Dimensional reduction, Seiberg-Witten map, and supersymmetry

E. Ulas Saka1,* and Kayhan Ulker2,†

1Physics Department, Istanbul University, TR-34134 Vezneciler, Istanbul, Turkey2Feza Gursey Institute, P.O. Box 6, TR-34684 Cengelkoy, Istanbul, Turkey

(Received 26 January 2007; published 17 April 2007)

It is argued that dimensional reduction of the Seiberg-Witten map for a gauge field induces Seiberg-Witten maps for the other noncommutative fields of a gauge invariant theory. We demonstrate thisobservation by dimensionally reducing the noncommutative N � 1 super Yang-Mills (SYM) theory in6 dimensions to obtain noncommutative N � 2 SYM in 4 dimensions. We explicitly derive Seiberg-Witten maps of the component fields in 6 and 4 dimensions. Moreover, we give a general method to definethe deformed supersymmetry transformations that leave the actions invariant after performing Seiberg-Witten maps.

DOI: 10.1103/PhysRevD.75.085009 PACS numbers: 11.10.Nx, 11.15.�q, 11.30.Pb, 12.60.Jv

I. INTRODUCTION

Quantum field theories on noncommutative (NC) space-times recently attracted much attention mainly due to theirrelation to string theory [1], although the idea is quite old[2]. However, this relation to string theory leads to theinteresting result that certain NC gauge theories can bemapped to commutative ones [1]. This map is commonlycalled as the Seiberg-Witten (SW) map.

The generalization of the SW map to supersymmetricNC gauge theories was considered in Refs. [3,4] by usingsuperfields in canonically deformed superspace.1 The ap-proach of [3] is to generalize the equations that lead to SWmaps to superfields in canonically deformed superspace.Though these equations can be solved directly they arecumbersome. Indeed, their solutions are not presented in[3]. On the other hand, the solution given in [4] is nonlocaland does not yield the original solution of Seiberg andWitten.

For the component field formalism of supersymmetrictheories, SW maps of the component fields of the Abeliantheory were given in [6] by using approaches given in [3,4].For NC super Yang-Mills (SYM) theory these maps werealready found in [7] by using a completely different ap-proach. In both [6,7] it is assumed that the NC componentfields that are superpartners of a NC gauge field function-ally depend on their ordinary counterparts and on theordinary gauge field.

As noted in [7], when one expands the NC action up tofirst order in the deformation parameter � after performingthe SW map, the resulting action is not invariant underclassical supersymmetry transformations. This fact sug-gests that supersymmetry (SUSY) transformations should

also be deformed when these transformations are written interms of ordinary fields after the SW map. Such deforma-tions are studied in [6,8] for Abelian NC gauge theories.

However, note also that by relaxing the assumption thatthe NC component fields are just functionals of theirordinary counterparts and gauge fields, in [9] SW mapsare studied for Abelian NC supersymmetric gauge theoryby solving the respective equations directly for superfieldsin order that the yielding action is invariant under classicalSUSY transformations.

On the other hand, use of extra dimensions in a trivialway is a fruitful method to construct theories with largersymmetries. One of the classical examples is to constructSYM theories with extended SUSY in four dimensionsfrom higher dimensional N � 1 SYM theories by usingdimensional reduction [10]. Therefore, it is natural to ask ifdimensional reduction of gauge theories can also shedsome light on the aforementioned approaches.

In this paper, we first note that dimensional reduction ofthe SW map for a NC gauge field AM gives directly the SWmap of NC scalar fields2 � in the lower dimensions bychoosing the deformation parameter in a suitable way. Thisobservation leads to the general form of the equationswhose solutions are SW maps for the NC fields of a gaugeinvariant theory. As a direct consequence, one can general-ize this result to the component fields of SYM theories. Theassumption used in Refs. [6,7] that the NC componentfields depend on their ordinary counterparts and on theordinary gauge field, i.e. � � ��;A�, arises naturally byusing the dimensional reduction of the original SW map.

