining the indices affected to one line. For more
details, see [51].
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JHEP12(2015)045
holds true for α= 0,1. The case V= 1 corresponds to the 2-point function for the
truncated model whose finiteness when j is almost apparent from the rightmost
quantity in (3.31). Note that this can be obtained from simple topological consideration for
the planar and non planar contributions to this 2-point function. One obtains B= 2 and
B= 1 respectively so that (3.50) holds true whenever V= 1 for α= 0,1. Summarizing the
above analysis, we conclude that the truncated model in finite to all orders in perturbation.
Let us go back to the gauge model (3.16). As far as finiteness of the diagrams is
concerned8one observes that (3.16) differs from the truncated model only through the
propagator. Hence, for a given diagram D, the amplitude computed within the gauge
model (3.16)Aj
Dsatisfies
|Aj
D|≤|Aj
D|,(3.51)
thanks to the estimate (3.32). Indeed, by using the general expression for any ribbon
amplitudes of NC φ4theory, one infers Aj
Dhas the generic structure
Aj
D=X
IY
λ
Pj
mλ(I)nλ(I);kλ(I)lλ(I)Fj(δ)mλ(I)nλ(I);kλ(I)lλ(I),(3.52)
where Iis some set of (internal) indices, all belonging to {−j, . . . j }so that all the sums
in PIare finite, λlabels the internal lines of D,Pj
mn;kl is the (positive) propagator given
in (3.22) and Fj(δ)mn;kl collects all the delta’s plus vertex weights depending only on j.
One has
|Aj
D| ≤ X
IY
λ(G1)j
mλ(I)nλ(I);kλ(I)lλ(I)Fj(δ)mλ(I)nλ(I);kλ(I)lλ(I).(3.53)
From (3.50), one then obtains
|Aj
D| ≤ K0w(j)VI(2j+ 1)2(FB)
(j2+ρ2)I<(3.54)
where the last inequality stems from (3.50) which has been shown to hold true.
One concludes that all the ribbon amplitudes stemming from (3.16) are finite so that
Sf
Ω=1 is perturbatively finite to all orders.
4 Discussion
Natural families of gauge invariant actions supporting a gauge invariant harmonic term
can be constructed on R3
λ. This last property, which does not hold true on Moyal spaces,
stems from the fact that the gauge invariant factor xµxµ=x2of the harmonic term,
linked to the sum of the squares of the components of the gauge invariant canonical 1-form
connection as defined in (2.26) belongs, actually to the non trivial center of the algebra R3
λ.
Restricting ourselves to positive functional actions depending on the covariant coordinates
8We consider only the finiteness of the loop contributions and not the nature of the various vertices
generated by loop corrections (i.e external legs) which simply amounts to analyze planar and non-planar
contribution for a φ4theory either with propagator (3.22) or with (3.43).
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JHEP12(2015)045
(says Φµdefined e.g by (2.20)) which support a trivial global vacuum, a suitable BRST
gauge-fixing gives rise to a family of matrix models with quartic interactions and kinetic
operator with compact resolvant while the ghost sector decouples. The resulting functional
action is given by Sf
(Φ) (3.15) where is the real coefficient of {Φµ,Φν}2involved in the
classical gauge-invariant action.
Note that in the Moyal case, a harmonic term can be generated into the action as
resulting from a gauge-fixing through the introduction of a suitable BRST-exact term [61].
This yields a gauge propagator with the spectral properties needed to deal with the UV/IR
mixing. Whether or not this interesting modification leads ultimately to a renormalisable
gauge theorie on R4
θremains to be seen.
We have considered the case Ω=1with 2 different types of traces, one being related to
the canonical trace on R3
λand the other one reproducing the usual behavior of the Lebesgues
integral on R3as discussed in the subsection 2.1. We have first computed the 2-point and 4-
point functions at the 1-loop order and have found finite expressions. Perturbative finiteness
of all the amplitudes has been then extended to all orders. This perturbative finiteness of
Sf
Ω=1 may be viewed as the result of the conjunction of 3 features:
i) a sufficient rapid decay of the propagator at large indices (large j) so that correlations
at large separation indices disappear,
ii) the special role played by j, the radius of the fuzzy sphere components as a (UV/IR)
cut-off,
iii) the existence of an upper bound for the (positive) propagator that depends only of the
cut-off.
