Maps for currents and anomalies in noncommutative gauge theories

PHYSICAL REVIEW D 71, 045013 (2005)

Maps for currents and anomalies in noncommutative gauge theories

Rabin Banerjee* and Kuldeep Kumar†

S. N. Bose National Centre for Basic Sciences, JD Block, Sector 3, Salt Lake, Kolkata 700098, India(Received 2 November 2004; published 24 February 2005)

*E-mail: rab†E-mail: ku

1550-7998=20

We derive maps relating currents and their divergences in non-Abelian U�N� noncommutative gaugetheory with the corresponding expressions in the ordinary (commutative) description. For the U(1) theory,in the slowly-varying-field approximation, these maps are also seen to connect the star-gauge-covariantanomaly in the noncommutative theory with the standard Adler-Bell-Jackiw anomaly in the commutativeversion. For arbitrary fields, derivative corrections to the maps are explicitly computed up to O��2�.

DOI: 10.1103/PhysRevD.71.045013 PACS numbers: 11.10.Nx, 11.15.–q

1This is in fact identical to the ordinary Adler-Bell-Jackiw

I. INTRODUCTION

The Seiberg-Witten (SW) map [1] ensures the stabilityof classical gauge transformations for theories defined onnoncommutative and usual (commutative) spacetime. Thefield redefinition contained in this map thus provides analternative method of studying noncommutative gaugetheories by recasting these in terms of their commutativeequivalents. Maps for the matter sector [2–5] as well as forcurrents and energy-momentum tensors [6] have also beenprovided.

An intriguing issue is the validity of such classical mapsat the quantum level. Studies in this direction [7–9] haveprincipally focused on extending the purported classicalequivalence of Chern-Simons theories (in 2� 1 dimen-sions) in different descriptions [10,11] to the quantumformulation.

In this paper, we provide an alternative approach tostudy these quantum aspects by relating the current-divergence anomalies in the noncommutative and commu-tative pictures through a SW-type map. Taking a cue froman earlier analysis involving one of us [6], we first derive amap connecting the star-gauge-covariant current in thenoncommutative gauge theory with the gauge-invariantcurrent in the �-expanded gauge theory, where � is thenoncommutativity parameter. From this relation, a map-ping between the (star-) covariant divergence of the cova-riant current and the ordinary divergence of the invariantcurrent in the two descriptions, respectively, is deduced.We find that ordinary current-conservation in the�-expanded theory implies covariant conservation in theoriginal noncommutative theory, and vice versa. The resultis true irrespective of the choice of the current to be vectoror axial vector. This is also to be expected on classicalconsiderations.

The issue is quite nontrivial for a quantum treatment dueto the occurrence of current-divergence anomalies for axial(chiral) currents. Since the star-gauge-covariant anomaly is

[email protected]@bose.res.in

05=71(4)=045013(12)$23.00 045013

known [12,13] and the gauge-invariant anomaly in the�-expanded theory is also known,1 it is possible to testthe map by inserting these expressions. We find that theclassical map does not hold in general. However, if weconfine to a slowly-varying-field approximation,2 thenthere is a remarkable set of simplifications and the classicalmap holds. We also give a modified map, that includes thederivative corrections, which is valid for arbitrary fieldconfigurations.

The paper is organized as follows. After briefly summa-rizing the standard SW map in Sec. II, the map for currentsand their divergences is derived in Sec. III. Here the treat-ment is for the non-Abelian gauge group U�N�. In Sec. IV,we discuss the map for anomalous currents and theirdivergences. The Abelian U(1) theory is considered andresults are given up to O��2�. As already mentioned, themap for the axial anomalies (in two and four dimensions)holds in the slowly-varying-field limit. A possible schemeis discussed in Sec. V, whereby further higher-order resultsare confirmed. Especially, O��3� computations are done insome detail. Our concluding remarks are given in Sec. VIwhere we also briefly discuss the implications of thisanalysis on the definition of effective actions.

II. A BRIEF REVIEW OF THESEIBERG-WITTEN MAP

Let us begin by briefly reviewing the salient features ofthe SW map. The ordinary Yang-Mills action is given by

SYM � �1

4

Zd4x Tr�F��F

��; (1)

where the non-Abelian field strength is defined as

F�� @�A� � @�A� � i�A�; A�� (2)

(ABJ) anomaly [14].2This approximation is also used in Ref. [1] to show the

equivalence of Dirac-Born-Infeld (DBI) actions in the twodescriptions.

-1 2005 The American Physical Society

A lower gauge index is equivalent to a raised one—whether agauge index appears as a superscript or as a subscript is a matterof notational convenience.

RABIN BANERJEE AND KULDEEP KUMAR PHYSICAL REVIEW D 71, 045013 (2005)

in terms of the Hermitian U�N� gauge fields A��x�. Thenoncommutativity of spacetime is characterized by thealgebra

�x ; x��? � x ? x� � x� ? x � i� �; (3)

where the noncommutativity parameter � � is real andantisymmetric. The star product of two fields A�x� andB�x� is defined as

�A ? B��x� � exp�i

2� �@ @0�

�A�x�B�x0�

��x0�x; (4)

where @0� � @@x0�

. In noncommutative spacetime, the usualmultiplication of functions is replaced by the star product.The Yang-Mills theory is generalized to

SYM � �1

4

Zd4x Tr�F�� ? F

�� (5)

with the noncommutative field strength

F�� @�A� � @�A� � i�A�; A��?: (6)

This theory reduces to the conventional U�N� Yang-Millstheory for �! 0.

