Derivations of the Moyal Algebra and Noncommutative Gauge Theories

0.46 MB·English

by  Jean-Christophe Wallet

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 013, 25 pages
Derivations of the Moyal Algebra
and Noncommutative Gauge Theories(cid:63)
Jean-Christophe WALLET
Laboratoire de Physique Th´eorique, Bˆat. 210, CNRS, Universit´e Paris-Sud 11,
F-91405 Orsay Cedex, France
E-mail: [email protected]
Received October 29, 2008, in final form January 17, 2009; Published online January 30, 2009
doi:10.3842/SIGMA.2009.013
9
0
Abstract. The differential calculus based on the derivations of an associative algebra un-
0
derliesmostofthenoncommutativefieldtheoriesconsideredsofar. Wereviewtheessential
2
properties of this framework and the main features of noncommutative connections in the
n
case of non graded associative unital algebras with involution. We extend this framework
a
J to the case of Z2-graded unital involutive algebras. We show, in the case of the Moyal alge-
bra or some related Z -graded version of it, that the derivation based differential calculus
0 2
3 is a suitable framework to construct Yang–Mills–Higgs type models on Moyal (or related)
algebras, the covariant coordinates having in particular a natural interpretation as Higgs

] fields. We also exhibit, in one situation, a link between the renormalisable NC ϕ4-model
h
p with harmonic term and a gauge theory model. Some possible consequences of this are
- briefly discussed.
h
t Key words: noncommutative geometry; noncommutative gauge theories
a
m
2000 Mathematics Subject Classification: 81T75; 81T13
[

2
v
0 1 Introduction
5
8
Aclassofnoncommutative(NC)fieldtheories[1,2](forgeneralreviewsonnoncommutativegeo-
3
. metry, see [3, 4]) came under increasing scrutiny after 1998 when it was argued [5, 6] that string
1
1 theory might have effective regimes related to noncommutative field theories (NCFT) defined
8 on a NC version of flat 4-dimensional space. This latter is the Moyal space (see e.g. [7, 8]) which
0
has constant commutators between space coordinates. It was further noticed [9, 10] that the
:
v simplest NC ϕ4 model, (ϕ real-valued) on the 4-dimensional Moyal space is not renormalisable
i
X due to the Ultraviolet/Infrared (UV/IR) mixing [9, 10, 11]. This phenomenon stems from the
r existence of nonplanar diagrams that are UV finite but nevertheless develop IR singularities
a
which when inserted into higher order diagrams are not of renormalisable type [1, 2]. A solution
to this problem was proposed in 2004 [12, 13]. It amounts to supplement the initial action with
a simple harmonic oscillator term leading to a fully renormalisable NCFT (for recent reviews,
see e.g. [14, 15]). This result seems to be related to the covariance of the model under the so
called Langmann–Szabo duality [16]. Other renormalisable NC matter field theories have then
been identified [17, 18, 19, 20] and studies of the properties of their renormalisation group flows
have been carried out [21, 22, 23].
So far, the construction of a renormalisable gauge theory on 4-D Moyal spaces remains
a problem. The naive NC version of the Yang–Mills action has UV/IR mixing, which stems
from the occurrence of an IR singularity in the polarisation tensor. From a standard one-loop
(cid:63)This paper is a contribution to the Proceedings of the XVIIth International Colloquium on Integrable Sys-
tems and Quantum Symmetries (June 19–22, 2008, Prague, Czech Republic). The full collection is available at
http://www.emis.de/journals/SIGMA/ISQS2008.html
2 J.-C. Wallet
calculation, we easily infer that
(cid:18) (cid:19)
D p˜ p˜
ω (p) ∼ (D−2)Γ µ ν +··· , p → 0, (1.1)
µν 2 πD/2(p˜2)D/2
where p˜ ≡Θ p , Θ is the symplectic matrix of the 4-D Moyal algebra and Γ(z) denotes
µ µν ν µν
the Euler function. This singularity, albeit transverse in the sense of the Slavnov–Taylor–Ward
identities, does not correspond to some gauge invariant term. This implies that the recent alter-
native solution to the UV/IR mixing proposed for the NC ϕ4 model in [24], cannot be extended
straighforwardly (if possible at all) to the gauge theories. Note that an attempt to reach this
goal has been proposed in [25]. It amounts to add to the naive Yang–Mills action a counterterm
whichremainsinvariantunderadefinedBRSTsymmetrybuthowevercanbeviewedasaformal
infinite series in the gauge potential A . The actual effect of this counterterm on the UV/IR
µ
mixing of the modified gauge action is still unclear and deserves further investigations.
