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arXiv:hep-th/0409248v4 8 Nov 2005
A new approach to the analysis of a noncommutative
Chern–Simons theory
Pradip Mukherjee∗†and Anirban Saha
Department of Physics, Presidency College
86/1 College Street, Kolkata - 700 073, India
February 1, 2008
Abstract
A novel approach to the analysis of a noncommutative Chern–Simons gauge theory
with matter coupled in the adjoint representation has been discussed. The analysis is
based on a recently proposed closed form Seiberg–Witten map which is exact in the
noncommutative parameter.
PAC codes: 11.10.Nx, 11.15-q
Keywords: Noncommutativity, Chern–Simons gauge field, Energy-momentum tensor,
Solitons
The idea of fuzzy space time where the coordinates xµsatisfy the noncommutative
(NC) algebra
[xµ,xν] = iθµν
(1)
where θµνis a constant anti-symmetric tensor, was mooted long ago [1]. This idea has
been revived in the recent past and field theories defined over this NC space are currently
the subject of very intense research [2]. One approach of analysis of the NC field theories is
to work in a certain Hilbert space which carries a representation of the basic NC algebra.
The fields are defined in this Hilbert space by the Weyl–Wigner correspondence. The
operator approach is easily extended to the abelian and nonabelian gauge groups [3].
An alternative approach of treating NC theories is to work in the deformed phase space
where the ordinary product is replaced by the star product. In this formalism the fields
are defined as functions of the phase space variables with the product of two fieldsˆφ(x)
andˆψ(x) given by the star product
ˆφ(x) ⋆ˆψ(x) =
?ˆφ ⋆ˆψ
?
(x) = e
i
2θαβ∂α∂′
β ˆφ(x)ˆψ(x
′)|x′=x.
(2)
∗pradip@bose.res.in
†Also Visiting Associate, S. N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake City,
Calcutta -700 098, India
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An important breakthrough in the investigation of the NC gauge theories has been
achieved by Seiberg and Witten [4] from their study of the Dirac–Born–Infeld (DBI) ac-
tion of open string dynamics on a D-Brane obtained in the limit of slowly varying fields
[5]. It was observed that depending on the regularization scheme one can have alter-
native descriptions of the theory in terms of commutative and noncommutative models.
Since physics must be independent of the particular regularization scheme a space time
redefinition between the ordinary and noncommutative gauge fields is indicated.
Prompted by Seiberg and Witten’s seminal work a new approach towards the study of
NC gauge theories has originated in the literature both for the abelian [6] and non-abelian
[7] gauge groups where the NC gauge theories have been analysed from their commutative
equivalent counterparts. Various NC models have been analysed in the recent past from
this point of view [8]. The essence of this approach is to formulate the theory on the
phase space and expand the star products (2) with appropriate Seiberg–Witten maps
implemented individually on the fields of the model. Consequently, the outcome is in the
form of a perturbative expansion in the noncommutative parameter. It will be very much
desirable if this analysis can be done in a closed form such that results exact to all orders
in θ are obtained. Naturally, the possibility of this rests on the availabilibility of the SW
maps in a closed form.
Recently, a method of obtaining SW maps for certain models has been devised which
is exact in the NC parameter [9, 10]. This is based on the change of variables between
open and closed string parameters and connection of the approach with the deformation
quantization technique [11] has been demonstrated [10]. Specifically, an exact map for an
adjoint scalar field has been found [10], consistent with that deduced from RR couplings
of unstable non-BPS D-branes [12]. In the present letter we will use this map to analyze
a U(1)⋆Chern–Simons (C–S) coupled scalar field theory in 2 + 1 dimensional flat space
time where the scalar field is in the adjoint representation of the gauge group. Models
with the NC scalar field in the adjoint representation have been considered earlier from
the operator approach with the gauge field dynamics governed solely by the Maxwell term
[13] and also by a combination of the Maxwell and the C–S term [14]. Our selection of the
model is motivated by the fact that in the commutative limit the scalar field decouples
from the gauge interaction. In other words any non-trivial result of our analysis comes
from the NC features only. For the same reason we chose the C–S coupling because it
remains form-invariant under SW map [15]. Apart from this the C–S theories have been
studied both in the commutative [16] and noncommutative settings [17] principally due
to their inherent interest in connection with the theory of fractional spin and statistics.
The action of our theory is given by
ˆS
=
?
d3x
?1
2
?ˆDµ⋆ˆφ
?
⋆
?ˆDµ⋆ˆφ
?
+k
2ǫµνλ
?
ˆAµ⋆ ∂νˆAλ−2i
3
ˆAµ⋆ˆAν⋆ˆAλ
??
