fayet-iliopoulos term and nonlinear self-duality

6
Fayet-Iliopoulos term and nonlinear self-duality Sergei M. Kuzenko * School of Physics M013, The University of Western Australia, 35 Stirling Highway, Crawley W.A. 6009, Australia (Received 10 December 2009; published 26 April 2010) The N ¼ 1 supersymmetric Born-Infeld action is known to describe the vector Goldstone multiplet for partially broken N ¼ 2 rigid supersymmetry, and this model is believed to be unique. However, it can be deformed by adding the Fayet-Iliopoulos term without losing the second nonlinearly realized supersym- metry. Although the first supersymmetry then becomes spontaneously broken, the deformed action still describes partial N ¼ 2 ! N ¼ 1 supersymmetry breaking. The unbroken supercharges in this theory correspond to a different choice of N ¼ 1 subspace in the N ¼ 2 superspace, as compared with the undeformed case. Implications of the Fayet-Iliopoulos term for general models for self-dual nonlinear supersymmetric electrodynamics are discussed. The known ubiquitous appearance of the Volkov-Akulov action in such models is explained. We also present a two-parameter duality-covariant deformation of the N ¼ 1 supersymmetric Born-Infeld action as a model for partial breaking of N ¼ 2 supersymmetry. DOI: 10.1103/PhysRevD.81.085036 PACS numbers: 12.60.Jv Recently, there have been interesting discussions [1,2] of the supercurrent [3] (i.e. the multiplet of currents contain- ing the energy-momentum tensor and the supersymmetry current) in N ¼ 1 supersymmetric gauge theories with a Fayet-Iliopoulos (FI) term [4]. These works are primarily targeted at phenomenological applications of supergravity theories. In the present paper, we would like to elaborate on somewhat different and more formal aspects such as im- plications of the FI term for partial supersymmetry break- ing, and more generally in the context of models for self- dual nonlinear supersymmetric electrodynamics. In par- ticular, we will revisit the conclusion of [5,6] about unique- ness of the Goldstone-Maxwell multiplet model for partially broken N ¼ 2 supersymmetry. At the end of this paper, we will also comment on the claim made in [2] that ‘‘no supercurrent supermultiplet exists for globally supersymmetric gauge theories with nonzero Fayet- Iliopoulos terms.’’ The N ¼ 1 supersymmetric Born-Infeld (BI) action is 1 S SBI ½W¼ 1 4 Z d 6 zW 2 þ 1 4 Z d 6 " z " W 2 þ g 2 4 Z d 8 z W 2 " W 2 1 þ 1 2 A þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ A þ 1 4 B 2 q ; A ¼ g 2 8 ðD 2 W 2 þ " D 2 " W 2 Þ; B ¼ g 2 8 ðD 2 W 2 " D 2 " W 2 Þ: (1) Here g is the coupling constant, W the chiral field strength of an Abelian vector multiplet, W ¼ 1 4 " D 2 D V; V ¼ " V; (2) with V the gauge prepotential. The action (1) was intro- duced for the first time in Refs. [9,10] as a supersymmetric extension of the BI theory [11], and as such it is not unique. Bagger and Galperin [5], and later Roc ˇek and Tseytlin [6], using alternative techniques, discovered that S SBI is the action for a Goldstone-Maxwell multiplet associated with N ¼ 2 ! N ¼ 1 partial supersymmetry breaking. This action was argued to be unique [5,6]. Being manifestly N ¼ 1 supersymmetric, S SBI also proves to be invariant under a second, nonlinearly realized supersymmetry trans- formation W ¼ þ g 2 4 1 4 " D 2 " X þ i@ _ X " _ ; (3) with a constant spinor parameter. Here X is a chiral superfield, " D _ X ¼ 0, satisfying the nonlinear constraint [5,6] X þ g 2 16 X " D 2 " X ¼ W 2 : (4) The proof of the invariance is based on the observations [5,6] that (i) the functional (1) can be rewritten in the form S SBI ½W¼ 1 4 Z d 6 zX þ 1 4 Z d 6 " z " X; (5) and (ii) the chiral scalar X transforms under (3) as X ¼ 2 W : (6) Consider now the N ¼ 1 supersymmetric FI term [4] S FI ¼ 2$ Z d 8 zV ¼ $ 2 Z d 6 z W þ $ 2 Z d 6 " z " _ " W _ : (7) It is easy to see that S FI is also invariant under the second * [email protected] 1 We follow the notation and conventions adopted in [7,8]. The superspace integration measures in (1) are defined as follows: d 6 z :¼ d 4 xd 2 , d 6 " z :¼ d 4 xd 2 " and d 8 z :¼ d 4 xd 2 d 2 " . PHYSICAL REVIEW D 81, 085036 (2010) 1550-7998= 2010=81(8)=085036(6) 085036-1 Ó 2010 The American Physical Society