Recently, an extension of the harmonic solution to the case of gauge theories has been pro-
posed in [26] and [27] (for various reviews, see [28, 15, 29]). These works have singled out, as
potential candidate for renormalisable gauge theory on 4-D Moyal space, the following generic
action
(cid:90) (cid:18)1 Ω2 (cid:19)
S = d4x F (cid:63)F + {A ,A }2+κA (cid:63)A , (1.2)
4 µν µν 4 µ ν (cid:63) µ µ
where F = ∂ A − ∂ A − i[A ,A ] , A is the gauge potential, (cid:63) denotes the associative
µν µ ν ν µ µ ν (cid:63) µ
product on the Moyal algebra, [a,b] ≡ a (cid:63) b − b (cid:63) a, {a,b} ≡ a (cid:63) b + b (cid:63) a and A is the
(cid:63) (cid:63) µ
covariant coordinates, a natural gauge covariant tensorial form stemming from the existence
of a canonical gauge invariant connection in the present NC framework, as we will recall in
a while (see [26, 15]). The 2nd and 3rd terms in (1.2) may be viewed as “gauge counterparts”
of the harmonic term of [12]. This action has interesting properties [26, 27] deserving further
studies. For instance, gauge invariant mass terms for the gauge fields are allowed even in the
absence of Higgs mechanism. Besides, the covariant coordinates appear to bear some similarity
with Higgs fields. This feature will be examined more closely in the course of the discussion.
Unfortunately, the action (1.2) has a non-trivial vacuum which complicates the study of its
actual renormalisability [30]. Notice that non trivial vacuum configurations also occur within
NC scalar models with harmonic term as shown in [31].
Therelevantalgebraicframeworkdescribing(mostof)theclassicalfeaturesoftheNCactions
consideredsofarisprovidedbythedifferentialcalculusbasedonthederivations. Thisframework
has been established in [32, 33, 34] and [35]. For an exhaustive review, see [36] and references
therein. The derivation based differential calculus underlies the first prototypes of NC (matrix-
valued)fieldtheories[37,38,39]. Forareview,see[40]. AsfarastheMoyalspacesareconcerned,
the “minimal” derivation-based differential calculus generated by the “spatial derivations” ∂
µ
underlies (most of) the works that appeared in the literature, so far. This differential calculus is
not unique but can be modified in numerous ways. Among these, a simple modification of the
minimal differential calculus has been shown recently to give rise to interesting features [41].
The purpose of this paper is to show that the use of the differential calculus based on the
derivations of the Moyal algebra or some related Z -graded version of it permits one to exhibit
2
interesting features related to gauge theories defined on Moyal space. We show, at least within
twonontrivialexamples,thatthederivationbaseddifferentialcalculusisanaturalframeworkto
constructYang–Mills–HiggstypemodelsonMoyal(orrelated)algebras, theso-called“covariant
coordinates” [1, 2] having a natural interpretation as Higgs fields. We also exhibit, in one
situation, a link between the renormalisable NC ϕ4-model with harmonic term [12, 13] and
a gauge theory model built from the square of a curvature. Some possible consequences of this
are briefly discussed.
Derivations of the Moyal Algebra and Noncommutative Gauge Theories 3
The paper is organised as follows. In Section 2.1 of this paper, we review briefly the main
properties of the differential calculus based on the derivations of an associative unital alge-
bra and introduce a definition of a NC connection on a module over the algebra, as a natural
generalisation of ordinary connections. The specific properties and simplifications occurring
when the module is equal to the algebra, which is the case relevant for the NCFT, are detailed
in Section 2.2. In Section 2.3, we extend the analysis to Z -graded algebras. We construct
2
a Z -graded differential calculus based on the graded derivations of the algebra, set a definition
2
of NC connection, the corresponding curvature and (unitary) gauge transformations. In Sec-
tion 3, we focus on the Moyal algebra M. We consider the differential calculus based on the
maximal subalgebra of the derivations of M whose elements can be related to (infinitesimal)
symplectomorphisms. Then, a direct application of the results of Section 2.2 leads to a natural
construction of Yang–Mills–Higgs models defined on M where the “covariant coordinates” used
in the physics literature can be naturally interpreted as Higgs fields, due to the existence of
a gauge invariant canonical connection. In Section 4 we compare in detail the salient mathema-
tical features underlying the NC differential calculus of Section 3 to those for the NC geometry
stemming from the finite dimensional matrix algebra M (C) as well as for the algebra of matrix
n
valued functions C∞(M)⊗M (C). In some sense, the case considered in Section 3 interpolates
n
betweenthesetwolattersituations. TheclassicalpropertiesoftheNCYang–Mills–Higgsactions
are also analysed. Explicit one-loop computation of the vacuum polarisation tensor shows that
this latter still exhibits an IR singularity of the type given in (1.1). In Section 5 we consider
a Z -graded version of the Moyal algebra M built from two copies of M and apply the general
2
results derived in Section 2.3 to the construction of a differential calculus generated by a Z -
2
graded extension of the derivation algebra considered in Section 3. We show that the gauge
theory action built from the “square” of the resulting curvature involves as contributions both
the action (1.2) derived in [26] and [27] as well the renormalisable NC ϕ4-model with harmonic
term elaborated in [12, 13], therefore exhibiting a link between this latter renormalisable NC
scalar theory and gauge theories. Finally, a summary of all the main results is presented in
Section 6.
2 Differential calculus based on derivations
2.1 General properties
Let A be an associative ∗-algebra with unit I and center Z(A). We denote the involution by
a (cid:55)→ a†, ∀a ∈ A. The differential calculus based on the derivations of A is a natural NC
generalisation of the usual de Rham differential calculus on a manifold. Basically, the role of
the vector fields is now played by the derivations of the algebra. In this subsection, we collect
the main properties that will be used in this paper. More details can be found in [32, 34, 33, 35].