(3)
whereˆφ is the scalar field andˆAµis the NC C–S gauge field. We adopt the Minkowski
metric ηµν= diag(+,−,−,−). The covariant derivativeˆDµ⋆ˆφ is defined as
ˆDµ⋆ˆφ = ∂µˆφ − i
The action (3) is invariant under the ⋆-gauge transformation
?ˆAµ,φ
?
⋆
(4)
ˆδˆλˆAµ=ˆDµ⋆ˆλ,
ˆδˆλˆφ = −i
?ˆφ,ˆλ
?
⋆
(5)
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The commutative version of (3) can easily be obtained by using the exact SW map for
ˆDµ⋆ˆφ(x) given in [10] and noting that the C–S action retains its form under SW map.
Proceeding in this direction we write the commutative equivalent of (3) as
ˆS
SW map
=
?
d3x
?1
2
?
det(1 + Fθ)
?
1
1 + Fθ
1
1 + θF
?µν
∂µφ∂νφ +k
2ǫµνλAµ∂νAλ
?
(6)
In (6) we have used the matrix notation
(AB)µν= AµλBλν
(7)
Also (1 + Fθ) is to be interpreted as a mixed tensor in calculating the determinant. Note
that the quartic term in the C–S action vanishes in the commutative equivalent version.
The scalar field part of the action (6) can be written as an ordinary scalar field theory
coupled with a gravitational field induced by the dynamical gauge field. However, the
dynamics of the gauge field, being dictated by the Chern–Simons three-form, is unaffected
by the induced gravity. If we would instead consider Maxwell theory then the coupling
should equally affect the gauge field dynamics also [10].
From (6) we readily observe that in the commutative limit (θµν→ 0) the gauge field
decouples, leading to the well known fact that there is no non-trivial gauge coupling of
the neutral scalar field in the corresponding commutative field theory. The commutative
equivalent to the transformations (5) are
δλAµ= ∂µλ,δλφ = 0 (8)
Clearly, the action (6) is manifestly invariant under (8).
It is now straightforward to write down the equations of motion for the scalar field φ
and the gauge field Aµfrom (6) respectively as
∂α
??
det(1 + Fθ)
?
1
1 + Fθ
1
1 + θF
?αν
∂νφ
?
= 0 (9)
and
kǫανλ∂νAλ= jα
(10)
where, in (10),
jα
=
∂ξ
??
?
1 + Fθ
det(1 + Fθ)
?
1
4
?
θ
1
1 + Fθ+
1
1 + θFθ
?αξ?
1
1 + Fθ
1
1 + Fθ
1
1 + θF
?µν
+
1
1
1 + θFθ
?µα?
1
1 + θF
?ξν
+
?
?µα?
θ
1
1 + Fθ
1
1 + θF
?ξν?
∂µφ∂νφ
?
(11)
Certain observations about the above equations are in order. In the commutative limit
or (and) vanishing gauge field
?
1
1 + Fθ
1
1 + θF
?µν
→ ηµν
(12)
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Thus the equation of motion for φ in (9) reduces to the expected form ∂µ∂µφ = 0 in
these limits. Again going to the commutative limit we find that the gauge field equation
becomes trivial, which is also a characteristic feature of C–S theories without any matter
coupling. By direct computation, we get from (11)
∂αjα= 0 (13)
This exhibits the consistency of (10). Naturally, jαis interpreted as the matter current.
At this point it can be noted that the usual approach of obtaining the commutative
equivalent of (3) is to expand the star products and use separate maps for the gauge fields
and matter fields in the form of perturbative expansions in the NC parameter θ [8]. To
the lowest order in θ the explicit forms of the SW maps are known as [4, 6]
ˆψ
=
ψ − θmjAm∂jψ
Ai−1
ˆAi
=
2θmjAm(∂jAi+ Fji) (14)
Using these expressions and the star product (2) to order θ in (3) we get
ˆS
SW map
=
?
d3x
??1
2∂µφ∂µφ − θαβFµα∂βφ∂µφ − θαβAα∂µ∂βφ∂µφ
?
+k
2ǫµνλAµ∂νAλ
?
(15)
We can show explicitly that the first order approximation of (6) matches exactly with
(15). Naturally, the equations of motion (9) and (10) should agree upto the first order
with those following from the conventional first order action (15). Expanding (9) and (10)
to first order in θ we get
∂α[{1 + Tr(Fθ)}∂αφ − (Fθ + θF)αν∂νφ] = 0(16)
and
kǫανλ∂νAλ
=
∂ξ[1
2θαξ∂µφ∂µφ + θµα∂µφ∂ξφ + θξµ∂αφ∂µφ] (17)
respectively. One can verify easily that the same equations follow as Euler–Lagrange
equations from (15).