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Page 1: Fayet-Iliopoulos term and nonlinear self-duality

Fayet-Iliopoulos term and nonlinear self-duality

Sergei M. Kuzenko*

School of Physics M013, The University of Western Australia, 35 Stirling Highway, Crawley W.A. 6009, Australia(Received 10 December 2009; published 26 April 2010)

TheN ¼ 1 supersymmetric Born-Infeld action is known to describe the vector Goldstone multiplet for

partially brokenN ¼ 2 rigid supersymmetry, and this model is believed to be unique. However, it can be

deformed by adding the Fayet-Iliopoulos term without losing the second nonlinearly realized supersym-

metry. Although the first supersymmetry then becomes spontaneously broken, the deformed action still

describes partial N ¼ 2 ! N ¼ 1 supersymmetry breaking. The unbroken supercharges in this theory

correspond to a different choice of N ¼ 1 subspace in the N ¼ 2 superspace, as compared with the

undeformed case. Implications of the Fayet-Iliopoulos term for general models for self-dual nonlinear

supersymmetric electrodynamics are discussed. The known ubiquitous appearance of the Volkov-Akulov

action in such models is explained. We also present a two-parameter duality-covariant deformation of the

N ¼ 1 supersymmetric Born-Infeld action as a model for partial breaking of N ¼ 2 supersymmetry.

DOI: 10.1103/PhysRevD.81.085036 PACS numbers: 12.60.Jv

Recently, there have been interesting discussions [1,2] ofthe supercurrent [3] (i.e. the multiplet of currents contain-ing the energy-momentum tensor and the supersymmetrycurrent) in N ¼ 1 supersymmetric gauge theories with aFayet-Iliopoulos (FI) term [4]. These works are primarilytargeted at phenomenological applications of supergravitytheories. In the present paper, wewould like to elaborate onsomewhat different and more formal aspects such as im-plications of the FI term for partial supersymmetry break-ing, and more generally in the context of models for self-dual nonlinear supersymmetric electrodynamics. In par-ticular, we will revisit the conclusion of [5,6] about unique-ness of the Goldstone-Maxwell multiplet model forpartially broken N ¼ 2 supersymmetry. At the end ofthis paper, we will also comment on the claim made in[2] that ‘‘no supercurrent supermultiplet exists for globallysupersymmetric gauge theories with nonzero Fayet-Iliopoulos terms.’’

TheN ¼ 1 supersymmetric Born-Infeld (BI) action is1

SSBI½W� ¼ 1

4

Zd6zW2 þ 1

4

Zd6 �z �W2

þ g2

4

Zd8z

W2 �W2

1þ 12Aþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Aþ 1

4B2

q ;

A ¼ g2

8ðD2W2 þ �D2 �W2Þ;

B ¼ g2

8ðD2W2 � �D2 �W2Þ: (1)

Here g is the coupling constant,W� the chiral field strengthof an Abelian vector multiplet,

W� ¼ � 1

4�D2D�V; V ¼ �V; (2)

with V the gauge prepotential. The action (1) was intro-duced for the first time in Refs. [9,10] as a supersymmetricextension of the BI theory [11], and as such it is not unique.Bagger and Galperin [5], and later Rocek and Tseytlin [6],using alternative techniques, discovered that SSBI is theaction for a Goldstone-Maxwell multiplet associated withN ¼ 2 ! N ¼ 1 partial supersymmetry breaking. Thisaction was argued to be unique [5,6]. Being manifestlyN ¼ 1 supersymmetric, SSBI also proves to be invariantunder a second, nonlinearly realized supersymmetry trans-formation

�W� ¼ �� þ g2

4

�1

4�D2 �X�� þ i@� _�X �� _�

�; (3)

with �� a constant spinor parameter. Here X is a chiralsuperfield, �D _�X ¼ 0, satisfying the nonlinear constraint[5,6]