After performing these SW maps to the componentfields, the resulting actions are not invariant under the(ordinary) SUSY transformations. In order to have super-symmetric actions after performing the SW maps it is clearthat NC SUSY transformations � should also be deformedafter the SW map. Following a similar approach given in

*Email address: ulassaka@istanbul.edu.tr†Email address: kulker@gursey.gov.tr1Here, canonically deformed superspace means x-x deforma-

tion. For more general discussion of deformation of superspaceincluding x-� and �-� deformations where � are the Grassmanncoordinates of the superspace; see for instance [5].

2An SW map of an adjoint scalar field via dimensionalreduction is studied also in [11].

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[6] for NC supersymmetric Abelian gauge theory, wepresent a general method to define deformed SUSY trans-formations � such that

� ���!SW map� � �0 � �1

where, �0 is the ordinary SUSY transformations and �1 isthe deformed part of SUSY that depends on the deforma-tion parameter �. These transformations are consistentwith the respective SW maps of the component fields andleave the actions invariant that are found after performingthe Seiberg-Witten maps.

In order to demonstrate the aforementioned ideas andmethods, we study the dimensional reduction of NCN � 1SYM theory in 6 dimensional NC Minkowski space whichis deformed with the help of a constant deformation pa-rameter �, to obtain NCN � 2 SYM in 4 dimensions. Ourresults are presented up to first order in �.

This paper is organized as follows: In Sec. II, we studythe dimensional reduction of NC N � 1 SYM theory in sixdimensions to four dimensions. As expected, we show thatthe resulting theory is NC N � 2 SYM theory in fourdimensions. In Sec. II, we also give the NC N � 2 (on-shell) SUSY transformations of the NC component fieldsthat leave the NC N � 2 SYM action invariant in fourdimensions.

In Sec. III, we study the relation between the dimen-sional reduction procedure and SW maps. We constructexplicitly the SW maps for component fields of super-multiplets in 6 and 4 dimensions up to first order in thedeformation parameter.

In Sec. IV, we write the 6 dimensional NC N � 1 SYMand 4 dimensional NC N � 2 SYM actions in terms ofordinary fields by using SW maps. We then give a generalmethod to define deformed supersymmetry transforma-tions that leave the actions invariant after performing theSeiberg-Witten maps. We construct explicitly the N � 1deformed SUSY transformations in six dimensions and theN � 2 ones in four dimensions.

Our conclusions are presented in Sec. V.

II. DIMENSIONAL REDUCTION OF NC SYMTHEORY

The simplest noncommutative (NC) space that is exten-sively studied in the literature is the deformation ofD-dimensional Minkowski or Euclidean space RD:

�xM; xN� � i�MN

with the help of a real constant antisymmetric parameter�. This NC space is characterized by the Moyal � product:

f�x� � g�x� � f�x�g�x� �i2

�MN@Mf�x�@Ng�x� �O��2�:

The action of NC N � 1 SYM in six dimensions can bewritten by replacing the ordinary product with the Moyal� product,3:

S 6 � trZd6x

��

1

4FMNF

MN �i2

���MDM��; (1)

where FMN is the field strength of the NC gauge field AMand � is a Weyl spinor that belongs to the same (on-shell)supermultiplet with AM in six dimensions.

The action (1) is invariant under NC SUSY transforma-tions

�AM � �i2� ���M�� ���M��; � � � �MN�FMN;

� �� � � ���MNFMN; (2)

where � and �� are the constant parameters of N � 1 SUSYin 6 dimensions.

Dimensional reduction of NC SYM can be obtained in asimilar way as it is done for the ordinary case [10]. For thispurpose, we let the space-time to be decomposed as xM ��x�; xi� such that the coordinates xi are the compactifiedones. We let any function f�x� be a function of onlyuncompactified coordinates x�, i.e. @if�x� � 0. Then, thedimensional reduction of the D-dimensional NC space canbe performed by choosing the deformation parameter � as

�MN ���� 0

0 0

� �; (3)

so that the lower dimensional space-time still has the samecanonical deformation

�x�; x�� � i���; �xi; xj� � 0:

This choice of the deformation parameter � leads triviallyto the Moyal product in four dimensions:

f�x� � g�x� � f�x�g�x� �i2���@�f�x�@�g�x� �O��2�:

Note that one could also choose some other form for �.This choice would also lead to the same Moyal � product infour dimensions, since any function is considered to beindependent of the compactified coordinates xi. However,as it will be clear in the next section, to be able to deriveconsistent SW maps via dimensional reduction the choice(3) is mandatory.