Definition 2.1. The vector space of derivations of A is the space of linear maps defined by
Der(A) = {X : A → A/X(ab) = X(a)b+aX(b), ∀a,b ∈ A}. The derivation X ∈ Der(A) is
called real if (X(a))† = X(a†), ∀a ∈ A.
The essential properties of the spaces of derivations of A can be summarised in the following
proposition.
Proposition 2.1. Der(A) is a Z(A)-module for the product (fX)a = f(Xa), ∀f ∈ Z(A),
∀X ∈ Der(A) and a Lie algebra for the bracket [X,Y]a = XYa−YXa, ∀X,Y ∈ Der(A). The
vector subspace of inner derivations is defined by Int(A) = {Ad : b (cid:55)→ [a,b]/a ∈ A} ⊂ Der(A).
a
It is a Z(A)-submodule and Lie ideal. The vector subspace of outer derivations is Out(A) =
Der(A)/Int(A). The following canonical short exact sequence of Lie algebras and Z(A)-modules
holds: 0 −→ Int(A) −→ Der(A) −→ Out(A) −→ 0.
4 J.-C. Wallet
The main features of the differential calculus based on Der(A) are involved in the following
proposition. Notice that both the Lie algebra structure and the Z(A)-module structures for
Der(A) are used as essential ingredients in the construction.
Proposition 2.2. Let Ωn (A) denote the space of Z(A)-multilinear antisymmetric maps from
Der
Der(A)n toA, withΩ0 (A) = AandletΩ• (A) = (cid:76) Ωn (A). Then(Ω• (A), ×, dˆ)isaN-
Der Der n≥0 Der Der
gradeddifferentialalgebrawiththeproduct×onΩ• (A)anddifferentialdˆ: Ωn (A) → Ωn+1(A)
Der Der Der
satisfying dˆ2 = 0, respectively defined for ∀ω ∈ Ωp (A), η ∈ Ωq (A) by:
Der Der
1 (cid:88)
(ω×η)(X ,...,X ) = (−1)sign(σ)ω(X ,...,X )η(X ,...,X ),
1 p+q p!q! σ(1) σ(p) σ(p+1) σ(p+q)
σ∈Sp+q
p+1
(cid:88)
dˆω(X ,...,X ) = (−1)i+1X ω(X ,···∨ ...,X )
1 p+1 i 1 i p+1
i=1
(cid:88)
+ (−1)i+jω([X ,X ],···∨ ···∨ ...,X ), (2.1)
i j i j p+1
1≤i<j≤p+1
where σ denotes permutation and the symbol ∨ indicates that X is omitted.
i i
It turns out that a differential calculus can also be built from suitable subalgebras of Der(A).
The following proposition holds [32, 33]
Proposition 2.3. Let G ⊂ Der(A) denote a Lie subalgebra which is also a Z(A)-submodule.
Then, a restricted derivation-based differential calculus Ω•(A) can be built from G. It is obtained
G
from Proposition 2.2 by replacing the set of Ωn (A), ∀n ∈ N by the set of Z(A)-multilinear
Der
antisymmetric maps from Gn to A for n ≥ 0 and still using (2.1).
From now on, we will not write explicitly the product of forms ×. It should be obvious from
the context when the relevant type of product is used.
In this paper, we will consider a natural NC generalisation of ordinary connections, as intro-
duced in [32, 34, 33] to which we refer for more details. It uses left or right finite projective
modules on the associative algebra. Notice that alternative NC extensions of connections based
on bimodules were considered in [35]. From now on, we denote by M a right A-module. Let
h : M ⊗ M → A denote a Hermitian structure1 on A. The connection, curvature and gauge
transformations are given as follows:
Definition 2.2. A NC connection on M is a linear map ∇ : M×Der(A) → M satisfying:
∇ (ma) = mX(a)+∇ (m)a, ∇ (m) = f∇ (m),
X X fX X
∇ (m) = ∇ (m)+∇ (m), (2.2)
X+Y X Y
∀X,Y ∈ Der(A),∀a ∈ A,∀m ∈ M,∀f ∈ Z(A). AHermitianNCconnectionisaNCconnection
satisfying in addition X(h(m ,m )) = h(∇ (m ),m )+h(m ,∇ (m )), ∀m ,m ∈ M, and
1 2 X 1 2 1 X 2 1 2
for any real derivation in Der(A). The curvature of ∇ is the linear map F(X,Y) : M → M
defined by
F(X,Y)m = [∇ ,∇ ]m−∇ m, ∀X,Y ∈ Der(A).
X Y [X,Y]
Definition 2.3. ThegaugegroupofMisdefinedasthegroupofautomorphismsofMasaright
A-module.
1Recall that a Hermitian structure is a sesquilinear map such that h(m ,m )† = h(m ,m ), h(ma ,ma ) =
1 2 2 1 1 2
a†h(m1,m2)a , ∀m ,m ∈M, ∀a ,a ∈A.