To first order in �, it is possible to relate the variables inthe noncommutative spacetime with those in the usual oneby the classical maps [1]

A� � A� �1

4� �fA ; @�A� � F��g �O��2�; (7)

F�� F�� 1

4� ��2fF� ; F��g

� fA ;D�F�� @�F��g� �O��2�; (8)

where f ; g appearing on the right-hand sides standsfor the anticommutator and D� denotes the covariant de-rivative defined as D�� @�� i��; A��. A further mapamong gauge parameters,

� � ��1

4� �f@ �; A�g �O��2�; (9)

ensures the stability of gauge transformations

��A� � @�� i��; A��? � D� ? �; (10)

��A� � @�� i��; A�� D��: (11)

That is, if two ordinary gauge fields A� and A0� are equiva-

lent by an ordinary gauge transformation, then the corre-sponding noncommutative gauge fields, A� and A0

�, willalso be gauge-equivalent by a noncommutative gaugetransformation. It may be noted that the map (8) is aconsequence of the map (7) following from the definition(6) of the noncommutative field strength. The fieldstrengths F�� and F�� transform covariantly under the

045013

usual and the star-gauge transformations, respectively:

��F�� i��;F��; ��F�� i��; F��?: (12)

The gauge fields A��x� may be expanded in terms of theLie-algebra generators Ta of U�N� as A��x� � Aa��x�Ta.These generators satisfy

�Ta; Tb� � ifabcTc; fTa; Tbg � dabcTc;

Tr�TaTb� � �ab:(13)

We shall take the structure functions fabc and dabc to be,respectively, totally antisymmetric and totally symmetric.The Yang-Mills action (1) can now be rewritten as3

SYM � �1

4

Zd4x Fa��F

��a ; (14)

where

Fa�� @�Aa� � @�Aa� � fabcAb�Ac�: (15)

In view of relations (13), the maps (7)–(9) can also bewritten as

Ac� � Ac� �1

4� �dabcAa �@�Ab� � Fb�� O��2�; (16)

Fc�� Fc�� 1

2� �dabc

�Fa� Fb�� Aa @�Fb��

�1

2fbdeAa A

e�F

d��

��O��2�; (17)

�c � �c �1

4� �dabc@ �aAb� �O��2�; (18)

and the gauge transformations (10)–(12) as

��Aa� � @��a � fabcAb��c; (19)

��Fa�� fabcFb��c; (20)

��Aa� � @��

a �i

2dabc��b; Ac��? �

1

2fabcf�b; Ac�g?

� @��a � fabcAb��

c �1

2� �dabc@ A

b�@��

c

�O��2�; (21)

��Fa��

i

2dabc��b; Fc��? �

1

2fabcf�b; Fc��g?

� fabcFb��c �

1

2� �dabc@ F

b��@��

c �O��2�:

(22)3

-2

MAPS FOR CURRENTS AND ANOMALIES IN . . . PHYSICAL REVIEW D 71, 045013 (2005)

III. THE MAP FOR NON-ABELIAN CURRENTS:CLASSICAL ASPECTS

In order to discuss noncommutative gauge theories withsources, it is essential to have a map for the sources also, sothat a complete transition between noncommutative gaugetheories and the usual ones is possible. Such a map was firstbriefly discussed in Ref. [6] for the Abelian case. Weconsider the non-Abelian case in this section.

Let the noncommutative action be defined as

S�A; � � SYM�A� � SM� ; A�; (23)

where are the charged matter fields. The equation ofmotion for Aa� is4

�SYM�Aa�

� D� ? F��a � �J�a ; (24)

where

J�a ��SM�Aa�

�� : (25)

Equation (24) shows that J�a transforms covariantly underthe star-gauge transformation:

��J� � �i�J�; ��?;

��J�a � fabcJ�b �

c �1

2� �dabc@ J

�b @��

c �O��2�:(26)

Also, it satisfies the noncommutative covariant conserva-tion law

D� ? J�a � 0; (27)

which may be seen from Eq. (24) by taking the noncom-mutative covariant divergence.

The use of SW map in the action (23) gives its�-expanded version in commutative space:

S�A; � ! S��A; � � S�YM�A� � S�M� ; A�; (28)

where S�YM�A� contains all terms involving Aa� only, and isgiven by

S�YM � �1

4

Zd4x

�Fa��F

��a � � �dabcF��a

�Fb� F

c��

�1

4Fb� F

c��

��O��2�

�; (29)

and we have dropped a boundary term in order to express itsolely in terms of the field strength. The equation of motionfollowing from the action (28) is

4We mention that the noncommutative gauge field A� is ingeneral an element of the enveloping algebra of the gauge group.Only for specific cases, as for instance the considered case ofU�N� gauge symmetry, it is Lie-algebra valued.

045013

�S�YM�Aa�

� �J�a ; (30)

where

J�a ��S�M�Aa�

�� : (31)

Expectedly, from these relations, it follows that J�a trans-forms covariantly,

��J� � �i�J�; ��; ��J�a � fabcJ�b �

c; (32)

and satisfies the covariant conservation law

D�J�a � 0: (33)

Now the application of SW map on the right-hand sideof Eq. (25) yields the relation between J�a and J�a :

J�a �x� �Zd4y

��S�M�Ac��y�

�� Ac��y�

�Aa��x��

�S�M� c �y�

��A� c �y�

�Aa��x�

�

�Zd4y J�c �y�

�Ac��y�

�Aa��x�; (34)

where the second term obtained in the first step has beendropped on using the equation of motion for a .