After performing the compactification, the action of NCN � 2 SYM in four dimensions can be obtained as4

3Our conventions and some useful formula are given inAppendix A.

4Our definitions of lower dimensional fields via dimensionalreduction are given in Appendix B.

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S4 � trZd4x

��

1

4F��F

�� � i���D���� i ��D�

�

� D��D��y � i

���2p� ��; �y�� � ��� � ; ����

�1

2��; �y�2�

�(4)

The action (4) is invariant under the (on-shell) NC N � 2SUSY transformations that can also be obtained from (2)by dimensional reduction:

�A� � i1�� ��� i2�� � � i �1 ����� i �2 ��� ;

�� � ���1F�� � i1��; �y�� � i

���2p�� �2D��;

� � ���2F�� � i2��; �y�� � i

���2p�� �1D��;

�� ����2p1 �

���2p2�;

(5)

where 1, 2 are the constant parameters of N � 2 super-symmetry. Note that the above action (4) and supersym-metry transformations (5) are the same with the wellknown ones in the undeformed space after replacing theordinary product with the � product as expected.

III. SEIBERG-WITTEN MAP VIA DIMENSIONALREDUCTION

Based on the fact that one can derive both conventionaland noncommutative gauge theories from the same twodimensional field theories by using different regularizationprocedures, Seiberg and Witten [1] showed that there existsa map from a commutative gauge field to a noncommuta-tive one that exhibits the equivalence between the twotheories. This map is commonly called as Seiberg-Wittenmap (SW map) and arises from the requirement that gaugeinvariance should be preserved in the following sense:

A�A� � �g�A�A� � A�A� �g�A�; (6)

where �g�, �g�

are gauge transformations with infinitesimal

parameters � and �, respectively, such that � �

��AM;��. The solutions of Eq. (6) up to first order in �are found to be [1],

A M�A� � AM �14�

KLfAK; @LAM � FLMg �O��2�; (7)

��A;�� � �� 14�

KLf@K�; ALg �O��2�: (8)

The SW map of noncommutative supersymmetric gaugetheories is studied in various ways in order to get theSeiberg-Witten map for the other fields that are in thesame supermultiplet with a gauge field. However, dimen-sional reduction procedure gives directly the desired SWmaps of the component fields in a supermultiplet.

In order to get these maps, note that when one dimen-sionally reduces the gauge field AM, the components on the

compactified dimensions Ai behave as scalar fields.Therefore, the dimensional reduction of the SW map (6)gives the original map

A ��A� � A� �14��fA; @�A� � F��g �O��2� (9)

and also the SW map of the scalar fields5 � in the lowerdimensional space-time that are in the adjoint representa-tion of the gauge group6

� � �� 14��fA; �@� �D���g �O��2� (10)

by choosing the noncommutativity parameter to have theform given in (3).

Here, the choice (3) for the deformation parameter �becomes clear. Since Eq. (6) and its solution (7) is valid inany dimension [1], the dimensional reduction of the SWmap (7) should also have the same form in lower dimen-sion, i.e. like (9). This is possible only if one chooses thedeformation parameter � as (3).

Since the dimensional reduction of Yang-Mills actiongives Yang-Mills coupled to scalar fields, the aforemen-tioned observation leads to the fact that a noncommutativescalar field � that couples to gauge fields in a gaugeinvariant way should be written in terms of ordinary gaugefields and ordinary scalar fields. In other words, � ���A;�� and just like the original case [1], to preservethe gauge invariance of the theory � should satisfy

��A;�� � �g���A;�� � ��A� �g�A;�� �

g���; (11)

which generates directly the SW map of the scalar fields(10).

One can deduce that a similar argument should also holdfor any field that couples to gauge fields in a gauge invari-ant theory. Therefore, for instance for the NC Weyl spinors� in (1), one can write a similar condition like Eq. (11) thatgives the SW map of �:

� � �� 14�

KLfAK; �@L �DL��g �O��2�: (12)

Clearly, SW maps of the Weyl spinors and � of NCN � 2 SYM in 4 dimensions have the same form:

� � 14��fA; �@� �D�� g �O��2�;

� � �� 14��fA; �@� �D���g �O��2�:

(13)

These SW maps (12) and (13) are also consistent witheach other in the sense that (13) can be obtained from thedimensional reduction of (12) when the deformation pa-rameter is chosen as (3).