1 2 1 2 1 2
Derivations of the Moyal Algebra and Noncommutative Gauge Theories 5
Proposition 2.4. For any g in the gauge group of M and for any NC connection ∇, the
map ∇g = g−1 ◦ ∇ ◦ g : M → M defines a NC connection. Then, one has F(X,Y)g =
X X
g−1◦F(X,Y)◦g.
It is convenient to require that the gauge transformations are compatible with the Hermitian
structure, that is h(g(m ),g(m )) = h(m ,m ). This defines a NC analogue of unitary gauge
1 2 1 2
transformations. From now on, we will only consider unitary gauge transformations.
2.2 The free module case
In the special case where M = A, that will be the case of interest for the ensuing discussion,
additional simplifications occur. It is further convenient to choose the canonical Hermitian
structure h (a ,a ) = a†a .
0 1 2 1 2
Proposition 2.5. Assume that M = A and h (a ,a ) = h(a ,a ) = a†a . Then:
0 1 2 1 2 1 2
i) Any NC connection is entirely determined by ∇ (I) via ∇ (a) = ∇ (I)a+X(a), ∀X ∈
X X X
Der(A), ∀a ∈ A. The 1-form connection A ∈ Ω1 (A) is defined by A : X → A(X) =
Der
∇ (I), ∀X ∈ Der(A).
X
ii) Assume that the derivations X are real. Then, a NC connection is Hermitian when
∇ (I)† = −∇ (I).
X X
iii) The gauge group can be identified with the group of unitary elements of A, U(A) and one
has ∇ (I)g = g†∇ (I)g+g†X(g), F(X,Y)g = g†F(X,Y)g, ∀X,Y ∈ Der(A), ∀a ∈ A.
X X
Proof. i) follows directly fromDefinition 2.2 (setm = Iin the 1stof (2.2)). Note that therela-
tion ∇ (a) = ∇ (I)a+X(a) can be reobtained from the map ∇ : Ω0 (A) → Ω1 (A), ∇(a) =
X X Der Der
Aa+dˆa with A : X → A(X) = ∇ (I). For ii), one has (∇ (a )†a +a†∇ (a )=X(a†a )+
X X 1 2 1 X 2 1 2
a†(∇ (I)† +∇ (I))a where the last equality stems from the expression for ∇ (a) given in i)
1 X X 2 X
and the fact that X is assumed to be real. From this follows ii). For iii), use Definition 2.3 and
compatibility of gauge transformations with h which gives g(a) = g(I)a and h (g(a ),g(a )) =
0 0 1 2
a†g(I)†g(I)a = h (a ,a ). Then, the gauge transformations for ∇ (I) and the curvature stems
1 2 0 1 2 X
from Proposition 2.4, the expression for ∇ (a) in i) and the expression for F(X,Y). (cid:4)
X
Definition 2.4. A tensor 1-form is a 1-form having the following gauge transformations:
Ag = g†Ag, ∀g ∈ U(A).
There is a special situation where canonical gauge invariant connections can show up, as
indicated in the following proposition.
Proposition 2.6. Assume that there exists η ∈ Ω1 (A), such that dˆa = [η,a], ∀a ∈ A.
Der
Consider the map ∇inv : Ω0 (A) → Ω1 (A), ∇inv(a) = dˆa−ηa, ∀a ∈ A, so that ∇inv(a) =
Der Der X
X(a)−η(X)a. Then, the following properties hold:
i) ∇inv defines a connection which is gauge invariant, called the canonical connection.
ii) For any NC connection ∇, A ≡ ∇−∇inv = A+η defines a tensor form. A(X), ∀X ∈
Der(A) are called the covariant coordinates of ∇.
Proof. Sinceany1-formcanserveasdefiningaconnectioninviewofProposition2.5,∇inv(a) =
dˆa−ηa is a connection. Notice that it reduces to ∇inv(a) = −aη, since dˆa = [η,a]. Then, one
has (∇inv)g(a) = g†∇inv(ga) = g†(d(ga)−ηga) = g†(−gaη) = −aη = ∇inv(a), which shows i).
The property ii) stems simply from Definition 2.4 and the gauge transformations of a NC
connection. (cid:4)
6 J.-C. Wallet
The existence of canonical connections translates into some rather general properties of the
curvatures, in particular the curvature for the canonical connection. Gauge theories defined on
Moyal spaces are a particular example of this, as shown in the next section.
Lemma 2.1. Let Finv(X,Y) ≡ η[X,Y]−[η(X),η(Y)] denote the curvature for the canonical
connection. Assume again that there exists η ∈ Ω1 (A), such that dˆa = [η,a], ∀a ∈ A. Then,
Der
the following properties hold:
i) Finv(X,Y) ≡ η[X,Y]−[η(X),η(Y)] ∈ Z(A).
ii) The curvature of any NC connection defined by the tensor 1-form A can be written as
F(X,Y) = ([A(X),A(Y)]−A[X,Y])−([η(X),η(Y)]−η[X,Y]), ∀X,Y ∈ Der(A).