We consider Eq. (34) as a closed form for the mapamong the sources. To get its explicit structure, the map(16) among the gauge potentials is necessary. Since themap (16) is a classical result, the map for the sourcesobtained in this way is also classical.

Let us next obtain the explicit form of this map up to firstorder in �. Using the map (16) and its inverse,

Ac� � Ac� �1

4� �dabcAa �@�A

b� � Fb�� O��2�; (35)

we can compute the functional derivative

�Ac��y�

�Aa��x�� ac��x� y� �

1

4� �� 2dabcAb �y�

� @y��x� y� � dedcfbadAe �y�Ab��y��x� y��

�1

4� �fdabcAb �y�@

y��x� y� � �dabc@y Ab��y�

� dabcFb ��y� � dedcfdabAe �y�Ab��y��x� y�g

�O��2�; (36)

where @y� stands for @@y�

. Putting this in Eq. (34), we get

J�a � J�a �1

2� �

�dabc@��Ab J

�c � �

1

2dedcfbadAe Ab�J

�c

�

�1

2� �

�dabcFb �J

�c �

1

2�dcadfdbe

� dbcdfdae�Ab Ae�J�c �1

2dabdAb @�J�d

��O��2�:

(37)

Since D�J�a � @�J�a � fabcJ�bAc�, we can use Eq. (33) to

-3


substitute@�J

�d � fdceJ�cA

e� (38)

in the last term on the right-hand side of Eq. (37) to obtain

J�a � J�a �1

2� �

�dabc@��A

b J

�c � �

1

2dedcfbadAe A

b�J

�c

�

�1

2� �dabcFb �J�c �O��2�; (39)

where we have used the identity

dabdfdce � dbcdfdae � dcadfdbe � 0: (40)

As a simple yet nontrivial consistency check, we show thestability of the map under gauge transformations. Underthe ordinary gauge transformations given by Eqs. (19) and(20), and using the covariant transformation law (32) forJ�a , the right-hand side of Eq. (39) transforms as

��J�a � fabcJ�b �

c �1

2� �

�dabc@�J

�c @ �b

� dcdbfbea@��Ad J�c �e� �

1

2�decbfbda

� dcdbfbea�Ad J�c @��

e �1

2dgcd�fabefedh

� fdaefebh�Ag Ab�J�c �h

�

�1

2� �dcdbfbaeFd �J�c�e �O��2�; (41)

where we have used the relation (40). On the other hand,using the maps (18) and (39), and the identity

fabefedh � fbdefeah � fdaefebh � 0; (42)

the right-hand side of the second relation in Eq. (26)reproduces the right-hand side of Eq. (41). Hence,

��J�a � ��J

�a ; (43)

thereby proving the stability of the map (39) under thegauge transformations. This statement is equivalent to theusual notion of stability which ensures that the star-gauge-transformed noncommutative current is mapped to theusual-gauge-transformed ordinary current, as may be veri-fied by performing a Taylor expansion of the right-handside of J�a �J; A� � ��J

�a �J; A� � J�a �J� ��J; A� ��A�

and comparing both sides.5

5Exactly the same thing happens when discussing the stabilityof the map (7) for the potentials.

045013

It is worthwhile to mention that the use of Eq. (38) inobtaining the map (39) is crucial to get the correct trans-formation property of J�a . This is because issues of gaugecovariance and covariant conservation are not independent.In an ordinary Abelian gauge theory, for example, currentconservation and gauge invariance are related. Likewise, inthe non-Abelian case, covariant conservation and gaugecovariance are related. This intertwining property is apeculiarity of the mapping among the sources and is notto be found in the mapping among the potentials or the fieldstrengths.

From these results, it is possible to give a map for thecovariant derivatives of the currents. Recall that

D� ? J�a � @�J

�a �

i

2dabc�J�b ; A

c��? �

1

2fabcfJ�b ; A

c�g?

� @�J�a � fabcAb�J

�c �

1

2� �dabc@ A

b�@�J

�c

�O��2�; (44)

which, using the maps (16) and (39), gives

D� ? J�a � D�J

�a �

1

2� �

�dabc@��Ab D�J

�c �

�1

2dedcfbadAe A

b�D�J

�c

��O��2�; (45)

where we have used the Jacobi identities (40) and (42), andthe relation (38). Thus we see that covariant conservationof the ordinary current, D�J

�a � 0, implies that J�a given

by Eq. (39) indeed satisfies the noncommutative covariantconservation law, D� ? J

�a � 0. This is also to be expected

from classical notions.At this point, an intriguing issue arises. Is it possible to

use Eq. (45) to relate the anomalies in the different de-scriptions? Indeed the analysis presented for the vectorcurrent can be readily taken over for the chiral current.Classically everything would be fine since the relevantcurrents are both conserved. At the quantum level, how-ever, the chiral currents are not conserved. We would liketo ascertain whether the relation (45) is still valid bysubstituting the relevant chiral anomalies in place ofD� ? J

�a and D�J

�a . Since the main aspects get highlighted

for the Abelian theory itself, we confine to this case, andpresent a detailed analysis in the remainder of this paper.

IV. THE ABELIAN CASE: CLASSICAL ANDQUANTUM ASPECTS

Some discussion on the use of the map (45) (in theAbelian case) for relating anomalies [up to O��] wasearlier given in Ref. [6]. In order to gain a deeper under-standing, it is essential to consider higher orders in �.Keeping this in mind, we present a calculation up toO��2� for two- and four-dimensional theories.