X þ g2

16X �D2 �X ¼ W2: (4)

The proof of the invariance is based on the observations[5,6] that (i) the functional (1) can be rewritten in the form

SSBI½W� ¼ 1

4

Zd6zX þ 1

4

Zd6 �z �X; (5)

and (ii) the chiral scalar X transforms under (3) as

�X ¼ 2��W�: (6)

Consider now the N ¼ 1 supersymmetric FI term [4]

SFI ¼ 2�Z

d8zV ¼ �

2

Zd6z��W� þ �

2

Zd6 �z �� _�

�W _�:

(7)

It is easy to see that SFI is also invariant under the second

*[email protected] follow the notation and conventions adopted in [7,8]. The

superspace integration measures in (1) are defined as follows:d6z :¼ d4xd2�, d6 �z :¼ d4xd2 �� and d8z :¼ d4xd2�d2 ��.

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nonlinearly realized supersymmetry (3), as pointed outrecently in [12].2 Therefore, the theory with action

S ¼ SSBI½W� þ SFI (8)

is manifestly N ¼ 1 supersymmetric, and is invariantunder the second nonlinearly realized supersymmetry (3).We are going to show that the deformed action also de-scribes partial N ¼ 2 ! N ¼ 1 supersymmetrybreaking.

Let us study the bosonic sector of the componentLagrangian. The component fields contained in W� are

��ðxÞ ¼ W�j�¼0; (9a)

F��ðxÞ ¼ � i

4ðD�W� þD�W�Þj�¼0; (9b)

DðxÞ ¼ � 1

2D�W�j�¼0; (9c)

with

F� _�� _� � ð�aÞ� _�ð�bÞ� _�Fab ¼ 2"�� �F _� _� þ 2" _� _�F��

(10)

the electromagnetic field strength. Setting the photino tozero, �� ¼ 0, the bosonic Lagrangian can be shown to be

Lboson ¼ 1

g2

�1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ g2ðuþ �uÞ þ 1

4g4ðu� �uÞ2

s �þ �D;

(11)

where we have defined

u :¼ 1

8D2W2j��¼��¼0 ¼ !� 1

2D2;

! :¼ 1

4FabFab þ i

4Fab ~Fab;

(12)

with ~Fab the Hodge-dual of Fab. From Lboson we read offthe equation of motion for the auxiliary field:

Dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ g2ðuþ �uÞ þ 1

4g4ðu� �uÞ2

q ¼ ��: (13)

Its solution is

D ¼ � �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ g2�2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ g2ð!þ �!Þ þ 1

4g4ð!� �!Þ2

s:

(14)

Here the second factor on the right is essentially the BILagrangian [11]:

LBI ¼ 1

g2

�1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ g2ð!þ �!Þ þ 1

4g4ð!� �!Þ2

s �

¼ 1

g2f1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detð�ab þ gFabÞ

qg: (15)

Upon elimination of the auxiliary field, the bosonicLagrangian (11) becomes

Lboson ¼ 1

g2f1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ g2�2

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detð�ab þ gFabÞ

qg: (16)

Modulo an irrelevant constant term and an overall normal-ization factor, this is again the BI Lagrangian.Looking at the expression for the auxiliary field,

Eq. (14), we see that it acquires a nonvanishing expectationvalue

hDi ¼ � �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ g2�2

p : (17)

Therefore, the manifestly realized supersymmetry of thetheory (8) becomes spontaneously broken. The corre-sponding supersymmetry transformation of the photino isnow

��� ¼ iðQþ � �QÞW�j�¼0

¼ hDi� þ field-dependent terms (18)

and therefore the photino turns into a Goldstino. As aresult, the situation is now the following. The model underconsideration, Eq. (8), possesses two supersymmetries, ofwhich one (Q) is linearly realized, and the other (S) isnonlinearly realized, as described by Eq. (3). On the massshell, both Q and S supersymmetries become nonlinearlyrealized. Clearly, this does not mean that the N ¼ 2supersymmetry is completely broken, for there is onlyone Goldstino in the theory. Therefore, a special combina-tion of the Q and S supersymmetries must remain unbro-ken. This can be seen explicitly as follows. The Q- andS-supersymmetry transformations of the theory (8) formtheN ¼ 2 super-Poincare algebra without central charge,

fQ�; �Q _�g ¼ 2ð�cÞ� _�Pc; fS�; �S _�g ¼ 2ð�cÞ� _�Pc; (19a)

fQ�; S�g ¼ 0; fQ�; �S _�g ¼ 0; (19b)

see [5,6] for more detail. Now, using the transformationlaws (3) and (18), one can readily check that N ¼ 1supersymmetry transformations generated by