Proof. First, from the definition of ∇inv(a) in Proposition 2.6, one infers that the 2-form
curvature associated to the canonical connection is Finv(a) ≡ ∇inv(∇inv(a)) = −(dˆη −ηη)(a),
∀a ∈ A. Then,oneobtainsFinv(X,Y) = η[X,Y]−[η(X),η(Y)]. Then,fromdˆa = [η,a]anddˆ2 =
0, one has dˆ(dˆa)=dˆ(ηa−aη)=[dˆη,a]−[η,dˆa]=[dˆη,a]−[η,[η,a]]=[dˆη−ηη,a]. From this follows
thepropertyi). Next,onehas∇ (a) = A(X)a−aη(X)sothat[∇ ,∇ ](a) = [A(X),A(Y)]a−
X X Y
a[η(X),η(Y)]. ThereforeF(X,Y)(a)=([A(X),A(Y)]−A([X,Y]))a−a([η(X),η(Y)]−η([X,Y])).
This last expression, combined with the property i) implies ii). (cid:4)
Notice that this proposition will be relevant when considering the NC gauge theories on
Moyal algebras. A somewhat similar (but not identical!) situation occurs for the algebra of
matrices M (C) as well as for the algebra of matrix valued functions.
n
2.3 Graded differential calculus
In this subsection, we extend the previous algebraic scheme to the case of differential calculus
based on the derivations of a graded algebra. We consider only the Z -graded case. The
2
extension to graduations based on additive groups as well as to more general structures [43] will
be reported in a separate work [44].
To fix the notations, let A• be a Z -graded associative unital ∗-algebra, namely A• =
2
(cid:76) Aα where Aα = {a ∈ A•/|a| = α}, |a| denoting the homogeneous degree of a, and
α= 0,1
AαAβ ⊆ A(α+β)(mod2). The graded bracket on A• is [a,b] ≡ ab−(−1)|a||b|ba, ∀a,b ∈ A•. Let
•
Z(A•) denote the center of A•, a graded-commutative algebra (a ∈ Z(A•) ⇐⇒ [a,b] = 0,
•
∀b ∈ A•). Gradedmoduleswillbeneededinthepresentframework. Recallthattheyaregraded
vector spaces M• = (cid:76) Mα satisfying MαAβ ⊆ Mα+β and AβMα ⊆ Mα+β respectively for
α=0,1
a right- and left-A•-module.
Definition 2.5. TheZ -gradedvectorspaceofderivationsofA• isDer(A•) = (cid:76) Derα(A•)
2 α=0,1
where Derα(A•) is the vector space of linear maps X of homogeneous degree |X| = α defined
by Derα(A•) = {X : Aβ → Aα+β/X(ab) = X(a)b+(−1)αβaX(b),∀a ∈ Aβ,∀b ∈ A•}.
Most of the properties of the spaces of derivations in the non-graded case can be extended
to the graded situation. In the rest of this paper, we will consider the case where the graded
module is the graded algebra itself, i.e. M• = A•.
Proposition 2.7. Der(A•) is a graded right Z(A•)-module for the product (zX)(a) = z(X(a)),
a graded left Z(A•)-module for the product (Xz)(a) = (−1)|a||z|(X(a))z, ∀z ∈ Z(A•), ∀X ∈
Der(A•) of homogeneous degree and a Z -graded Lie algebra of the graded bracket [X,Y] =
2 •
XY −(−1)|X||Y|YX, ∀X,Y ∈ Der(A•). The Z -graded vector subspaces of inner derivations
2
is defined by Int•(A) = {Ad : b (cid:55)→ [a,b] /a ∈ A•} ⊂ Der•(A). It is a Z -graded Z(A•)-
a • 2
submodules and Lie subalgebra. The Z -graded vector subspace of outer derivations is defined as
2
Out(A•) = Der(A•)/Int(A•).
Derivations of the Moyal Algebra and Noncommutative Gauge Theories 7
Proof. The properties follow from straightforward calculations. (cid:4)
Definition 2.6. The space Ωn,α(A•), α ∈ {0,1}, n ∈ N denotes the vector space of n-linear
Der
maps ω with homogenous degree α, ω : (Der(A•))×n → A• defined by
i) Ω0,α(A•) = Aα, ∀α ∈ {0,1} and, ∀ω ∈ Ωn,α(A•), ∀X ∈ Der(A•) of homogeneous degree,
Der Der i
ii) ω(X1,...,Xn) ∈ Aα+Pnl=1|Xl|,
iii) ω(X ,...,X z) = ω(X ,...,X )z, ∀z ∈ Z(A•), where (Xz) is given in Proposition 2.7,
1 n 1 n
iv) ω(X ,X ,...,X ,X ,...,X ) = (−1)|Xi||Xi+1|ω(X ,X ,...,X ,X ,...,X ).
1 2 i i+1 n 1 2 i+1 i n
Proposition 2.8. Let us define Ω (A•) = (cid:76) Ωn,α(A•).
Der n∈N,α=0,1 Der
Then, the triplet (cid:0)Ω (A•),×,dˆ(cid:1) is a bigraded differential algebra with the product × on
Der
Ω (A•) and differential dˆ: Ωn,α(A•) → Ωn+1,α(A•) such that dˆ2 = 0 defined by:
Der Der Der
i) For any ω ∈ Ωp,α(A•), η ∈ Ωq,α(A•)
Der Der
ω×η(X ,X ,...,X )
1 2 p+q
= 1 (cid:88) (−1)sign(σ)(−1)Pk<l,σ(k)>σ(l)|Xk||Xl|+|η|Pσ(k)≤p|Xk|
p!q!