The above analysis can be extended to the case 6= 1 for which the relevant action is
given by (3.11)–(3.13). The relevant kinetic operator is defined by
Qj1j2
mn;kl = 8πλ3w(j1)δj1j2Λj1(k, l)δmnδkl (4.1)
Λj(k, l) = M+λ2µj(j+ 1) +
2λ2(k+l)2+43Ω
2λ2(kl)2,(4.2)
for any jN
2,jm, n, k, l j. Note that the spectrum of Qis positive, which is
obvious from (4.2). The corresponding propagator is given by
(Q1)j1j2
mn;kl =δj1j2δml δkn
8πλ3w(j1)M+λ2µj1(j1+ 1) +
2λ2(k+l)2+43Ω
2λ2(kl)2.(4.3)
As for the case Ω=1the propagator (4.3) verifies the following estimate
0(Q1)j1j2
mn;kl (G1)j1j2
mn;kl ,j1, j2N
2,jm, n, k, l j . (4.4)
Thanks to this estimate, the analysis carried out above for the amplitudes of the Ω=1
theory can be reproduced for Sf
6=1 in a way similar to the one followed in the subsection 3.4
showing finiteness of the corresponding amplitudes to all orders in perturbation. As a
– 19 –
JHEP12(2015)045
remark, we note that from the parameter dimensions (2.31) and the general expressions
for the trace (2.6) and kinetic terms SKin 1
g2Tr(ΦKΦ), the large j(large indices) limit
j→ ∞ can be interpreted naturally as the UV regime while j= 0 corresponds to the IR
regime. Hence, all the gauge theories on R3
λconsidered in this paper are UV finite with no
IR singular behavior insured by condition (2.35).
The gauge theories considered here describe fluctuations of the covariant coordi-
nate (2.20) around the vacuum A0
µ= 0 (or alternatively the fluctuations of a gauge potential
Aµaround the gauge potential A0
µ=θµdefined by the gauge-invariant connection, in view
of (2.20)). The gauge theories considered in [55] correspond to a choice A0
µ6= 0 (or A0
µ= 0).
Then, expanding the classical gauge-invariant action S(A)around this vacuum generates
cubic interaction terms responsible for the occurrence of a non-zero tadpole showing up at
the one-loop order leading to a vacuum instability. This is one major difference between the
present work and [55] (apart from more technical differences such as gauge choice and/or
parameter choice). Note that the generic action for the family of gauge models in [55] when
truncated to a single fuzzy sphere component of the orthogonal sum in R3
λ(2.4) is the
action for the Alekseev-Recknagel-Schomerus model [62] describing the low energy action
for brane dynamics on S3. It would be interesting to see if a similar relation still exists with
the family of gauge models considered here.
One aspect which deserves further study is to investigate carefully the commutative/
semi-classical limit of the gauge theories considered in this paper and in [55] in the spirit
of what has been done e.g in [63]. Recall that the commutative limit of one of the traces
considered here (the one for which w(j) = j+ 1) has been already investigated in [55,59]
and formally shown to reproduce the usual Lebesgue integral on R3while the fate of (gauge-
fixed) kinetic operators that may occur in these gauge theories is not known so far.
As pointed out in the subsection 3.1, the gauge-fixed action Sf
bears some similarity
with the so-called duality-covariant LSZ model [24]. In fact, one observes that Sf
Ω= 1
3
(3.15)
coincides formally with one of the actions investigated in [24] leading to an exactly solvable
model. Whenever Ω = 1
3, the quartic interaction potential in (3.15) depends only on the
monomial Φ) while the (positive) kinetic operator is somewhat different from the one
of [24]. In fact, the partition function can be factorized in obvious notations as
Z(Q) = Y
jN
2
Zj(Q),(4.5)
with
Zj(Q) = ZDΦjDΦjexp w(j)
g2Trj2ΦjQjΦj+ ΦjQjΦj+64
3ΦjΦjΦjΦj,
(4.6)
where
DΦjDΦj:= Y
jm,njDΦj
mnDΦj
mn ,(4.7)
and Qjis given by (4.1)–(4.2), with however the weight w(j)factored out from (4.2) as it
appears in front of the argument of the exponential and Trjand the matrix ΦjM2j+1(C)
– 20 –
JHEP12(2015)045
have been defined in (2.7). By combining a singular value decomposition of Φjwith the
Harish-Chandra/Itzykson-Zuber measure formula, a standard computation allows us to put
any factor Zj(Q)under the form
Zj(Q) = 1
2(Qj)Nj(g2) det
jm,njf(ωj
m+ωj
n),(4.8)
where Nj(g2)is a prefactor which is not essential here, ∆(Qj)is the Vandermonde deter-
minant associated with the matrix Qj,ωj
kare the eigenvalues of the real symmetric matrix
defined by Λj(m, n)(4.2) and
f(z) = sπg2
128w(j)erfc zsw(j)
64g2!ezw(j)
64g2(4.9)
where erfc is the complementary error function defined by
erfc(z) = 2
πZ
z
dx ex2,zR.(4.10)
The ratio of determinants appearing in the r.h.s. of (4.8) already signals that any Zjcan
be related to a τ-function such as those occurring in integrable hierarchies. The relevant
one here is the 2-Toda hierarchy. The complete analysis will be presented elsewhere [64].