-4

6This is essential to ensure the stability of map (53) underappropriate gauge transformations. A similar manipulation wasneeded for getting the non-Abelian expression (39).


The maps to the second-order in � in the Abelian caseare given by [15]

A� � A� �1

2� �A �@�A� � F��

�1

6� �� !A �@��A @!A� � 2A F!��

� F� �@!A� � 2F!�� O��3�; (46)

F�� F�� A @�F�� F� F��

�1

2� �� !�A @��A @!F�� 2F� F!��

� F� �A @!F�� 2F� F!�� O��3�; (47)

� � ��1

2� �A @��

1

6� �� !A �@��A @!��

� F� @!�� O��3�; (48)

which ensure the stability of gauge transformations

��A� � D� ? � � @�� i��; A��?

� @�� @ A�@��O��3�; (49)

��A� � @��: (50)

Analogous to the non-Abelian theory, the map for currentsis consistent with the requirements that while the currentJ� is gauge-invariant and satisfies the ordinary conserva-tion law, @�J� � 0, the current J� is star-gauge-covariantand satisfies the noncommutative covariant conservationlaw, D� ? J

� � 0. Now the currents J� and J� are relatedby the Abelian version of Eq. (34) [6],

J��x� �Zd4y J��y�

�A��y�

�A��x�; (51)

which, using the map (46) and its inverse,

A� � A� �1

2� �A �@�A� � F��

�1

6� �� !A

�1

2@��A @!A� � A F!��

�1

2F� �@!A� � 5F!��

3

2�2A @�F!�

� @�A @!A� � @�A F!��O��3�; (52)

yields the explicit O��2� form of the source map:

045013

J� � J� � � ��A @�J� �

1

2F �J�

�� F �J�

�1

2� �� !@

�A F�!J

� � A�A @!J�

�1

2A�F !J�

�� @ �A�F �J�� O��3�;

(53)

where we have used @�J� � 0 to simplify the integrand.6

The above map, up to O��, was earlier given in Ref. [6].Now let us check explicitly the stability under the gaugetransformations. Under the ordinary gauge-transformation,��A� � @��, ��F�� 0, and ��J� � 0. Hence theright-hand side of Eq. (53) transforms as

��J� � � �@ J�@�� @ �F �J��@��

�1

2� �� !�2@�@!�A J��@ �

� @��A @!��@ J�� O��3�: (54)

On the other hand,

��J� � i��; J��? � � �@ J

�@��O��3�: (55)

Next, using the maps (48) and (53) in the above equation,one finds that the right-hand side of Eq. (54) is reproduced.Hence,

��J� � ��J

�; (56)

thereby proving the gauge-equivalence, as observed ear-lier. Furthermore, using the maps (46) and (53), the cova-riant divergence of J�,

D� ? J� � @�J

� � i�J�; A��?

� @�J� � � �@ J

�@�A� �O��3�; (57)

can be expressed as

D� ? J� � @�J

� � � �@ �A�@�J��

�1

2� �� !@ �A F�!@�J�

� A�@!�A @�J�� O��3�; (58)

where each term on the right-hand side involves @�J�, so

-5

8Contrary to the four-dimensional example, the map holds for


that the covariant conservation of J� follows from theordinary conservation of J�. This is the Abelian analogueof Eq. (45), but valid up to O��2�.

We are now in a position to discuss the mapping ofanomalies. Since the maps have been obtained for thegauge currents, the anomalies refer to chiral anomaliesfound in chiral gauge theories. Moreover, we implicitlyassume a regularization which preserves vector-currentconservation so that the chiral anomaly @�� "��

1�"5

2 � �is proportional to the usual ABJ anomaly @�J

�5 [16]. The

first step is to realize that the standard ABJ anomaly[17,18] is not modified in �-expanded gauge theory [14].In other words,

A � @�J�5 �

1

16#2 "��%F��F�% (59)

still holds. The star-gauge-covariant anomaly is just givenby a standard deformation of the above result [12,13]:

A � D� ? J�5 �

1

16#2 "��%F�� ? F�%: (60)

The expected map for anomalies, obtained by a lift fromthe classical result (58), follows as

A � A� � �@ �A�A� �1

2� �� !@ �A F�!A

� A�@!�A A�� O��3�: (61)

Let us digress a bit on this map. The starting point is theclassical map (53) with the vector current replaced by theaxial one. Although current conservation is used to derivethe map (53), the analysis still remains valid since the axialcurrent is also classically conserved. Also, as discussedearlier, the retention of the term proportional to the diver-gence of the current would spoil the stability of the gaugetransformations, which must hold irrespective of whetherthe current is vector or axial. From the map (53) one is ledto the relation (58). Now we would like to see whether thisclassical map persists even at the quantum level, written inthe form (61).7 As far as gauge-transformation propertiesare concerned, it is obviously compatible since the anoma-lies in the different descriptions transform exactly as thecorresponding currents. Corrections, if any, would thusentail only gauge-invariant terms, involving the field tensorF��. We now prove that the relation (61) is indeed valid forthe slowly-varying-field approximation, which was alsoessential for demonstrating the equivalence of DBI actions[1]. Later on we shall compute the corrections that appearfor arbitrary field configurations. In the slowly-varying-field approximation, since derivatives on F�� can beignored, the star product in Eq. (60) is dropped. Usingthe map (47), we write this expression as

7See also the discussion in the last paragraph of Sec. III.

045013

A �1

16#2 "��%F��F�%

�1

16#2 "��%

�F��F�% � � �fA�@ �F��F�%�

� 2F��F� F�%g�� !