We now turn to the construction of an energy momentum (EM) tensor of our model
(6). The issue of energy momentum tensor for a noncommutative gauge theory involves
many subtle points as evidenced in the literature [18]. It is thus instructive to address
the question from different approaches, which in the context of commutative models are
known to lead to equivalent conclusions but the same is not true apriori for NC gauge
theories. Indeed, the commutative equivalent model offers an appropriate platform to
discuss these aspects.
We begin with the construction of the Noether EM tensor. Consider the infinitesimal
space time translation xµ→ (xµ+ aµ) under which the fields φ and Aµtransform as
δφ = aµ∂µφ, δAµ= aν∂νAµ
(18)
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From the invariance of the theory we get the following form of the EM tensor in the usual
way,
Θc
ρσ
=
?
det(1 + Fθ)
??
1
1 + Fθ
1
1 + θF
?α
?
?
ρ
ν∂νφ∂σφ
+
?1
?
?
4
?
θ
1
1 + Fθ+
1
1 + θFθ
?µα?
1
1 + Fθ
1
1 + θF
ρ
1
1 + Fθ
?
ρ
?
ρ
?
1
1 + θF
?µν
+
1
1 + Fθ
1
1 + θFθ
1
1 + θF
ν
+
1
1 + Fθ
?µα?
1
1 + Fθ
θ
1
1 + θF
?µν
ν
?
(∂µφ∂νφ)(∂σAα)
−1
−k
2ηρσ
?
∂µφ∂νφ
2(ǫρµαAµ∂σAα+ ηρσǫµναAµ∂νAα) (19)
The Noether E–M tensor is useful to construct the generators of space time transfor-
mations. However, it is neither gauge invariant nor symmetric. One would then like to
improve it to get a gauge invariant EM tensor using Belinfante’s method. A better alter-
native is to considr a subsequent gauge transformation with the spatial translation (18)
so that the gauge field transform covariantly,
δAµ= aνFνµ
(20)
and obtain an improved EM tensor by Noether’s method [19] using the modified trans-
formation. This leads to
Tρσ
=
∂L
∂ (∂ρφ)∂σφ +
∂L
∂ (∂ρAα)Fσα− ηρσL
??
1 + Fθ
1
1 + θFθ
1
1 + θFθ
=
?
?1
?
?
det(1 + Fθ)
1
1
1 + θF
?α
1
1 + θF
?
ρ
1
ν∂νφ∂σφ
+
4
?
θ
1
1 + Fθ+
ρ
?
1 + Fθ
?
ρ
?
ρ
?
1
1 + θF
?µν
+
1
1 + Fθ
?µα?
1
1 + Fθ
1
1 + θF
ν
+
1
1 + Fθ
?µα?
1
1 + Fθ
θ
1
1 + θF
?µν
ν
?
(∂µφ∂νφ)(Fσα)
−1
−k
2ηρσ
?
∂µφ∂νφ
2(ǫρµαAµFσα+ ηρσǫµναAµ∂νAα) (21)
Apart from the contribution from the C–S part this expression is gauge invariant. How-
ever, it is not symmetric. In the commutative theories this part of the improved EM tensor
becomes simultaniously symmetric. The exception in the context of NC gauge theories
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has already been mentioned and is due to the fact that Lorentz and classical conformal
invariance are broken in such theories [18].
We have observed that the canonical procedures do not lead to a satisfactory EM
tensor. An alternative procedure is to vary the action (6) with respect to a background
metric and finally keeping the metric flat. We thus extend the action (6) as
S =
?
d3x√−gL (22)
where g = detgµνand gµνis the background metric. The pure C–S part of (6) is generally
covariant irrespective of any metric. Thus the Lagrangean L in (22) is taken to be the
Lagrangean of (6) without the C–S kinetic term. The EM tensor is obtained from
Θ(s)
αβ= 2∂L
∂gαβ− Lgαβ
(23)
in the limit gµν→ ηµν. Explicitly
Θ(s)
αβ
=
1
2
?
?
det(1 + Fθ)
?
1
2
?
θF
1
1 + θF+
1
1 + FθFθ
?
1
αβ
?
?
1
1 + Fθ
1
1 + θF
?µν
∂µφ∂νφ
+
1
1 + Fθ
1
1 + θF
?
α
ν∂βφ∂νφ +
?
1
1 + Fθ
1 + θF
β
ν∂αφ∂νφ
?
− Lηαβ
(24)
Note that by construction this EM tensor is both symmetric and gauge invariant. We can
conclude that of the various expressions given above this form is the most satisfactory
and can be identified as the physical EM tensor.