Q � :¼ cos’Q� þ sin’S�; tan’ ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ g2�2

p(20)

remain unbroken. We see that the result of adding the FIterm to theN ¼ 1 supersymmetric BI action amounts to aU(1) rotation of the unbroken N ¼ 1 supersymmetrygenerators in the N ¼ 2 super-Poincare algebra.

2This property is directly related to the fact that theN ¼ 1 FIterm preserves also N ¼ 2 supersymmetry [13].

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Therefore, the dynamical system (8) is a (deformed) vectorGoldstone multiplet model for partial supersymmetrybreaking N ¼ 2 ! N ¼ 1. Similarly to the considera-tions in [6,12], the action (8) can be obtained by (i) startingfrom a nonrenormalized model for an Abelian N ¼ 2vector multiplet with two types (electric and magnetic)of N ¼ 2 FI terms [14,15], and then (ii) integrating outa massive N ¼ 1 scalar multiplet.3

The supersymmetric BI theory (1) is known to be in-variant under U(1) duality rotations [17,18]. It is in fact aspecial member of the family of models for self-dual non-linear N ¼ 1 supersymmetric electrodynamics con-structed in [18] and described by actions of the form,

S½W� ¼ 1

4

Zd6zW2 þ 1

4

Zd6 �z �W2

þ 1

4

Zd8zW2 �W2�ðu; �uÞ;

u :¼ 1

8D2W2; (21)

where �ðu; �uÞ is a real analytic function of one complexvariable. In accordance with [18], this theory possessesU(1) duality invariance provided � obeys the self-dualityequation

Im f�� �u�2g ¼ 0; �ðu; �uÞ :¼ @

@uðu�ðu; �uÞÞ: (22)

The fermionic sector of such models turns out to possess aremarkable structure [19].

As demonstrated in [19], under the only additional re-striction

�u �uð0; 0Þ ¼ 3�3ð0; 0Þ; (23)

the component fermionic action coincides, modulo a non-linear field redefinition, with the Volkov-Akulov action[20].4 At first sight, this ubiquity of the Goldstino actionin the framework of nonlinear self-duality looks somewhatmiraculous. It can be explained, however, if we let the FIterm enter the game and consider the following model:

S½W� þ SFI: (24)

In the purely bosonic sector, the equation of motion for theauxiliary field is

D ½1� �u�ðu; �uÞ � u ��ðu; �uÞ� ¼ ��; (25)

with u defined as in (12). This equation should be used toexpress D in terms of the electromagnetic field strength,D ¼ fð!; �!Þ. Generically, the auxiliary field develops anonvanishing expectation value, hDi � 0, which must sat-

isfy an algebraic nonlinear equation that follows from (25)by setting ! ¼ 0.5 As a result, the supersymmetry be-comes spontaneously broken, and thus the photino actionshould be related to the Goldstino action, due to the uniqueof the latter.It is worth briefly recalling the structure of the bosonic

sector in the model (21). The corresponding equation ofmotion for D is obtained from (25) by setting � ¼ 0, andhence it always has the solutionD ¼ 0. With this solutionchosen, the dynamics of the electromagnetic field is de-scribed by the Lagrangian

L ¼ � 1

2ð!þ �!Þ þ! �!�ð!; �!Þ; (26)

with the interaction � obeying the self-duality equa-tion (22). This is a model for self-dual nonlinear electro-dynamics in the sense of [22–24], of which the BI theory(15) is a special case. Such theories and their generaliza-tions possess very interesting properties; see, e.g., thesecond reference in [18] and [25] for reviews. It is naturalto ask the following question: Is self-duality preserved insome form in the case of deformed theory (24) with anonzero D obeying the Eq. (25)? We now turn to answer-ing this question.The model (21) can be generalized to include couplings

to supermultiplets containing the dilaton and axion, theNeveu-Schwarz and Ramond-Ramond two-forms, B2 andC2, and the Ramond-Ramond four-form, C4, as presentedin [18] building on the bosonic constructions of [26–28].The extended action has the form