5Here, � denotes any component Ai of the gauge field on acompactified coordinate or any linear combination of Ai.

6A similar way to obtain the SW map of the scalar fields is alsostudied in [11].

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Obviously, maps (7) and (12) and maps (9), (10), and(13) give the desired SW maps for the component fields ofN � 1 NCSYM in 6 dimensions and N � 2 NCSYM in 4dimensions, respectively. Moreover, the above derived SWmaps are in agreement with the ones that are found for thecomponents of four dimensional N � 1 supersymmetricgauge theories in [6] for the U(1) case and in [7] for thenon-Abelian case. However, our method, which is consid-erably simpler, and the methods studied in [6,7] to findthese maps are completely different from each other.

On the other hand, the assumption that the NC compo-nent fields in a supersymmetric gauge theory depend ongauge field and their ordinary counterparts [6,7], ariseshere naturally as a direct consequence of the dimensionalreduction of the SW map of the gauge field AM. Indeed,one can generalize the above observation for any NC fieldthat couples to a gauge field whether the theory is super-symmetric or not.

IV. DEFORMED SUSY TRANSFORMATIONS

After obtaining the SW maps (7) and (12) of the com-ponent fields of NC SYM theory in six dimensions one canwrite the action (1) in terms of ordinary component fields�AM;�; ��� up to order � as

S6 � trZd4x

��

1

4FMNFMN �

i2

���MDM�

�1

4�KL

�FMNfFMK; FNLg �

1

4FKLfFMN; F

MNg

�i2

���M�fFKL;DM�g � 2fFMK;DL�g���

(14)

N � 2 SYM action in 4 dimensions can be derived intwo different ways: either by dimensionally reducing theaction (14) or by applying the SW maps (9), (10), and (13)to the action (4). One can show that both ways of obtainingthe action give exactly the same result:

S4 � trZd4x

��

1

4F��F�� � i���D�

��� i ��D�� �D��D��y � i

���2p� ��;�y� � ��� � ;��� �

1

2��;�y�2

�

� trZd4x��

��

1

4F��fF�; F��g �

1

16F�fF��; F

��g �i4����fF�;D�

��g � 2fF�;D���g�

�i4 ���fF�;D�

� g � 2fF�;D�� g� �

1

2�D��yfFk�;D��g �D��fF�;D��yg� �

1

4F�fD��y; D��g

�i���2p

4fF�; g��;�

y� �

���2p

2 fD�;D��

yg �i���2p

4fF�; ��g� � ;�� �

���2p

2��fD

� ;D��g

�1

8��y; ��fF�; ��y; ��g �

i2��y; ��fD�y; D��g

�(15)

It is clear that, neither the action (14) nor (15) is invari-ant under the respective classical SUSY transformations.7

This was first noted in [7]. To overcome this circumstanceone can deform the NC SUSY generators �. Such defor-mations are studied in [6,8] for Abelian NC supersymmet-ric gauge theories.

However, following [6], one can attain a general methodto construct deformed SUSY transformations that keep thedeformed supersymmetric gauge theory actions invariant.To achieve this goal, we let the generator of the SUSYtransformation � be

� ���!SW map� � �0 � �1;

where �0 is the ordinary SUSY transformations and �1 isthe deformed part at the order of �. The invariance of a NCaction S under NC SUSY transformations � is then mappedto the invariance of the action S under the new deformedSUSY transformations � after implementing SW maps:

� S��� � 0 ���!SW map�S��; �� � 0;

where � and � denote collectively all the NC componentfields and their ordinary counterparts, respectively, includ-ing the gauge field.

In order to construct consistent SUSY transformationswith SW maps of the component fields, let us denote theNC supersymmetry transformations as

� � � X;

where X denotes SUSY transformation of a NC field �.After performing the SW map on both sides of the trans-formation one can write up to first order in �,

� � � �0�� �0��1� � �1��O��2�

� X� X�1� �O��2�;

which leads to

�0� � X; �0��1� � �1� � X�1�;

where � � ����1� � , X � X� X�1� � , and��1�, X�1� denote the first order terms in � after the SWmap. It is clear that �0� � X are ordinary SUSY trans-

7The ordinary SUSY transformations can be read easily fromthe transformations (2) and (5) by replacing the NC fields withthe ordinary ones and � product with the ordinary product.