σ∈S(p+q)
×ω(X ,...,X )η(X ,...,X ).
σ(1) σ(p) σ(p+1) σ(p+q)
ii) The differential is defined, ∀ω ∈ Ωn,α(A•) by:
Der
dˆω(X ,X ,...,X )
1 2 n+1
n+1
= (cid:88)(−1)k+1(−1)|ω||Xk|(−1)Pnk−=11|Xk||Xn|Xkω(X1,X2,...,∨k,...,Xn+1)
k=1
+ (cid:88) (−1)k+l(−1)|Xk||Xl|(−1)Pkn−=11|Xk||Xn|
1≤k<l≤n+1
×(−1)Pln−=11|Xl||Xn|ω([Xk,Xl]•,...,∨k,···∨l,...,Xn+1),
where the symbol ∨ indicates that X is omitted.
i i
iii) A restricted differential calculus is obtained as follows. Let G• ⊂ Der(A•) denote a Z -
2
graded Lie subalgebra and Z(A•)-submodule. A restricted differential calculus based on G•
can be constructed by replacing in i) and ii) Der(A•) by G• and each Ωn,α(A•), α ∈ {0,1},
Der
n ∈ N by the vector space of n-linear maps ω with homogenous degree α, ω : (G•)×n → A•.
Proof. The proposition can be verified from direct calculation. (cid:4)
As in Section 2.2, we will not write explicitly the symbol × for the product of forms as the
product to be used in each case is obvious from the context.
We now define the connection on A• and its curvature as follows:
Definition 2.7. Let Ωn,• ≡ (cid:76) Ωn,α(A•), ∀n ∈ N.
Der α=0,1 Der
i) The NC connection on A• is defined as a linear map of homogeneous degree 0 ∇ : Ω0,• →
Der
Ω1,• such that
Der
∇(a) = dˆa+Aa, ∀a ∈ A•, A ∈ Ω1,0 (A•),
Der
where dˆis the differential defined in ii) of Proposition 2.8. A ∈ Ω1,0 (A•) is the 1-form
Der
connection.
8 J.-C. Wallet
ii) Let h be a Hermitian structure on A•. The NC connection on A• is Hermitian if it
satisfiesX(h(a ,a )) = h(∇ (a ),a )+(−1)|X||a1|h(a ,∇ (a )),∀a ∈ A• ofhomogenous
1 2 X 1 2 1 X 2 1
degree |a |, ∀a ∈ A•, and for any real derivation X in Der(A•) with homogeneous deg-
1 2
ree |X|.
iii) The curvature is defined as a linear map F : Ω0,α(A•) → Ω2,α(A•), such that
Der Der
F(a) = ∇(∇(a)) = (dˆa+AA)(a), ∀a ∈ A•.
Note that the connection defined here preserves the degree of the Z -graduation,
2
∇ : Ω0,α(A•) → Ω1,α(A•).
Der Der
Then, one has for any X ∈ Der•(A) of homogeneous degree |X|, |A(X)| = |∇ | = |X|. Notice
X
that we do not use a NC analogue of a graded connection. This latter appears to have inter-
esting relationship with the concept of superconnection introduced by Quillen [45]. This will be
reported elsewhere [46].
From Definition 2.7, one obtains the following relations that will be used in the sequel
Proposition 2.9. For any X,Y ∈ Der(A•), any z ∈ Z(A•) and any a ∈ A• of homogeneous
degree |X|, |Y|, |z| and |a|, one has
∇ (a) = X(a)+A(X)a; ∇ (ab) = ∇ (a)b+(−1)|X||a|aX(b), ∀b ∈ A•, (2.3)
X X X
∇ (a) = z∇ (a), ∇ (a) = (−1)|a||z|(∇ (a))z, (2.4)
zX X Xz X
F(X,Y)(a) = ([∇ ,∇ ] −∇ )(a), ∀a ∈ A•. (2.5)
X Y • [X,Y]•
Let † denote the involution on A•, (ab)† = (−1)|a||b|b†a† and let h (a ,a ) = a†a be the Her-
0 1 2 1 2
mitian structure on A•. A NC connection on A• as given by Definition 2.7 is Hermitian when
for any real derivation X of homogeneous degree, the 1-form connection A ∈ Ω1,0 (A•) satisfies
Der
A(X)† = −A(X).
Proof. The relations (2.3)–(2.5) can be simply obtained from standard calculation. Then,
assume that the Hermitian structure is given by h (a ,a ) = a†a and consider a Hermitian NC
0 1 2 1 2
connection ∇. For any real derivation in Der(A•) with homogeneous degree |X| one has, on one
hand, X(h (a ,a )) = X(a†)a +(−1)|X||a1|a X(a ) and on the other hand h (∇ (a ),a )+
0 1 2 1 2 1 2 0 X 1 2
(−1)|X||a1|h (a ,∇ (a )) = (X(a†)+(−1)|X||a1|a†A†(X))a +(−1)|X||a1|a†(X(a )+A(X)a ),
0 1 X 2 1 1 2 1 2 2
using Proposition 2.9 and the reality of X. Then, the relation given in Definition 2.7
X(h (a ,a ))A(X) = h (∇ (a ),a )+(−1)|X||a1|h (a ,∇ (a ))
0 1 2 0 X 1 2 0 1 X 2
is fulfilled provided A†(X) = −A(X). (cid:4)
The gauge transformations are defined in a way which preserves the degree of the NC con-
nection. In the following, we will focus on unitary gauge transformations that are compatible
with the Hermitian structure.
