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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 [10–12].

For reviews, see for instance [13–15].

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[16–18], 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 [25–29], 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 [33–35]. 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 [39–42]. 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 [44–46]. 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 φ∈R3

λhas the following blockwise expansion

φ=X

j∈N

2X

−j≤m,n∈N≤j

φj

mn vj

mn ,(2.3)

where φj

mn ∈C, and the family {vj

mn , j ∈N

2,−j≤m, n ≤j}is the natural orthogonal

basis of R3

λintroduced in [51], stemming from the direct sum decomposition

R3

λ=M

j∈N

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 ,∀j∈N

2,−j≤m, 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(a†b), for any a, b ∈R3

λ. Here, the trace functional Tr can be deﬁned [55] for any

Φ,Ψ∈R3

λas

Tr(ΦΨ) := 8πλ3X

j∈N

2

w(j)trj(ΦjΨj)(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)−j≤m,n≤j(resp.

Ψj:= (ψj

qp)−j≤q,p≤j). Therefore we have

Tr(ΦΨ) = 8πλ3X

j∈N

2

w(j)

X

−j≤m,n≤j

φj

mnψj

nm

,(2.7)

and

trj(vj

mn) = δmn ,hvj1

mn, vj2

pqi= 8πλ3X

j1∈N

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,n−1

x−vj

mn =λF(j, −m)vj

m−1,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)(j−m).(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 [27–29].

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 n∈N, by Ωn

Gthe space of n−(Z(R3

λ))-linear) antisymmetric maps

ω:Gn→R3

λ, the corresponding N-graded diﬀerential algebra is (Ω•

G=⊕n∈NΩn

G, d, ×),

with nilpotent diﬀerential d: Ωn

G→Ωn+1

Gand product ×on Ω•

Gdeﬁned for any ω∈Ωp

G

and ρ∈Ωq

Gby

dω(X1, . . . , Xp+1) =

p+1

X

k=1

(−1)k+1Xkω(X1,...,∨k, . . . , Xp+1 )

+X

1≤k<l≤p+1

(−1)k+lω([Xk, Xl],...,∨k,...,∨l, . . . , Xp+1),(2.14)

ω×ρ(X1, . . . , Xp+q) = 1

p!q!X

σ∈Sp+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 σ∈Sp+q.

Let Mdenotes a right-module over R3

λ. Recall that a connection on Mcan be deﬁned

as a linear map ∇:G × M→Mwith

∇X(ma) = ∇X(m)a+mXa , ∇zX (a) = z∇X(a),∇X+Y(a) = ∇X(a) + ∇Y(a),

for any a∈R3

λ, any m∈M,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=C⊗R3

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, a2∈R3

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 a∈R3

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]=DµAν−DνAµ+ [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

φ∈R3

λ, one has g†g=gg†=I,φg=gφ. From the deﬁnition of the gauge transformations

of the connection given by ∇g

µ=g†∇µ◦g, for any g∈ U(R3

λ), one infers

Ag

µ=g†Aµg+g†Dµg , and Fg

µν =g†Fµν 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) = Dµa−iθµa=−iaθµ,∀a∈R3

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+iθµ,∀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 a∈R3

λ, and g∈ U(R3

λ), the following gauge transformations

hold true

(∇inv

µ(a))g=∇inv

µ(a),Ag

µ=g†Aµg , ∀µ= 1,2,3.(2.22)

4Given a hermitean structure, says h:M×M→R3

λ,∇is hermitean if Xh(m1, m2) = h(∇X(m1), m2) +

h(m1,∇X(m2)), for any X∈ G,m1, m2∈M.

– 6 –

JHEP12(2015)045

Deﬁne the real invariant 1-form Θ∈Ω1

Gby

Θ∈Ω1

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:= dΘ∈Ω2

Gcan be viewed as the natural

symplectic form on the algebra R3

λin the setting of [25,26] with Ham(a) = Adia for any

a∈R3

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=−iθµ(2.26)

as it can be readily obtained from (2.16) and (2.19). One easily checks that

(Ainv

µ)g= (−iθµ)g=−iθµ,(2.27)

– 7 –

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

(−iθµ)(−iθµ) = −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(−iθµ)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

(−iθµ)g(−iθµ)g(Ag

αAg

α)

= Tr

g

3

X

µ=1

(−iθµ)(−iθµ)g†(AαAα)