�1

2A @��A @!�F

��F�%��

�1

2A F� @!�F��F�%� � 2A @��F��F� F!%�

� 2F��F� F� F!% � F� F��F� F!%

��O��3�

�:

(62)

Next, using the identities [11]

"��%� ��F��F�%F � � 4F��F� F�%� � 0; (63)

"��%� �� !�F� F�

�F� F!%

� 2F��F� F� F!% �1

2F��F� F!%F �

�1

4F��F�%F F!�� 0; (64)

and the usual Bianchi identity, we can write down

A �1

16#2 "��%�F��F�% � � �@ �A�F

��F�%�

�1

2� �� !@ fA F�!F

��F�%

� A�@!�A F��F�%�g �O��3��: (65)

The identities (63) and (64) are valid in four dimensionsand, in fact, hold not only for just F�� but for any anti-symmetric tensor, in particular, for F�� also. This gives adefinite way for obtaining the identity (64) starting from(63). The identity (64) may be obtained from the identity(63) by doing the replacement F�� ! F�� followed byusing the map (47) and retaining O��2� terms.Alternatively, one can check it by explicitly carrying outall the summations. Now substituting for the anomaly (59)on the right-hand side of Eq. (65), we indeed get back ourexpected anomaly map (61).

It is easy to show that the map (61) is equally valid intwo dimensions,8 in which case,

arbitrary fields. This is because the anomaly does not involve any(star) product of fields and hence the slowly-varying-field ap-proximation becomes redundant.

-6


A2d � @�J�5 �

1

2#"��F��;

A2d � D� ? J�5 �

1

2#"��F

��:(66)

It follows from the map (47) for the field strength that

A2d �1

2#"��F

��

�1

2#"��

�F�� A @�F�� F� F�

��

�1

2� �� !fA @��A @!F

�� A F� @!F��

� 2A @��F� F!

�� 2F� F� F!�g �O��3�

�:

(67)

In two dimensions, we have the identities

"�� F �F�� 2F� F�� 0; (68)

"�� !�F F!�F�� F �F� F!�

� 4F� F� F!�� 0; (69)

which are the analogue of the identities (63) and (64).Likewise, these identities hold for any antisymmetricsecond-rank tensor, and the second identity can be ob-tained from the first by replacing the usual field strengthby the noncommutative field strength and then using theSW map. Using these identities, Eq. (67) can be rewrittenas

A2d �1

2#"��

�F�� @ �A�F��

�1

2� �� !@ fA F�!F

�� A�@!�A F��g

�O��3��; (70)

which, substituting for the usual anomaly on the right-handside, reproduces the map (61) with A and A replaced byA2d and A2d, respectively.

For arbitrary fields, the derivative corrections to the mapin the four-dimensional case are next computed. Now thenoncommutative anomaly takes the form

045013

A �1

16#2 "��%F�� ? F�%

�1

16#2 "��%

�F��F�% � � �@ �A�F��F�%�

�1

2� �� !@ fA F�!F

��F�%

� A�@!�A F��F�%�g��

1

128#2 "��%� �� !

� @ @ F��@�@!F�% �O��3�: (71)

The last term is the new piece added to Eq. (65). Thus, themap (61) gets modified as

A � A� � �@ �A�A� �1

2� �� !@ �A F�!A

� A�@!�A A�� 1

128#2 "��%� �� !

� @ �@ F��@�@!F�%� �O��3�: (72)

This is reproduced by including a derivative correction tothe classical map (53) for currents:

J�5 � J�5 � � ��A @�J

�5 �

1

2F �J

�5

�� F �J

�5

�1

2� �� !@

�A F�!J

�5 � A�A @!J

�5

�1

2A�F !J

�5

�� @ �A�F �J�5 �

�1

128#2 "!��%� �� @ F

!�@ @�F�% �O��3�:

(73)

The correction term is given at the end. It is straightforwardto see the contribution of this derivative term. Since this isan O��2� term and we are restricting ourselves to thesecond order itself, taking its noncommutative covariantderivative amounts to just taking its ordinary partial de-rivative. Then taking into account the antisymmetric natureof � � it immediately yields the corresponding term inEq. (72). We therefore interpret this term as a quantumcorrection for correctly mapping anomalies for arbitraryfields.

It is to be noted that Eq. (72) can be put in a form so thatthe �-dependent terms are all expressed as a total deriva-tive. This implies

Zd4x D� ? J

�5 �

Zd4x @�J

�5 ; (74)

reproducing the familiar equivalence of the integratedanomalies [11–13,19].

We shall now give some useful inverse maps. From maps(52) and (53), the inverse map for the currents follows:

-7

PHYSICAL REVIEW D 71, 045013 (2005)

J� � J� � � ��A @�J

� �1

2F �J

�� F �J

�

�1

2� �� !

�A @�F! J

� � A A @�@!J�

� 2A @�A @!J� �

1

2A F �@!J

�

�3

2A @��F !J

�� 1

2F F!�J

� �1

4F �F !J

��

� � �� @ �A�F �J�� O��3�: (75)

Taking the ordinary derivative and doing some simplifica-tions yields

@�J� � D� ? J� � � �@ �A��D� ? J

��

�1

2� �� !@ @ �A�A!�D� ? J

�� O��3�;

(76)

which may be regarded as the inverse map of (58). Indeed,use of this relation reduces the expression on the right-handside of Eq. (58) to that on its left-hand side which shows theconsistency of the results. This also proves that the cova-riant conservation of J� implies the ordinary conservationof J�, as expected.