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formations. On the other hand, since ��1� is a polynomialof (ordinary) component fields � and their derivatives, andsince �0 transformation of ��1� is already known, one canread the action of the generator �1 on the fields � as�1� � X�1� � �0��1�. Deformed SUSY transformations� of the (ordinary) component fields can then be written as

�� � X� X�1� � �0��1�: (16)

Deformed SUSY transformations of six dimensional NCN � 1 SYM can be derived by following the above givensteps. The resulting transformations are then found to be

�AM � �i2� ���M�� ���M�� �

i8

�KL�fAK;DM� ���L�� ���L��g � f ���L�� ���L�; @KAM � FKMg�;

�� � �MN�FMN �1

2�KL

��MN�fFMK; FNLg �

i4f ���L�� ���L�; �DK � @K��g �

1

4fAK; � ���L�� ���L�;��g

�;

� �� � � ���MNFMN �1

2�KL

����MNfFMK; FNLg �

i4f ���L�� ���L�; �DK � @K� ��g �

1

4fAK; � ���L�� ���L�; ���g

�:

(17)

The deformed SUSY transformations � that are con-structed by using the aforementioned procedure automati-cally guarantee that the action S��; �� is invariant under�. This is due to the fact that the transformations � arederived directly from the NC SUSY transformations � thatleave a NC action invariant, � S��� � 0.

Indeed, one can check explicitly that the above givendeformed SUSY transformations in six dimensions (17)leave the deformed N � 1 SYM action (14) invariant.

Note also that the above transformations (17) are alsoconsistent with the SW maps (7) and (12) by construction.

This consistency can also be checked explicitly, by apply-ing directly SW maps on both sides of the NC transforma-tions (2).

On the other hand, one can follow two equivalent waysto obtain the deformed SUSY transformations of N � 2NC SYM that leave (15) invariant in four dimensions:either by dimensionally reducing the transformations (17)or from the NC N � 2 SUSY transformations (5) by usingthe aforementioned method. Both approaches give thesame result:

�A� � i1�� ��� i2�� � � i �1 ����� i �2 ��� �i4���f1� ��� 2� � � �1 ���� �2 �� ; @�A� � F��g

� fA�;D��1� ��� 2� � � �1 ���� �2 �� �g�;

�� � ���1F�� � i1��;�y� � i

���2p�� �2D���

1

2��

�1�

��fF�; F��g � i���2p

�2fF�;D��g � 1fD�y; D��g

�i2f1� ��� 2� � � �1 ���� �2 �� ; �@� �D���g �

1

2fA; �1�� ��� 2�� � � �1 ����� �2 ��� ; ��g

�;

� � ���2F�� � i2��;�y� � i

���2p�� �1D���

1

2��

�2�

��fF�; F��g � i���2p

�1fF�;D��g � 2fD�;D��yg

�i2f1� ��� 2� � � �1 ���� �2 �� ; �@� �D�� g �

1

2fA; �1�� ��� 2�� � � �1 ����� �2 ��� ; �g

�;

�� ����2p1 �

���2p2��

i4���f1� ��� 2� � � � ���� �2 �� ; �@� �D���g

� ifA; �1�� ��� 2�� � � �1 ����� �2 ��� ;��g�: (18)

V. CONCLUSION

We argued that the use of extra dimensions in a trivialway for noncommutative gauge theories leads to the equa-tions

A�A� � �g�A�A� � A�A� �g�A�; (19)

��A;�� � �g�

��A;�� � ��A� �g�A;�� �g���;

(20)

where the first equation is the original one for gauge fieldsderived in [1] and the second one (20) is for any NC field �which is not the gauge field in a gauge invariant theory.Note that, in order to keep the form of the SW map of thegauge field A the same before and after performing the

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dimensional reduction, the deformation parameter has tobe chosen as in (3).

As a direct consequence of the dimensional reduction ofthe original SW map, one can prove that the noncommu-tative fields � depend on their ordinary counterparts � andalso on the gauge field A. The solutions of these equationsare SW maps of the respective fields. It is clear that thisresult lets one write the SW maps of the component fieldsin NC SYM theories. We gave these SW maps for thecomponent fields of NC N � 1 SYM in 6 and the onesof NC N � 2 SYM in 4 dimensions.