= Tr

3

X

µ=1

(−iθµ)(−iθµ)(AαAα)

(2.29)

where we used Pµ(−iθµ)(−iθµ)∈ 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 [27–29]), 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+iζµνρΦµΦνΦρ+ (M+µx2)ΦµΦµ

=1

g2Tr η−κ

4[Φµ,Φν]2+η+κ

4{Φµ,Φν}2+iζµνρΦµΦνΦρ

+ (M+µx2)ΦµΦµ,(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)

– 8 –

JHEP12(2015)045

where the real parameter Ωis dimensionless, thus ﬁxing for convenience the overall nor-

malization of the term ∼[Φµ,Φν]2in (2.30). This latter action can be rewritten as

S(Φ) = 1

g2Tr Fµν −i

λµνρ Φρ†Fµν −i

λµνρ Φρ+ Ω {Φµ,Φν}2+iζµνρΦµΦνΦρ

+ (M+µx2)ΦµΦµ

=1

g2Tr F†

µν Fµν + Ω {Φµ,Φν}2+iζ 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

Ω≥0, µ > 0, ζ = 0, M > 0(2.35)

or

Ω≥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

SΩ=1

g2Tr Fµν −i

λµνρ Φρ†Fµν −i

λµνρ Φρ+Ω {Φµ,Φν}2+(M+µx2)ΦµΦµ.(2.37)

The equation of motion for (2.37) given by

4(Ω + 1)(ΦρΦµΦµ+ ΦµΦµΦρ) + 8(Ω −1)ΦµΦρΦµ+ 2(M+µx2)Φρ= 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±qη2λ2−32[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)SΩis given by

SKin(Φ) = 1

g2Tr(Φµ(M+µx2)Φµ)(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 SΩ(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 SΩa BRST invariant gauge-ﬁxing term given

by (2.37)

Sﬁx =δ0Tr ¯

C(Φ3−θ3)= Tr b(Φ3−θ3)−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

Ω=S2+S4,(3.11)

with

S2=1

g2Tr (Φ1,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)(Φ1Φ2Φ1Φ2−Φ2

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

2(Φ1+iΦ2),Φ†=1

2(Φ1−iΦ2),(3.14)

so that the gauge-ﬁxed action Sf

Ωcan be expressed alternatively into the form

Sf

Ω=2

g2Tr ΦQΦ†+ Φ†QΦ+16

g2Tr (Ω + 1)ΦΦ†ΦΦ†+ (3Ω −1)ΦΦΦ†Φ†.(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

j∈N

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 (Φ1,Φ2) K0

0K! Φ1

Φ2!!+4

g2Tr (Φ2

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(a†b)).

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 DΦαe−P((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,−j≤m, n ≤j(3.24)

for any j∈N

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

e−S4(Φ1,Φ2)e−Tr(JαΦα)=e−S4δ

δJ1,δ

δJ2e−P(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)=e−S4δ

δJ1,δ

δJ2eW0(J)

to obtain

W(J) = W0(J) + ln 1 + e−W0(J)e−S4δ

δJ1,δ

δJ2−1eW0(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

2(Φα)given by

Γ1

2(Φα) = 32πλ3

g2X

j∈N

2

X

−j≤m,n,r,p≤j

(φα)j

pr w(j)Pj

rm;np(φα)j

mn

+X

−j≤p,r,n≤j

3(φα)j

pr

j

X

m=−j

w(j)Pj

rm;mn

(φα)j

np

,(3.28)

– 13 –

JHEP12(2015)045

in which the 1st (resp. 2nd) term corresponds to the non-planar (resp. planar) contribution.

Writing generically Γ1

2(Φα) = 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 j∈N

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):

0≤Pj1j2

mn;kl ≤Π(M, j1)

w(j1)δj1j2δmlδnk ,(3.32)

for any j1, j2∈N

2,−j1≤m, 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

4(Φα) = 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

4∼X

X

−j≤p,q≤j

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

−j≤p,q≤j

w2(j)Pj

n1p;qr2Pj

pm2;s1q≤δn1r2δs1m2(2j+ 1)Π(M, j )2,(3.36)

which is ﬁnite for any value of jand decays to 0as j−3when j→ ∞.