Definition 2.8. The gauge group is defined as Aut0(A•), the group of the automorphisms with
degree 0 of A•. Let aϕ ≡ ϕ(a) ∀ϕ ∈ Aut0(A•), ∀a ∈ A•. Let h be a Hermitian structure
on A•. The unitary gauge group is defined as the subgroup U(A•) ⊂ Aut0(A•) such that
U(A•) = {ϕ ∈ Aut0(A•)/h(aϕ,aϕ) = h(a ,a ), ∀a ,a ∈ A•}.
1 2 1 2 1 2
When h(a ,a ) = h (a ,a ) = a†a , it follows from this definition that the gauge group
1 2 0 1 2 1 2
U(A•) can be identified obviously with the group of unitary elements of A•.
Derivations of the Moyal Algebra and Noncommutative Gauge Theories 9
Proposition 2.10. For any NC connection ∇ as given in Definition 2.7 and for any ϕ ∈
Aut0(A•), the linear map of homogeneous degree 0 ∇ϕ : Ω0,• → Ω1,• such that ∇ϕ(a)(X) =
Der Der
(ϕ−1 ◦ ∇ ◦ ϕ)(a)(X), ∀a ∈ A•, ∀X ∈ Der(A•) defines a NC connection. The corresponding
1-form connection is defined by Aϕ = ϕ−1(I)dˆϕ(I)+ϕ−1(I)Aϕ(I). The gauge transformation of
the curvature is given by Fϕ(a) = (ϕ−1◦F ◦ϕ)(a), ∀a ∈ A•.
Proof. First, as for iii) of Proposition 2.5, one has aϕ = ϕ(a) = ϕ(I)a, ∀ϕ ∈ Aut0(A•). To
simplify the notations, we set g = ϕ(I). Then, from Definition 2.7, one obtains ϕ−1(∇(ϕ(a))) =
g−1∇(ga) = g−1dˆ(ga)+g−1Aga = dˆa+(g−1dˆg+g−1Ag)a. The expression for Aϕ follows. The
gauge transformation for the curvature can be obtained by a simple calculation. (cid:4)
Proposition 2.6 and Lemma 2.1 must be slightly modified to take into account the total
gradingN×Z . Infact,thiscanbecompactlyexpressedintermsofdifferentialforms. Notethat
2
the bracket on Ω (A•) is [ω ,η ] = ωη−(−1)pq(−1)|ω||η|ηω, ∀ω ∈ Ωp,|ω|(A•), ∀η ∈ Ωq,|η|(A•).
Der p q • Der Der
The following lemma holds
Lemma 2.2. Assume that ∃η ∈ Ω1,0 (A•)/dˆa = [η,a] = ηa − aη, ∀a ∈ A• where dˆ is the
Der •
differential given in Proposition 2.8. Then, one has the properties:
i) The map ∇inv : Ω0,α(A•) → Ω1,α(A•), α = 0,1 such that ∇inv(a) = dˆa − ηa = −aη,
Der Der
∀a ∈ A•, defines a connection in the sense of Definition 2.7. It is called the canonical
connection. ∇inv is gauge invariant: (∇inv)g(a) = ∇inv(a), ∀g ∈ U(A•).
ii) The curvature for the canonical connection is defined by the linear map Finv : Ω0,α(A•) →
Der
Ω2,α(A•) such that Finv(a) = (−dˆη+ηη)(a), ∀a ∈ A•.
Der
iii) For any NC connection ∇ in the sense of Definition 2.7, A ∈ Ω1,0 (A•) such that A(a) =
Der
(∇−∇inv)(a) = (A+η)(a), ∀a ∈ A• defines a tensor form. A(X), ∀X ∈ Der(A•) are
called the covariant coordinates. The curvature for any NC connection ∇ is defined by the
linear map F : Ω0,α(A•) → Ω2,α(A•) such that F(a) = Finv(a)+(AA+dˆA−[A,η] ))(a),
Der Der •
∀a ∈ A•.