It is also worth mentioning here that one can performdimensional reduction from a NC space to another beforeor after performing the SW map. We pointed out thisobservation for the dimensional reduction of the NC N �1 SYM in 6 dimensions that can be expressed by thefollowing commuting diagram:

S 6

Dim.Red.

SW mapS 6

Dim.Red.

S 4 SW mapS 4

On the other hand, when one considers NC SYM on thelevel of the ordinary component fields, to keep the SUSYinvariance of NC SYM actions after SW map, one mustalso deform the SUSY transformations after the SW map:

� ���!SW map� � �0 � �1:

We gave a general method to construct such deformedSUSY transformations once SW maps of the componentfields are known. These transformations are consistent withSW maps and leave the NC action invariant after perform-ing the SW map by construction. We gave two such ex-amples, namely, the deformed (on-shell) SUSYtransformations of NC N � 1 SYM in 6 dimensions andNC N � 2 SYM in 4 dimensions. We have explicitlyderived the deformed SUSY transformations for thesetwo cases that leave the respective actions invariant afterthe SW map.

Finally, it is also worth to note that the method given inSec. IV to construct deformations of SUSY transforma-tions in NC space-times is quite general. We expect that

one can generalize this procedure to other NC supersym-metric models, such as NC supergravity [12].

ACKNOWLEDGMENTS

We thank Omer F. Day and Cemsinan Deliduman forvarious enlightening discussions.

APPENDIX A: CONVENTIONS

The covariant derivative and the field strength are givenin the noncommutative space as

D M� � @M�� i�AM; ���;

FMN � @MAN � @NAM � i�AM; AN�� :(A1)

The Moyal � product is defined as

f�x� � g�x� � f�x�g�x� �i2

�MN@Mf�x�@Ng�x� �O��2�;

(A2)

whereas �A;B�� � A � B� B � A is the � commutator.Our SUSY conventions in six dimensions are similar

with that of [13]. The gamma matrices in 6 dimensionssatisfy,

f�M;�Ng � �2�MN;

where �MN � ��1; 1; 1; 1; 1; 1�.We often use the identities,

�M�RS � �1

2

��MR�S � �MS�R

�i6�MRSABC�7�A�B�C

�

�RS�M � �1

2

���MR�S � �MS�R

�i6�MRSABC�7�A�B�C

�throughout the calculations where �MN �

14 ��M;�N� and �

is the totally antisymmetric tensor.In four dimensions, we use Wess-Bagger conventions

[14], i.e.

�� � �� � ; �� � �� � ; � � �� � � �; �� _� � � _� _ �� _ ;

�� _� � � _� _ �� _ ; �� � � �� _�

� _� � � ��; �12 � ��21 � ��12 � �21 � 1;

�0 ��1 00 �1

� �; �1 �

0 11 0

� �; �2 �

0 �ii 0

� �; �3 �

1 00 �1

� �:

��� _� � �� � _� _ ��� _ ; ��� _�� � � _� _ �� �� _ ;

��� � � 14��

�� _� �� _� � � ��� _� �� _� ��; ���� _�

�14� �� _�����

� _ � �� _�����

� _ �:

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APPENDIX B: DIMENSIONAL REDUCTION

To perform the dimensional reduction we use the fol-lowing parametrization of the � matrices:

�� � I ��; �4 � �1 �5;

�5 � �2 �5; �7 � �3 �5;(B1)

where �’s are Pauli matrices and �5 � �0�1�2�3. In thisparametrization, the Weyl spinor in six dimensions can bewritten in terms of four dimensional Dirac spinors:

� ��1�i�5

2 ���1�i�5

2 ��

" #; �� � ���1�i�5

2 � ���1�i�5

2 �h i

:

The Weyl spinors of the N � 2 supermultiplet are thenobtained as

� ����2p ��

� _�

� �; �� �

���2p

� �� _�

;

whereas the scalar field and its Hermitian conjugate aredefined in our notation as

� � �1���2p �A4 � iA5�; �y � �

1���2p �A4 � iA5�:

(B2)

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