Likewise, inverting the relation (47), we obtain

F�� F�� A @�F�� F� F��

� � �� !�A @�A @!F��

1

2A A @�@!F��

� A @��F� F!�� F� F� F!�

��O��3�: (77)

If we now write down the usual anomaly as

RABIN BANERJEE AND KULDEEP KUMAR

045013

1

16#2 "��%F��F�% �

1

16#2 "��%

�F�� ? F�%

�1

8� �� !@ @ F

��

� @�@!F�% �O��3�

�; (78)

and use Eq. (77) on the right-hand side, we get

1

16#2 "��%F��F�% �

1

16#2 "��%

�F�� ? F�% � � �

� @ fA��F�� ? F�%�g �

1

2� �� !

� @ @ fA�A!�F�� ? F�%�g

�

�1

128#2 "��%� �� !@ @ F

��

� @�@!F�% �O��3�; (79)

where we have used the identities (63) and (64) with thereplacement F�� ! F��. Thus we have the map for theanomalies:

@�J�5 � D� ? J

�5 � � �@ �A��D� ? J

�5 ��

�1

2� �� !@ @ �A�A!�D� ? J

�5 ��

�1

128#2 "��%� �� !@ @ F

��@�@!F�%

�O��3�: (80)

In the slowly-varying-field approximation, the last termdrops out. Then it mimics the usual map (76). Again, asbefore, it is possible to find the correction term for arbitraryfields and write down the map for anomalous current as

J�5 � J�5 � � ��A @�J

�5 �

1

2F �J

�5

�� F �J

�5 �

1

2� �� !

�A @�F! J

�5 � A A @�@!J

�5 � 2A @�A @!J

�5

�1

2A F �@!J

�5 �

3

2A @��F !J

�5 � �

1

2F F!�J

�5 �

1

4F �F !J

�5

�� @ �A�F �J

�5�

�1

128#2 "!��%� �� @ F

!�@ @�F�% �O��3�; (81)

which reproduces Eq. (80) correctly. Substituting this map,the expression on the right-hand side of Eq. (73) reduces tothat on its left-hand side, which shows the consistency ofthe results.

We conclude this section by providing a mapping be-tween modified chiral currents which are anomaly-free butno longer gauge-invariant. In the ordinary (commutative)theory, such a modified chiral current may be defined as

J� � J�5 �1

8#2 "��%A�F�%: (82)

By construction, this is anomaly-free (@�J� � 0) but nolonger gauge-invariant. It is possible to do a similar thingfor the noncommutative theory. We rewrite Eq. (73) byreplacing J�5 in favor of J�. The terms independent of J�,including the quantum correction, are then moved to theother side and a new current is defined as

J� � J�5 � X��A�; (83)

where all A�-dependent terms lumped in X� have beenexpressed in terms of the noncommutative variables using

-8


the SW map. Thus we have

J� � J� � � ��A @�J

� �1

2F �J

�� F �J

�

�1

2� �� !@

�A F�!J� � A�A @!J�

�1

2A�F !J

�� @ �A�F �J

��

�O��3�: (84)

Since the above equation is structurally identical toEq. (53), a relation akin to (58) follows:

D� ? J� � @�J� � � �@ �A�@�J��

�1

2� �� !@ �A F�!@�J

�

� A�@!�A @�J�� O��3�; (85)

which shows that @�J� � 0 implies D� ? J� � 0. Weare thus successful in constructing an anomaly-free currentwhich however does not transform (star-) covariantly. It isthe X�, appearing in Eq. (83), which spoils the covarianceof J�.

V. HIGHER-ORDER COMPUTATIONS

Results in the previous section were valid up toO��2�. Anatural question that arises is the validity of these resultsfor further higher-order corrections. Here we face a prob-lem. The point is that although the map (51) for sources isgiven in a closed form, its explicit structure is dictated bythe map involving the potentials. Thus one has to firstconstruct the latter map before proceeding. All these fea-tures make higher- [than O��2�] order computations veryformidable, if not practically impossible. An alternateapproach is suggested, which is explicitly demonstratedby considering O��3� calculations.

Consider first the two-dimensional example. The star-gauge-covariant anomaly, after an application of the SWmap, is given by

A2d �1

2#"��F

��

� A�0�2d �A�1�

2d �A�2�2d �A�3�

2d �O��4�; (86)

with A�0�2d , A�1�

2d and A�2�2d respectively being the zeroth-,

first- and second-order (in �) parts already appearing on the

045013

right-hand side of Eq. (70), and

A�3�2d � �

1

12#"�� !�'(�A @�fA @!�A'@(F��

� 3F�'F(�� 2F!'�A @(F�� 3F� F(

��g

� A F� @!�A'@(F�� 3F�'F(��

� 2F� F!'�A @(F�� 3F� F(

��; (87)

where the O��3� contribution to the map (47) has beentaken from Ref. [15].

Now our objective is to rewrite theO��3� contribution ina form akin toO�� andO��2� terms, namely, to recast it assomething proportional to the commutative anomaly("��F��), and also as a total derivative. Expressing it asa total derivative is necessary to preserve the equality of theintegrated anomalies (

Rd2x "��F

�� Rd2x "��F��)

[5,6,11–13].The O��3� contribution may be expressed as

A�3�2d � �

1

12#"��

�� !�'(�A @�

�A @!

�A'@(F

��

�3

2F'(F��

�� 2A F!'@(F��

3

4�F !F'(

� 2F 'F(!�F�� A F�

�@!