iv) For any X,Y ∈ Der(A•), ∀a ∈ A• with respective homogeneous degree |X|, |Y|, |a|, the
following relations hold:
∇inv(a) = −(−1)|X||a|aη(X), ∇ (a) = A(X)a−(−1)|X||a|aη(X), (2.6)
X X
F(X,Y) = [A(X),A(Y)] −A([X,Y] )+(η([X,Y] )−[η(X),η(Y)] ), (2.7)
• • • •
Finv(X,Y) = −[η(X),η(Y)] +η([X,Y] ). (2.8)
• •
Proof. From Definition 2.7, any 1-form in Ω1,0 (A•) defines a connection. In particular, pick
Der
A = −η. The property i) follows. Then, ∀g ∈ U(A•), (∇inv)g(a) = g†∇inv(ga) = −g†(gaη) =
−aη = ∇inv(a). The corresponding curvature is determined by the 2-form F = dˆA + AA
for A = −η. So, Finv = −dˆη +ηη. The property for A in iii) is a direct consequence of the
definitionofatensorform. TheexpressionforF intermsofAisobtainedbysettingA = A−ηin
Definition2.7andusingtheexpressionforFinv. Finally,therelationsgiveniniv)areobtainedby
usingthemainpropertiesofthegradeddifferentialcalculusgiveninProposition2.8. Onehasin
particularAA(X,Y) = [A(X),A(Y)] ,[A,η] (X,Y) = [η(X),A(Y)] −(−1)|X||Y|[η(Y),A(X)] ,
• • • •
dˆA(X,Y) = [η(X),A(Y)] −(−1)|X||Y|[η(Y),A(X)] −A([X,Y] ), dˆη(X,Y) = 2[η(X),η(Y)] −
• • • •
η([X,Y] ). Then, the relations (2.6)–(2.8) follow. (cid:4)
•
We will now apply the above non-graded and graded differential calculi based on the deriva-
tions of associative algebras to two cases. The first case deals with a non-graded differential
calculus on the Moyal algebra M stemming from the maximal subalgebra of Der(M) whose
10 J.-C. Wallet
elements can be interpreted as infinitesimal symplectomorphisms. This gives rise to a natural
construction of Yang–Mills–Higgs type models defined on M. The second case deals with a Z -
2
graded differential calculus on a Z -graded associative algebra built from two copies of M. Due
2
to the grading, the gauge theory built from the square of the corresponding curvatures involves
as contributions the action (1.2) derived in [26] and [27] as well the renormalisable NC ϕ4-model
with harmonic term elaborated in [12, 13].
Fromnowon,theHermitianstructureish (a ,a ) = a†a . Accordingtotheabovediscussion,
0 1 2 1 2
the gauge group is the group of unitary elements of the associative algebra.
3 Application: gauge theories on the Moyal algebra
3.1 General properties of the Moyal algebra
In this subsection, we collect the properties of the Moyal algebra that will be used in the
sequel. For more details, see e.g. [7, 8]). Let S(RD) ≡ S and S(cid:48)(RD) ≡ S(cid:48), with D = 2n,
be respectively the space of complex-valued Schwartz functions on RD and the dual space of
tempered distributions on RD. The complex conjugation in S, a (cid:55)→ a†, ∀a ∈ S, defines a
natural involution in S that can be extended to S(cid:48) by duality and that will be used in the rest of
this paper. Let Θ be an invertible constant skew-symmetric matrix which can be written as
µν
Θ = θΣ where Σ is the “block-diagonal” matrix, Σ = diag(J,...,J) involving n (2×2) matrix J
(cid:18) (cid:19)
0 −1
given by J = and the parameter θ has mass dimension −2. We use the notation
1 0
yΘ−1z ≡ y Θ−1zν. The following proposition summarises properties relevant for the ensuing
µ µν
discussion:
Proposition 3.1. Let the (cid:63)-Moyal product be defined as (cid:63) : S ×S → S by
(cid:90)
1
(a(cid:63)b)(x) = dDydDz a(x+y)b(x+z)e−i2yΘ−1z, ∀a,b ∈ S. (3.1)
(πθ)D
Then, (S,(cid:63)) is a non unital associative involutive Fr´echet algebra with faithful trace given by
(cid:82) (cid:82) (cid:82)
dDx (a (cid:63) b)(x) = dDx (b (cid:63) a)(x) = dDxa(x) · b(x), where the symbol “·” is the usual
commutative product of functions in S.
The (cid:63) product (3.1) can be further extended to S(cid:48) ×S → S(cid:48) upon using duality of vector
spaces: (cid:104)T (cid:63)a,b(cid:105) = (cid:104)T,a(cid:63)b(cid:105), ∀T ∈ S(cid:48), ∀a,b ∈ S. In a similar way, (3.1) can be extended to
S ×S(cid:48) → S(cid:48), via (cid:104)a(cid:63)T,b(cid:105) = (cid:104)T,b(cid:63)a(cid:105), ∀T ∈ S(cid:48), ∀a,b ∈ S. Then, the Moyal algebra is defined
as [7, 8].
Definition3.1. LetM andM berespectivelydefinedbyM = {T ∈ S(cid:48)/a(cid:63)T ∈ S, ∀a ∈ S}
L R L
and M = {T ∈ S(cid:48)/T (cid:63)a ∈ S, ∀a ∈ S}. Then, the Moyal algebra M is defined by
R
M = M ∩M .
L R
Notice that M and M are sometimes called in the literature respectively the left and
L R
right multiplier algebras. By construction, S is a two-sided ideal of M. The essential structural
properties of the Moyal algebra that we will need are summarised in the following proposition:
Proposition 3.2. M,(cid:63) is a maximal unitisation of (S,(cid:63)). It is a locally convex associative
unital ∗-algebra. It involves the plane waves, the Dirac distribution and its derivatives and the
polynomial functions.