Acknowledgments
This work is dedicated to the memory of Daniel Kastler. Discussions with Nicola Pinamonti
are gratefully acknowledged. J.-C. W. thanks Michel Dubois-Violette for discussions on the
role of canonical connections in noncommutative geometry and Patrizia Vitale for valuable
discussions on algebraic structures related to R3
λ. T. J. would like to thank LPT-Orsay
and J.-C. Wallet for the kind hospitality during his visit which was funded by the French
Government and Rudjer Boskovic Institute. T. J. would like to thank Stjepan Meljanac for
his encouragement and support during various stages of this work. The work by T. J. has
been partially by Croatian Science Foundation under the project (IP-2014-09-9582).
A Properties of the kinetic operators
To simplify the notations, we drop the overall factor 8πλ3in (3.5) and first assume w(j)=1.
Let L(a)denotes the left-multiplication operator by any element aof R3
λ. Self-adjointness
of the classical corresponding kinetic operator can be shown by using eq. (3.5) to define the
unbounded operator Gas
G:= MI+µL(x2),(A.1)
an element of L(H), the space of linear operators acting on
H=span vj
mn , j N
2,jm, n
– 21 –
JHEP12(2015)045
with natural Hilbert product ha, bi= Tr(ab)defined in (2.7). Obviously, Gis symmetric.
By using (2.10) and (2.3), one infers
x0=λX
j,m
j vj
mm ,and x2=λ2X
j,m
j(j+ 1) vj
mm.(A.2)
Therefore,
L(x2) = λ2X
j,m
j(j+ 1)L(vj
mm),(A.3)
i.e L(x2)is a sum of orthogonal projectors, hence a sum of self-adjoint operators, says
L(vj
mm) : R3
λM2j+1(C). This stems from vj
mmvj
mm =vj
mm (see (2.5)) and R3
λ=
jN
2M2j+1(C)(2.4). One concludes that the classical kinetic operator Gis self-adjoint.
The positivity of Gcan be realized from its spectrum given by
spec(G) = λj=M+λ2µj(j+ 1) >0,jN
2.(A.4)
The corresponding (2j+ 1)2-dimensional eigenspaces are
Vj=span vj
mn,jm, n j,(A.5)
for any jN
2. The extension of this analysis to arbitrary polynomial w(j)is easily achieved
by performing a simple rescaling at each step of the above discussion.
As a remark, we note that RG(z)=(GzI)1,z /spec(G), the resolvant operator of
G, is compact. Indeed, pick z= 0. Then, one easily realizes from spec(G)that the operator
RG(0) has decaying spectrum at j→ ∞, still with finite degeneracy for the eigenvalues at
finite j. Hence RG(0) is compact which extends to RG(z), z /spec(G)by making use of
the resolvant equation.
A similar analysis holds for the gauge-fixed kinetic operator K(3.10) when Ω=1. Its
spectrum is easily found to be given by
spec(K) = ρj,p =M+µλ2j(j+ 1) + 8
λ2p2>0,jN
2,jpj, (A.6)
using
x2
3=λ2X
j,m
m2vj
mm.
The corresponding eigenspaces are
Vj,k =span{vj
pq,jqj, |p|=k}, k = 1,2, . . . , j, (A.7)
for any jN
2and one has dim Vj,k6=0 = 2(2j+ 1),dim Vj,0= 2j+ 1 together with the
expected orthogonal decomposition Vj=kVj,k .
Self-adjointness of Kstill holds since it can be written as a sum of orthogonal pro-
jectors, in view of the above expression for x3while positivity of Kis obvious from the
spectrum (A.6).
– 22 –
JHEP12(2015)045
B Connected 2-point function at one-loop
One starts from the relevant contribution of (3.27) to the connected 2-point function at
one-loop written as
W(Jα) = W0(Jα)eW0(Jα)S4(Jα)eW0(Jα)+. . . (B.1)
with
S4(Jα) = X32πλ3
g2w(j) δ
δ(J1)j
mn
δ
δ(J1)j
np
δ
δ(J1)j
pr
δ
δ(J1)j
rm
+ 1 2!