�A'@(F��

�3

2F'(F

�� 2F!'@(F

��

�F 'F( F!�

�1

8F �F !F'( �

3

4F �F 'F(!

�F��

�; (88)

where, in addition to the identities (68) and (69), we havealso used

"�� !�'(�F��F 'F( F!� � F� F!

�F 'F(�

�F� F!'F(�F � � 6F� F� F!'F(

�� 0;

(89)

which follows from the identity (69) by doing the replace-ment F�� ! F�� followed by exploiting the SW map andretaining O��3� terms. We notice that each term on theright-hand side of Eq. (88) contains the usual anomaly, asdesired. After some algebra, the right-hand side of Eq. (88)can be written as a total divergence, which gives us the finalimproved version of the map (70) as

A2d �1

2#"��

�F�� @ �A�F��

1

2� �� !@ fA F�!F�� A�@!�A F��g

�1

6� �� !�'(@

�F��

�2A'F( F!� � 2A�A @!F'( �

3

2A�F 'F(! �

1

4A�F !F'( � A�@!�A'F( �

�1

2A F!�F'(

�� @(F��A�A �@!A' � 2F!'� � A'�A F�! � A�F !�� A�A A'@!@(F��

��O��4�

�: (90)

-9


Thus, in two dimensions, the noncommutative anomaly can be written in terms of the usual anomaly at O��3� also:

A2d � A2d � � �@ �A�A2d� �1

2� �� !@ fA F�!A2d � A�@!�A A2d�g �

1

6� �� !�'(@

�A2d

�2A'F( F!�

� 2A�A @!F'( �3

2A�F 'F(! �

1

4A�F !F'( � A�@!�A'F( � �

1

2A F!�F'(

�� @(A2d�A�A �@!A' � 2F!'�

� A'�A F�! � A�F !�� A�A A'@!@(A2d

��O��4�: (91)

If the anomalies in four dimensions also satisfy the above map, then clearly we have a general result, valid up to O��3�.Now it will be shown that, in the slowly-varying-field approximation, such a relation indeed holds. We have

1

16#2 "��%F��F�% �

1

16#2 "��%

�F��F�% � � �@ �A�F��F�%� �

1

2� �� !@ fA F�!F��F�% � A�@!�A F��F�%�g

�1

6� �� !�'(@

�F��F�%

�2A'F( F!� � 2A�A @!F'( �

3

2A�F 'F(! �

1

4A�F !F'(

� A�@!�A'F( � �1

2A F!�F'(

�� @(�F��F�%��A�A �@!A' � 2F!'� � A'�A F�! � A�F !��

� A�A A'@!@(�F��F�%�

��O��4�

�: (92)

In obtaining this equation, it is necessary to use the identities (63) and (64), and a new one (given below), which followsfrom the identity (64) by doing the replacement F�� ! F�� followed by using the SW map and retaining O��3� terms:

"��%� �� !�'(�6F� F�

�F� F!'F(% � 6F��F� F� F!'F(

% � F��F� F!%F 'F(� � F��F� F!'F(%F �

�1

2F�'F(

�F� F!%F � �

1

2F��F�%F 'F( F!�

�� 0:

(93)

Obviously, Eq. (92) reproduces the map (91), with A2d

and A2d replaced by the corresponding expressions in fourdimensions. This proves our claim.

Starting from the results in two dimensions, it is thusfeasible to infer the general structure valid in higher di-mensions. This is an outcome of the topological propertiesof anomalies. Proceeding in this fashion, the map for theanomalies can be extended to higher orders.

VI. DISCUSSIONS

We have provided a SW-type map relating the sources inthe noncommutative and commutative descriptions. In thenon-Abelian theory, the classical maps for the currents andtheir covariant divergences were given up to O��. Forinvestigating quantum aspects of the mapping, we appliedit to the divergence anomalies for the Abelian theory in thetwo descriptions. For the slowly-varying-field approxima-tion, the anomalies indeed got identified. Thus the classicalmap correctly accounted for the quantum effects inherentin the calculations of the anomalies. The results were

045013

checked up to O��2�. We also provided an indirect methodof extending the calculations and found an agreement up toO��3�. Our analysis strongly suggests that the classicalmapping would hold for all orders in �, albeit in theslowly-varying-field approximation. Our findings mayalso be compared with Refs. [8,9] where the classicalequivalence of the Chern-Simons theories in different de-scriptions was found to persist even in the quantum case.

For arbitrary field configurations, derivative correctionsto the classical source map were explicitly computed up toO��2�. Indeed, it is known that if one has to go beyond theslowly-varying-field approximation, derivative correctionsare essential. For instance, DBI actions with derivativecorrections have been discussed [20–22]. A possible ex-tension of this analysis would be to study the mapping ofconformal (trace) anomalies.

To put our results in a proper perspective, let us recallthat the SW maps are classical maps. A priori, therefore, itwas not clear whether they had any role in the mapping ofanomalies which are essentially of quantum origin. Thefirst hint that such a possibility might exist came from

-10


Eq. (58) [or Eq. (61)] where the covariant derivative of thenoncommutative covariant current was expressed in termsof the ordinary derivative of the commutative current.Indeed, to put the map in this form was quite nontrivial.Classically, such a map was trivially consistent, since boththe covariant divergence in the noncommutative descrip-tion and the ordinary divergence in the usual (commuta-tive) picture vanish. The remarkable feature, however, wasthat such a map remained valid even for the quantum casein the slowly-varying-field approximation which waschecked explicitly by inserting the familiar anomalies9 inthe different descriptions. Incidentally, the slowly-varying-field approximation is quite significant in discussions of theSW maps. For instance, it was in this approximation thatthe equivalence of the DBI actions in the noncommutativeand the commutative pictures was established [1] throughthe use of SW maps.