The equations (9) and (10) are a set of coupled nonlinear equations. It will thus be
instructive to investigate whether they admit any solitary wave solution. This can be seen
in a systemetic way by looking for the Bogomolnyi bounds of the equations. To this end
we require the energy functional which can appropriately be constructed from the physical
EM tensor (24). Note that until now our approach was completely general. Specifically,
we did not assume vanishing time-space noncommutativity i.e. θ0i= 0. The issue of non
zero time space noncommutativity is an involved subject in the literature. It has been
argued that noncommutativity in this sector spoils unitarity [20] and causality [21] but
there also exists counter examples [22]. However, assuming θ0i= 0 is almost conventional
in the study of NC solitons and in the context of odd dimensional theories it is always
possible to do so. A la this tradition we now assume that the noncommutativity exists
only in the spatial direction. Going over to this limit the energy functional becomes
E
=
?
−
d2xΘ(s)
00=
?
d2x1
2
?
det(1 + Fθ)
?
2
?
1
1 + Fθ
1
1 + θF
?
0
ν(∂0φ∂νφ)
?
1
1 + Fθ
1
1 + θF
?µν
(∂µφ∂νφ)
?
(25)
With the stated assumptions about NC tensor θµνthe form of the matrices appearing in
the above equation can be easily worked out. Explicitly, the matrix for
?
1
1+Fθ
1
1+θF
?µν
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can be written as
?
1 −
θ2(E2
(1−θB)2
θE2
(1−θB)2
−θE1
(1−θB)2
1+E2
2)
?
θE2
(1−θB)2
−1
(1−θB)2
0
−θE1
(1−θB)2
0
−1
(1−θB)2
(26)
Also det(1 + Fθ) = (1 − θB)2. While extracting the square root of this determinant one
has to take positive value only. So for θB < 1,
θB > 1 it is to be replaced by (θB − 1). The critical point θB = 1 is known to be a general
feature of the NC models, the origin of which can be traced back to noncommutativity in
planar quantum mechanics [23].
The static limit of the energy functional will now be worked out. First, we observe
from (11) that for θ0i = 0, j0vanishes in the static limit. This leads to vanishing B-
field, as can be seen from (10), making the coupling trivial. The expression of the energy
functional (25) becomes,
?
The energy functional is positive definite and trivially minimized. Clearly, there is no
non-trivial solutions. We thus observe that there is no BPS soliton of the model. Note
that nontrivial soliton solutions has been found in NC adjoint scalar field theories [13, 14]
with Maxwell coupling. However, these soliton solutions become singular in the θ ?→ 0
limit. Since our approach has a smooth commutative limit, based as it is on the SW map,
such singular solutions (if any) are not included in our model.
We have discussed a novel approach of analysing a Chern–Simons (C–S) coupled real
scalar field theory. A commutative equivalent of the model is obtained which is exact
in the noncommutative (NC) parameter θ. This is based on a recently proposed exact
Seiberg–Witten (SW) map [10] which does not require explicit expansion of ⋆-product. At
this point the approach is markedly different from the usual analysis of NC gauge theories
using SW fields where one uses the maps in the form of series expansions in θ [6, 7] along
with an expansion of the ⋆-product of the functions. Equations of motion satisfied by the
dynamical fields have been written down without any restriction on the noncommutative
structure. The resulting matter current has been shown to be conserved by explicit
calculation which again provides a consistency check of our equations of motion. We have
also demonstrated that upto first order in the NC parameters our commutative equivalent
action is mapped into the usual version. Different forms of the energy momentum (EM)
tensor have been worked out. It was observed that a satisfectory EM tensor can not be
obtained from the canonical prescriptions. A symmetric and gauge invariant EM tensor
is constructed by varying the action with respect to the a background metric and this has
been identified as the physical EM tensor for our model. Specializing this NC tensor by
neglecting noncommutativity in the time–space direction we have shown that the model
does not have any nontrivial BPS soliton. Note that this only indicates the absence of
such solutions in the sector which has a smooth commutative limit and any other singular
soliton solution is not ruled out. The NC Maxwell term can be straightforwardly added
in our model which will also be useful in the context of comparision with known results.
Also the exact commutative equivalent approach illustrated here may be extended to
?det(1 + Fθ) = (1 − θB) whereas for
E =
d2x
?
(∂1φ)2+ (∂2φ)2?
(27)
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Page 8
scalar fields in the fundamental representation if the corresponding exact SW map can be
devised. This and other related issues will be taken up subsequently.
Acknowledgment
PM likes to thank the Director, S. N. Bose National Centre for Basic Sciences for the
award of visiting associateship. AS wants to thank the Council of Scientific and Industrial
Research (CSIR), Govt. of India, for financial support and the Director, S. N. Bose
National Centre for Basic Sciences, for providing computer facilities.
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