+ 2 δ
δ(J1)j
mn
δ
δ(J1)j
np
δ
δ(J2)j
pr
δ
δ(J2)j
rm !!.(B.2)
To simplify the notations, it will be convenient to define
(Pα)j
mn := 1
2Pj
mn;kl(Jα)j
kl, α = 1,2,jm, n j(B.3)
for any jN
2which shows up naturally when using the Legendre transform to obtain the
counterpart of (B.1) in the effective action Γ(Φα). Indeed, one has
W(Jα) = Γ(Φα)X
j,m,n
(Jα)j
mn(φα)j
nm,δ W (Jα)
δ(Jα)j
mn
=(φα)j
nm,(B.4)
from which one realizes that the 1st order solution of the 2nd relation in (B.4), which is
needed in the present computation, is provided by (B.3), namely
δW0(Jα)
δ(Jα)j
mn
=(Pα)j
mn.(B.5)
After performing standard computation, we obtain (obvious summation indices not ex-
plicitely written)
W(Jα)) = 1
4X(Jα)j1
mnPj1j2
mn;kl(Jα)j2
kl 32πλ3
g2Xw(j)(P1)j
mp(P1)j
pr(P1)j
rn(P1)j
nm
+ (1 2) + 2(P1)j
mp(P1)j
pr(P2)j
rn(P2)j
nm)32πλ3
g2Xw(j)Pj
rm;pr Pj
np;mn
+1
4Pj
rm;npPj
pr;mn +1
2Pj
rm;mn(P1)j
pr(P1)j
np +1
2Pj
pr;mn(P1)j
rm(P1)j
np
+1
2Pj
np;mn(P1)j
pr(P1)j
rm +1
2Pj
rm;np(P1)j
pr(P1)j
mn
+1
2Pj
pr;np(P1)j
rm(P1)j
mn +1
2Pj
rm;pr (P1)j
np(P1)j
mn +Pj
rm;pr (P1)j
np(P1)j
mn
+ (1 2)+. . . . (B.6)
By making use of the Legendre transform (B.4) with
(Jα)j
sr =2Kj
rs;nm(φα)j
nm, α = 1,2,(B.7)
– 23 –
JHEP12(2015)045
and taking into account the symmetries of the propagator stemming from (3.20), we finally
obtain the expression for the relevant part of the effective action
Γ(Φα) = X(φα)j
mnKj
mn;kl(φα)j
kl +32πλ3
g2Xw(j) Trj((Φ2
1+ Φ2
2)2)
+32πλ3
g2Xw(j)(φα)j
prPj
rm;np(φα)j
mn + 3(φα)j
prPj
rm;mn(φα)j
np+. . . (B.8)
in which the last two terms corresponds to the one-loop corrections.
C Loop summation for the truncated model
Consider the loop built from from any N-point sub-diagram Am1,n1,...,mN,nNand a propa-
gator (3.43). This latter can be taken to be (G1)j
m1n1;m2n2without loss of generality. The
corresponding N2amplitude is
Am3,n3,...,mN,nN=X
jm1,n1,m2,n2jAm1,n1,...,mN,nN(G1)j
m1n1;m2n2,(C.1)
where the N-point part can be written generically as
Am1,n1,...,mN,nN=FN(j)
N
Y
p=1
δmpnσ(p)(C.2)
where σSNis some permutation of {1,2, . . . , N }and FN(j)is some function depending
on jand the other parameters of the model. Combining (C.1) with (C.2) and performing
two summations, one obtains
Am3,n3,...,mN,nN=FN(j)Π(J, M )
w(j)X
jn1,n2j
N
Y
p=3
δmpnσ(p)
δnσ(1)n2δnσ(2) n1.(C.3)
The actual value of (C.3) is rules by the permutation σ. If σ(1) = 2 and σ(2) = 1 or
σ(1) = 1 and σ(2) = 2, (C.3) yields obviously
Am3,n3,...,mN,nN= (2j+1)2FN(j)Π(J, M )
w(j)
N
Y
p=3
δmpnσ(p)
when σ(1) = 2, σ(2) = 1,(C.4)
Am3,n3,...,mN,nN= (2j+ 1)FN(j)Π(J, M )
w(j)
N
Y
p=3
δmpnσ(p)
when σ(1) = 1, σ(2) = 2 (C.5)
thanks to the 2 last delta functions. When σ(1) = 2 and σ(2) 6= 1, the summation over n2
can be readily performed to give
Am3,n3,...,mN,nN= (2j+ 1)FN(j)Π(J, M )
w(j)
j
X
n1=j
N
Y
p=3
δmpnσ(p)
δnσ(2)n1.(C.6)
– 24 –
JHEP12(2015)045
One further observes that σ(2) is valued in {3,4, . . . , N }so that there exists one p0
{3, . . . , N }such that σ(p0)=1. Hence, the last summation can be performed to give
Am3,n3,...,mN,nN= (2j+ 1)FN(j)Π(J, M )
w(j)
N
Y
p=3,p6=p0
δmp0nσ(2)
,(C.7)
so that the loop summation produces an overall factor (2j+ 1). A similar conclusion holds
true for the case σ(2) = 1 and σ(1) 6= 2. Finally, in the remaining case σ(1) 6= 1,2,
σ(2) 6= 1,2, the 2 summations simply yields a product of N2delta functions.
Open Access. This article is distributed under the terms of the Creative Commons
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