– 14 –

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;r1s2≤Π(M, j )2δm1r2δn2s1δn1s2δm2r1(3.40)

X

p

w2(j)Pj

m1p;s1r2Pj

pn2;r1s2≤Π(M, j )2δm1r2δr1n2(3.41)

X

p,q

w2(j)Pj

pm2;qs2Pj

n1p;s1q≤Π(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 j−4for 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)∼j−2for 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):

(G−1)j1j2

mn;kl =δj1j2δmnδk l

Π(M, j1)

w(j1)(3.43)

which depends only on a single j∈N

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

j∈N

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 (Q−1)j

m1n1;m2n2

without loss of generality. Namely, one has

Am3,n3,...,mN,nN=X

−j≤m1,n1,m2,n2≤jAm1,n1,...,mN,nN(Q−1)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 N−2remaining 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.

– 16 –

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=F−B. (3.46)

Let g∈Nbe the genus of the Riemann surface on which Dcan be drawn. Recall that gis

determined by the following relation

2−2g=V−I+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 F−Bloops

which, by the decoupling argument discussed above, give a net overall factor bounded by

(2j+ 1)2(F−B). Therefore, we can write

AD≤Kw(j)V−IΠ(M, j )I(2j+ 1)2(F−B)=K0w(j)V−I(2j+ 1)2(F−B)

(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(2g−2) + V(2 −α)≥0,(3.50)

where we used (3.47) and one has still α= 0,1. For g≥1, one has ω(D)>0. The case

g= 0, for which the ﬁniteness condition (3.50) becomes ω(D) = αI + 2B+V(2 −α)−4≥0

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].

– 17 –

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(G−1)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)V−I(2j+ 1)2(F−B)

(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).

– 18 –

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

Ω(Φ) (3.15) where Ωis 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 Ω6= 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) + Ω

2λ2(k+l)2+4−3Ω

2λ2(k−l)2,(4.2)

for any j∈N

2,−j≤m, n, k, l ≤j. Note that the spectrum of Qis positive, which is

obvious from (4.2). The corresponding propagator is given by

(Q−1)j1j2

mn;kl =δj1j2δml δkn

8πλ3w(j1)M+λ2µj1(j1+ 1) + Ω

2λ2(k+l)2+4−3Ω

2λ2(k−l)2.(4.3)

As for the case Ω=1the propagator (4.3) veriﬁes the following estimate

0≤(Q−1)j1j2

mn;kl ≤(G−1)j1j2

mn;kl ,∀j1, j2∈N

2,−j≤m, 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

Ω6=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

Ωbears 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†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

j∈N

2

Zj(Q),(4.5)

with

Zj(Q) = ZDΦjDΦ†jexp −w(j)

g2Trj2ΦjQjΦ†j+ Φ†jQjΦj+64

3ΦjΦ†jΦjΦ†j,

(4.6)

where

DΦjDΦ†j:= Y

−j≤m,n≤jDΦj

mnDΦ†j

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 Φj∈M2j+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

−j≤m,n≤jf(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 e−x2,∀z∈R.(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,−j≤m, n ≤

– 21 –

JHEP12(2015)045

with natural Hilbert product ha, bi= Tr(a†b)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

λ→M2j+1(C). This stems from vj

mmvj

mm =vj

mm (see (2.5)) and R3

λ=

⊕j∈N

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,∀j∈N

2.(A.4)

The corresponding (2j+ 1)2-dimensional eigenspaces are

Vj=span vj

mn,−j≤m, n ≤j,(A.5)

for any j∈N

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)=(G−zI)−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,∀j∈N

2,−j≤p≤j, (A.6)

using

x2

3=λ2X

j,m

m2vj

mm.

The corresponding eigenspaces are

Vj,k =span{vj

pq,−j≤q≤j, |p|=k}, k = 1,2, . . . , j, (A.7)

for any j∈N

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)−e−W0(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,−j≤m, n ≤j(B.3)

for any j∈N

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 (G−1)j

m1n1;m2n2without loss of generality. The

corresponding N−2amplitude is

Am3,n3,...,mN,nN=X

−j≤m1,n1,m2,n2≤jAm1,n1,...,mN,nN(G−1)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 σ∈SNis 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

−j≤n1,n2≤j

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 N−2delta functions.

Open Access. This article is distributed under the terms of the Creative Commons

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any medium, provided the original author(s) and source are credited.

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