Our analysis has certain implications for the mappingamong the effective actions (for chiral theories) obtainedby integrating out the matter degrees of freedom. The pointis that the anomalies are the gauge-variations of the effec-tive actions and if the anomalies get mapped then oneexpects that, modulo local counterterms, the effective ac-tions might get identified, i.e., it suggests that

W�A�A�� W�A� � local counterterms; (94)

where W and W denote the effective actions in the com-mutative and noncommutative formulations, respectively.Taking the gauge-variations (with parameters � and �),yields

Zd4x �D� ? J

�5 � ? � �

Zd4x �@�J

�5 ��

Zd4x �@��

��;

(95)

where

J�5 ��W

�A�; J�5 �

�W�A�

(96)

10

and �� accounts for the ambiguity (local counterterms) inobtaining the effective actions. Now Eq. (71) expresses thenoncommutative anomaly in terms of the commutativevariables. Using that result and the SW map (48) for thegauge parameter � simplifies the left-hand side of Eq. (95):

9The planar anomaly for the noncommutative description andthe ABJ anomaly for the commutative case.

045013

Zd4x �D� ? J

�5 � ? � �

Zd4x �D� ? J

�5 ��

�1

16#2 "��%Zd4x �F�� ? F�%��

�1

16#2 "��%Zd4x �F��F�%��

�Zd4x �@ �

��; (97)

where Eq. (71) and the map (48) have been used in the laststep, and

� �1

16#2 "��%

�1

2� �A�F

��F�% � � �� !

�

�1

3A F�!F��F�% �

1

6A�@ �A!F��F�%�

�1

8@ F

��@�@!F�%��; (98)

thereby proving Eq. (95) and establishing the claim (94).We further stress, to avoid any confusion, that the rela-

tion (94) was not assumed, either explicitly or implicitly, inour calculations.10 Rather, as shown here, our analysissuggested such a relation. Its explicit verification confirmsthe consistency of our approach. It should be mentionedthat the map among anomalies (61) follows from the map(53) for currents through a series of algebraic manipula-tions. This does not depend on the interpretation of theanomaly as gauge-variation of an effective action. If onesticks to this interpretation and furthermore assumes therelation (94), then it might be possible to get a relation [likeEq. (95)] involving the integrated version of the products ofanomalies and gauge parameters. Our formulation alwaysled to maps involving unintegrated anomalies or currents,which are more fundamental.

We also note that the map (61) for the unintegratedanomalies, which follows from the basic map (53) amongthe currents, was only valid in the slowly-varying-fieldapproximation. The suggested map (94) among the effec-tive actions, on the other hand, led to the map (95), involv-ing the integrated anomalies and the gauge parameters, thatwas valid in general. For the pure integrated anomalies wehave the familiar map (74) that has been discussed exten-sively in the literature [11–13,19].

ACKNOWLEDGMENTS

K. K. thanks the Council of Scientific and IndustrialResearch (CSIR), Government of India for financialsupport.

Indeed, as already stated, there cannot be any a priori basisfor such an assumption since the classical SW map need not bevalid for mapping effective actions that take into account loopeffects.

-11


[1] N. Seiberg and E. Witten, J. High Energy Phys. 09 (1999)032.

[2] A. A. Bichl, J. M. Grimstrup, L. Popp, M. Schweda, and R.Wulkenhaar, hep-th/0102103.

[3] V. O. Rivelles, Phys. Lett. B 558, 191 (2003).[4] H. S. Yang, hep-th/0402002.[5] R. Banerjee and H. S. Yang, Nucl. Phys. B708, 434 (2005).[6] R. Banerjee, C. Lee, and H. S. Yang, Phys. Rev. D 70,

065015 (2004).[7] B. Jurco, P. Schupp, and J. Wess, Nucl. Phys. B584, 784

(2000).[8] K. Kaminsky, Y. Okawa, and H. Ooguri, Nucl. Phys.

B663, 33 (2003).[9] K. Kaminsky, Nucl. Phys. B679, 189 (2004).

[10] N. Grandi and G. A. Silva, Phys. Lett. B 507, 345 (2001).[11] R. Banerjee, Int. J. Mod. Phys. A 19, 613 (2004).

045013

[12] F. Ardalan and N. Sadooghi, Int. J. Mod. Phys. A 16, 3151(2001).

[13] J. M. Gracia-Bondia and C. P. Martin, Phys. Lett. B 479,321 (2000).

[14] F. Brandt, C. P. Martin, and F. Ruiz Ruiz, J. High EnergyPhys. 07 (2003) 068.

[15] S. Fidanza, J. High Energy Phys. 06 (2002) 016.[16] H. Banerjee and R. Banerjee, Phys. Lett. B 174, 313

(1986).[17] S. L. Adler, Phys. Rev. 177, 2426 (1969).[18] J. S. Bell and R. Jackiw, Nuovo Cimento A 60, 47 (1969).[19] R. Banerjee and S. Ghosh, Phys. Lett. B 533, 162 (2002).[20] N. Wyllard, Nucl. Phys. B598, 247 (2001).[21] N. Wyllard, J. High Energy Phys. 08 (2001) 027.[22] S. R. Das, S. Mukhi, and N. V. Suryanarayana, J. High

Energy Phys. 08 (2001) 039.

-12

Documents

Maps for currents and anomalies in noncommutative gauge theories