S½W;�; �; ;�� ¼ S½W;��þ�Z

d6z

��þ 1

2�W�

�þ c:c:

�;

(27)

where

S½W;�� ¼ i

4

Zd6z�W2 � i

4

Zd6 �z �� �W2

� 1

16

Zd8zð�� ��Þ2W2 �W2�

��i

16ð�� ��ÞD2W2;

i

16ð�� ��Þ �D2 �W2

�(28)

describes SLð2;RÞ duality invariant coupling of the vectormultiplet to the dilaton-axion chiral multiplet �, and

W � :¼ W� þ i�� (29)

is the supersymmetrization of Fþ B. Here ��, � and �are unconstrained chiral superfields which include, among

3It would be interesting to understand how to integrate outmassive degrees of freedom in the non-Abelian extensions [16]of the Antoniadis-Partouche-Taylor model [14].

4The results in [19] extended the earlier component analysis[21] of the supersymmetric BI theory (1).

5In some cases, the algebraic nonlinear equation on hDi has nosolution, and then (24) is inconsistent.

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their components, the fields B2, C2 and C4, respectively.The resulting action is invariant under the following gaugetransformations:

��� ¼ i�W� ¼ i �D2D�K1; (30a)

�� ¼ i �D2D�K2; �� ¼ 1

2�� �D2D�K2; (30b)

�� ¼ i �D2K3; (30c)

with Ki real unconstrained superfields. The action (27)reduces to (21) by setting � ¼ �i and switching off theother chiral superfields ��, � and �.

As demonstrated in [18], the theory (27) is invariantunder SLð2;RÞ duality transformations

M0�

W 0�

� �¼ a b

c d

� �M�

W�

� �; �0 ¼ a�þ b

c�þ d;

a bc d

� �2 SLð2;RÞ;

(31)

provided the superfields ��, � and � transform as

0

�0

!¼ a b

c d

!

!; (32a)

�0 ¼ �� i

4bd�2 � i

2bc�� i

4ac2: (32b)

HereM� denotes a variational derivative of the action withrespect to the field strength,

iM� :¼ 2�

�W� S½W;�; �; ;��: (33)

A detailed discussion can be found in [18].Using the second form of the FI term, Eq. (7), the

action (24) is seen to be of the type (27) with the following‘‘frozen’’ values for background fields: � ¼ ���, � ¼�i and �� ¼ � ¼ 0. As a natural generalization, an an-satz compatible with duality transformations is � / ��,�� / �� and � / �2. A consistent with duality choice is

� ¼ ���; �� ¼ ���; � ¼ 0; �; � 2 R:

(34)

As follows from (32b), applying a duality transformationgenerates a purely imaginary nonzero value for � which,however, does not contribute to the action.

Let us now return to the model (24) and consider itsduality-covariant extension

S½W � þ SFI; W � :¼ W� þ i���: (35)

Here the deformed field strength W � obeys the modifiedBianchi identity

�D _��W _� �D�W � ¼ 4i�: (36)

This action is invariant under inhomogeneous supersym-metry transformations

�W� ¼ �i�� þ iðQþ � �QÞW�: (37)

If S½W� coincides with the supersymmetric BI action (1),then the resulting model

SSBI½W � þ SFI (38)

is also invariant under a second nonlinearly realized super-symmetry, which is a natural generalization of (3),

�W� ¼ �� þ g2

4

�1

4�D2 �X�� þ i@� _�X �� _�

�; (39)

where X is a chiral superfield, �D _�X ¼ 0, satisfying thenonlinear constraint

X þ g2

16X �D2 �X ¼ W 2; (40)

compare with (4). Our action (38) is a two-parameterdeformation of the supersymmetric BI theory (1). Thisaction appears to be the most general Goldstone-Maxwellmultiplet model for partial N ¼ 2 ! N ¼ 1 supersym-metry breaking. Requiring the first supersymmetry to bemanifest eliminates one of the deformations, � ¼ 0. Thesupersymmetric BI action (1) is indeed unique if the firstsupersymmetry is required to be unbroken.In conclusion, we would like to comment on the state-

ment made in [2] that ‘‘no supercurrent supermultipletexists for globally supersymmetric gauge theories withnonzero Fayet-Iliopoulos terms.’’ A general scheme tocompute supercurrents [3] in rigid supersymmetric theo-ries is by evaluating a variational derivative of the actionfunctional with respect to the gravitational superfield H� _�,

J� _� ¼ �S

�H� _� ; (41)

with the idea due to Ogievetsky and Sokatchev [29]. Morespecifically, the procedure is as follows: (i) one should liftthe theory to a curved superspace corresponding to one ofthe known off-shell supergravity formulations (realized interms of the gravitational superfield H� _� and an appropri-ate compensator); (ii) compute the (covariantized) varia-tional derivative �S=�H� _�; (iii) return to the flatsuperspace by switching off the supergravity prepotentials.The scheme is worked out in detail in two textbooks [8,30],including numerous examples. The explicit form of thesupercurrent conservation equation depends on the off-shell supergravity realization chosen. In the case of theold minimal (n ¼ �1=3) formulation for N ¼ 1 super-gravity [31] (see also [32]), the conservation equation is

�D _�J� _� ¼ D�T; �D _�T ¼ 0; (42)

with T called the supertrace. In the case of the newminimal(n ¼ 0) formulation for N ¼ 1 supergravity [33], theconservation equation is

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�D _�J� _� ¼ T�; �D _�T� ¼ 0; D�T� ¼ �D _��T _�:

(43)

The difference between (42) and (43) is due to the differenttypes of compensators used in these supergravity formula-tions. One can construct more general supercurrent con-servation equations; see, e.g., [34,35].

We wish to analyze the two versions of the supercurrentfor the theory (24). Within the old minimal formulation forN ¼ 1 supergravity, the supercurrent for the model (21)was computed in [36] and shown to be duality invariant.The supercurrent obtained is rather complicated to dealwith. So for simplicity, we restrict our consideration to theMaxwell case, by setting � ¼ 0 in (21), similarly to [1].The consideration below can naturally be generalized tothe case � � 0.

Using the old minimal formulation for N ¼ 1 super-gravity, the supermultiplets J� _� and T are computed to be[1]

J� _� ¼ 2W��W _� þ 2

3�½D�; �D _��V; T ¼ 1

3� �D2V:

(44)

Using the equation of motion,D�W� ¼ 2�, one can checkthat the conservation equation (42) holds. As pointed outby Komargodski and Seiberg [1], both J� _� and T are notgauge invariant. The reason for this is very simple: mini-mal coupling of the FI term to supergravity, which makesuse of the first expression in (7), is not gauge invariant. Oldminimal supergravity is not well suited to describe FIterms.

Using the new minimal formulation for N ¼ 1 super-gravity, the supermultiplets J� _� and T� can be computedto be

J � _� ¼ 2W��W _�; T� ¼ 4�W�: (45)

Both objects are gauge invariant. The reason for this is verysimple: minimal coupling of the FI term to supergravity,which makes use of the second expression in (7) and isobtained by replacing (the vacuum expectation value) ��

with the chiral spinor compensator, is gauge invariant. Newminimal supergravity is ideal for describing FI terms.The theory (24) is R invariant, and the corresponding R

current jð5Þ� _� must be conserved. If one identifies jð5Þ� _� withthe lowest component of the supercurrent in (44), it fails tobe conserved, for one finds

@� _�J� _� ¼ i

6�½D2; �D2�V � 2

3�@� _�½D�; �D _��V � 0: (46)

This result was interpreted in [2] as nonexistence of super-current for rigid supersymmetric gauge theories with non-zero FI terms. However, if one defines the R current by

jð5Þ� _�:¼

�J� _� � 2

3�½D�; �D _��V

�j�¼0; (47)

then it is clearly conserved.In the case of the supercurrent (45), the standard defini-

tion of the R current applies

jð5Þ� _�:¼ J� _�j�¼0: (48)

Our consideration shows that the supercurrent does existfor globally supersymmetric gauge theories in the presenceof FI terms.

The author is grateful to Ian McArthur and ArkadyTseytlin for useful comments. This work is supported inpart by the Australian Research Council.Note added.—The published version of [2], Ref. [37],

significantly differs from the original preprint, Ref. [2],discussed above. The conclusions of [37] are similar tothose given in the present paper.

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